2. Governing equations

The shear layer problem considered in figure 1 is assumed to be inviscid and irrotational and, thus, the governing equations are the two-dimensional non-dimensionalised Euler equations

(2.1) \begin{equation} \frac {\partial \boldsymbol{u}}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{u}=-\frac {1}{{\rho }}\boldsymbol{\nabla }\widehat {p}-F^{-2}\boldsymbol{e}_y,\,\,\,\,\,\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}=0,\,\,\,\,\,{\frac {\partial \rho }{\partial t}+\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla }\rho =0}, \end{equation}

where $F=(U^*_1+U^*_2)/(2\sqrt {gL})$ is the Froude number, $\boldsymbol{e}_y$ is a unit vector in the $y$ direction and $g$ is the gravitational constant. Here $\boldsymbol{u}=(\widehat {u},\widehat {v})$ are the velocity components in the $x$ and $y$ directions, and $\widehat {p}$ is the pressure. The function $\widehat {S}(y)$ is the non-dimensional density profile and is assumed to vary only in the cross-flow direction. The flow is confined above and below by rigid parallel plates at $y=h_1$ and $y=-h_2$ .

The form of the governing stability equations come from seeking two-dimensional normal-mode disturbances to the basic parallel velocity profile $\boldsymbol{u}=U(y)\boldsymbol{e}_x$ of the form

(2.2) \begin{align} \boldsymbol{u}=(\widehat {u}(x,y,t),\widehat {v}(x,y,t))&=(U(y),0)+\widehat {\delta }(u(y),v(y))\exp \left [ \textrm {i}(\alpha x-\omega t)\right ],\end{align}

(2.3) \begin{align} \widehat {p}(x,y,t)&= P_0-\int \widehat {S}(y)F^{-2}\,{\textrm {d}} y+\widehat {\delta } p(y)\exp \left [\textrm {i}(\alpha x-\omega t)\right ],\end{align}

(2.4) \begin{align} \rho (x,y,t)&= \widehat {S}(y)+\widehat {\delta }\sigma (y)\exp \left [\textrm {i}(\alpha x-\omega t)\right ],\end{align}

where $0\lt \widehat {\delta }\ll 1$ is a small parameter that allows for linearisation and $P_0$ is a constant. The disturbances are wave-like disturbances with wavenumber $\alpha$ and angular frequency $\omega =\alpha c$ , where $c$ is the phase speed. Substituting these forms into (2.1) and linearising leads to the system of equations

(2.5) \begin{align} \textrm {i} \alpha u+v' &= 0, \end{align}

(2.6) \begin{align} -\textrm {i}\omega u+\textrm {i}\alpha U u+U'v &= -\textrm {i}\alpha \frac {p}{\widehat {S}},\end{align}

(2.7) \begin{align} -\textrm {i}\omega v+\textrm {i}\alpha U v &= -\frac {p'}{\widehat {S}}-F^{-2}\frac {{\sigma }}{\widehat {S}},\end{align}

(2.8) \begin{align} -\textrm {i}\omega \sigma +\textrm {i}\alpha U\sigma +v\widehat {S}' &= 0, \end{align}

where the dash denotes a derivative with respect to $y$ . Eliminating the quantities $u$ , $\sigma$ and $p$ from these equations leads to a modified Taylor–Goldstein equation

(2.9) \begin{equation} (\alpha U-\omega )(v''-\alpha ^2v)-\alpha U''v+\frac {\widehat {S}'}{\widehat {S}}\left [ (\alpha U-\omega )v'-\alpha U' v-\frac {\alpha ^2F^{-2}}{(\alpha U-\omega )}v\right ]=0, \end{equation}

with the inertia terms retained (Drazin & Reid Reference Drazin and Reid1981; Drazin Reference Drazin2002). Here $F^{-2}\widehat {S}'/\widehat {S}$ is the Richardson number, a measure of the buoyancy to shear in the system. This equation is solved with the appropriate boundary conditions, which, for $h_1$ or $h_2$ finite, are

(2.10) \begin{equation} v(h_1)=v(-h_2)=0, \end{equation}

while, for $h_1$ or $h_2$ infinite,

(2.11) \begin{equation} v'(h_1)+\alpha v(h_1)=v'(-h_2)-\alpha v(-h_2)=0, \end{equation}

gives the correct exponential decay at $y=\pm \infty$ , when $\textrm {Re}(\alpha )\gt 0$ . It should also be noted that in this case we also require the wave to be unstable, i.e. $\omega _i\gt 0$ . In the case when the mode decays in time $\omega _i\lt 0$ , then the causal solutions could involve exponential growth in the far field (Le Dizès & Billant Reference Le Dizès and Billant2009; Riedinger, Le Dizès & Meunier Reference Riedinger, Le Dizès and Meunier2010; Zuccoli, Brambley & Barkley Reference Zuccoli, Brambley and Barkley2024). When using (2.11) in numerical calculations, $h_1$ and $h_2$ need to be chosen large enough such that the eigenmode is in the correct exponential decay regime, and we find that a value of $h_1=h_2=20$ is sufficient to guarantee this for the results in this paper.

In §§ 3–6 we consider a discontinuous density profile that consists of a jump in density at $y=d$ , and so in this case $\widehat {S}(y)$ takes the form

(2.12) \begin{equation} \widehat {S}(y)=\left \{\begin{array}{ll} 1, & y\geq d, \\ S, & y\lt d, \end{array} \right . \end{equation}

with $S\geqslant 1$ so that the stratification is stable. This profile is a model for the scenario of two immiscible fluids in the shear layer, such as the breakup of liquid–liquid jets (Kitamura, Mishima & Takahashi Reference Kitamura, Mishima and Takahashi1982; Richards, Lenhoff & Beris Reference Richards, Lenhoff and Beris1994; Homma et al. Reference Homma, Koga, Matsumoto, Song and Tryggvason2006) or the breakup of liquid jets in a coaxial injector (Juniper & Candel Reference Juniper and Candel2003; Juniper Reference Juniper2006). In this case we require two jump conditions at $y=d$ to ensure we have continuity of the normal displacement of the fluids and the continuity of pressure at the density interface, which are given by

(2.13) \begin{equation} \left [\frac {v}{\alpha U-\omega } \right ]_{y=d}=0\,\,\,\,\textrm {and}\,\,\,\,\left [\widehat {S}\left \{(\alpha U-\omega )v'-\alpha U'v-\frac {\alpha ^2 F^{-2}}{(\alpha U-\omega )}v\right \}\right ]_{y=d}=0, \end{equation}

where $[\boldsymbol{\cdot }]_{y=d}\equiv \lim _{y\to d^+}[\boldsymbol{\cdot }]-\lim _{y\to d^-}[\boldsymbol{\cdot }]$ (Pouliquen, Chomaz & Huerre Reference Pouliquen, Chomaz and Huerre1994; Schmid & Henningson Reference Schmid and Henningson2001).

The form of the basic shear layer flow we consider is given by (1.1). The flow shear rate, $r$ , can be positive or negative and this change corresponds to the faster stream being in the lighter or heavier fluid, respectively. In this paper we present new theoretical and numerical results on the temporal stability of a general shear layer where $f$ in (1.1) is a general anti-symmetric function with the required properties, and illustrate these findings specifically for the tanh shear layer where

(2.14) \begin{equation} f(y)=\tanh \left (\frac {y}{2}\right ). \end{equation}

The absolute instability results will solely be focused on the tanh shear layer.

In the results we present our interest is in considering a fixed anti-symmetric velocity profile (1.1) with $f(y)=-f(-y)$ for varying $r$ , but then breaking the remaining symmetry of the flow by considering both asymmetric confinement $h_1\neq h_2$ and by considering a density jump that is asymmetric about $y=0$ , i.e. $d\neq 0$ in (2.12). This second case has been observed in the ocean scenario (Anni & Farmer Reference Anni and Farmer1988; Wesson & Gregg Reference Wesson and Gregg1994; Yoshida et al. Reference Yoshida, Ohtani, Nishida and Linden1998; Tedford et al. Reference Tedford, Carpenter, Pawlowicz, Pieters and Lawrence2009) as well as in the laboratory (Koop & Browand Reference Koop and Browand1979; Lawrence et al. Reference Lawrence, Browand and Redekopp1991).

3. Temporal stability of confined shear layers

The theoretical results we present here are concerned with the position of the neutral curve as confinement parameters are varied for fixed system parameters $S,\,d,\,r,\,F^{-2}$ . Along the neutral curve $c={\omega /\alpha }{=}c_r+\textrm {i} c_i$ is purely real ( $c_i=0$ ) and we seek the value of $\alpha \in \mathbb{R}$ at the point where $c_i$ first equals $0$ . Inside the neutral curves $c\in \mathbb{C}$ and to determine information about growth rates, $c_i$ , for such disturbances requires a numerical solution of (2.9).

Before going on to explore theoretical results on the temporal stability neutral curve of these stratified shear flows, we motivate our analysis by considering the global temporal eigenvalue structure, inside the neutral curve ( $c_i\neq 0$ ), for the tanh shear layer for the value $\alpha =0.2$ and $r=1$ in figure 2 for the cases $S=1$ and $S=5$ (panel a) with $(F^{-2},d)=(0,1)$ (panel b), $(F^{-2},d)=(0,-1)$ (panel c), $(F^{-2},d)=(0.075,1)$ (panel d). In each case the flow is symmetrically confined with $h_1=h_2=10$ . The numerical scheme for this calculation, solving (2.9), is based on collocation with Chebyshev polynomials and is given in § A of the supplementary material. The density ratio $S=5$ is akin to that used in the experiments of Strykowski & Niccum (Reference Strykowski and Niccum1992), who used air and sulphur hexafluoride as two fluids to examine the stability of spatially developing mixing layers.

Figure 2.

Global eigenvalues $\omega =\alpha c$ for the tanh shear layer with $\alpha =0.2$ and $r=1$ for (a) $S=1$ and $S=5$ with (b) $(F^{-2},d)=(0,1)$ , (c) $(F^{-2},d)=(0,-1)$ , (d) $(F^{-2},d)=(0.075,1)$ . In each case $h_1=h_2=10$ . The modes denoted by the squares (Kelvin–Helmholtz modes) eventually become neutral at a finite value of $\alpha$ , while in panels (c) and (d) the circle modes (internal modes) persist to large $\alpha$ values. The modes along the $\textrm {Re}(\omega )$ axis are a discrete representation of the continuous spectrum.

Here, as we are looking at the temporal stability inside the neutral curve, $\alpha \in \mathbb{R}$ , and the flow is inviscid, therefore the temporal eigenvalues $c=c_r+\textrm {i} c_i$ are either real or complex conjugate pairs for unstable modes. The result for $S=1$ in figure 2 shows the stability of the homogeneous shear layer, and here the unstable modes, represented by the squares, are the Kelvin–Helmholtz (KH) modes that are driven by the form of the shear layer. When we consider $S\gt 1$ with $d\gt 0$ in panel (b), we again only observe the one unstable KH mode, but in panel (c), where $S\gt 1$ with $d\leq 0$ , i.e. the centre of the shear layer occurs in the lighter of the two fluids, the flow has two unstable modes. Following both these modes while increasing $\alpha$ we find that the modes represented by squares are again KH modes and eventually become neutral at a point where $(c_r=1)$ , while the modes identified by the circles remain unstable to large values of $\alpha$ (i.e. the short-wave limit). These modes indicated by the circles are internal modes, with their eigenmodes concentrated at the density interface. In the zero-Richardson-number limit ( $F^{-2}=0$ ), these internal modes are only unstable when $d\leq 0$ , but when $F^{-2}\neq 0$ (panel d) they can be shown to also be unstable for $d\gt 0$ . The KH modes have the nice property that they become neutrally stable at a value of $\alpha$ where $c=1$ , and hence, we are able to give theoretical results for these modes in § 3.1, while for the internal modes, we are restricted to examining these numerically (except for the $F^{-2}=0$ limit) in § 3.3. While we examine the properties of these two mode types separately, we note that the full flow structure can only be known if properties of both modes are considered, when they are both unstable. As the main interest in this paper is in high-speed jet flows, i.e. small Richardson numbers, we expect the $F^{-2}=0$ results to be informative of these flows, with the $0\lt F^{-2}\ll 1$ results being a perturbation on these results. Note that while our interest is in the $F^{-2}\ll 1$ case, we choose to use moderately large values of $F^{-2}$ such that its non-zero effect can be observed graphically.

The numerical global eigenvalue results for the stratified KH modes have the same structure as those observed in the homogeneous shear layer in Healey (Reference Healey2009). These neutral waves have a critical point at $y=y_c$ that occurs where $U(y_c)=c$ , which coincides with the inflection point of the flow, $U''=0$ . If we assume that $f$ is anti-symmetric and $f(-y)=-f(y)$ , this implies that $y_c=0$ and, thus, $c=1$ at the neutral curve, as we observed numerically earlier. The Taylor–Goldstein equation (2.9) then reduces in each fluid to the Rayleigh equation

(3.1) \begin{equation} f(v''-\alpha ^2 v)-f'' v=0, \end{equation}

with the matching conditions across the density jump at $y=d$ , i.e.

(3.2) \begin{equation} \left [\frac {v}{f}\right ]_{y=d}=0,\,\,\,\,\left [\widehat {S}\left (fv'-f'v-\frac {J_0}{(S-1)f}v\right )\right ]_{y=d}=0, \end{equation}

where

(3.3) \begin{align} J_0=\frac {(S-1)F^{-2}}{r^2} \end{align}

is a local Richardson number and $\widehat {S}$ is given by (2.12). Here we have chosen to include the $(S-1)$ factor in the definition of $J_0$ to simplify the form of the results presented in this section. The above system with the appropriate boundary conditions gives an eigenrelation, in terms of the system parameters, along the neutral curve. We note that the form of the neutral curve is not directly dependent on $r$ , but is only a function of $r$ via $J_0$ . However, the unstable growth rates are directly dependent on $r$ as we will show later. We first consider results in the long-wave limit, when $\alpha$ is small, for a general anti-symmetric flow, before applying these results to the tanh shear layer profile.

3.1. Confinement effects in the long-wave limit for the KH mode

We can investigate the position of the intercept of the neutral curve with $\alpha =0$ by noting that when $\alpha =0$ , $v=v_A=f$ is a solution of (3.1) as shown by Michalke (Reference Michalke1964). Healey (Reference Healey2009) then showed that by writing $v=v_B=f(y)V(y)$ a second linearly independent solution can be found. The general solution in each fluid is then a linear combination of these solutions, namely

(3.4) \begin{align} v_j=A_j f(y)+B_j f(y)V(y), \end{align}

where

(3.5) \begin{align} V(y)=\int \frac {1}{f(t)^2}\,{\textrm {d}} t \end{align}

for arbitrary constants $A_j$ , $B_j$ with $j=1,2$ as per the notation in figure 1. Satisfying the boundary conditions $v_1(h_1)=v_2(-h_2)=0$ for the case of asymmetric confinement in each layer leads to the two solutions

(3.6) \begin{align} v_1(y) &= B_1f(y)\left [\int _{h_1}^y g(t)\,{\textrm {d}} t+\frac {1}{f'(0)^2}\left (\frac {1}{h_1}-\frac {1}{y}\right )\right ], \end{align}

(3.7) \begin{align} v_2(y) &= B_2f(y)\left [\int _{-h_2}^y g(t)\,{\textrm {d}} t-\frac {1}{f'(0)^2}\left (\frac {1}{h_2}+\frac {1}{y}\right )\right ], \end{align}

where

(3.8) \begin{equation} g(t)=\frac {1}{f(t)^2}-\frac {1}{f'(0)^2t^2}. \end{equation}

Matching these two solutions at the density jump $y=d\neq 0$ and seeking a non-trivial solution leads to the condition

(3.9) \begin{eqnarray} &&SI(h_1)-I(-h_2)+\frac {1}{f'(0)^2}\left (\frac {1-S}{d}+\frac {h_1+Sh_2}{h_1h_2}\right ) \nonumber \\ &&-J_0\left [I(h_1)+\frac {1}{f'(0)^2}\left (\frac {1}{h_1}-\frac {1}{d}\right )\right ]\left [I(-h_2)-\frac {1}{f'(0)^2}\left (\frac {1}{h_2}+\frac {1}{d}\right )\right ]=0, \end{eqnarray}

where

(3.10) \begin{align} I(\xi )=\int _{\xi }^d g(t)\, {\textrm {d}} t. \end{align}

Due to its construction, condition (3.9) is an eigenrelation for the parameters $h_1,h_2,d,S,J_0$ , where given four of these parameters, the fifth can be identified such that the neutral curve intercepts $\alpha =0$ . The case $d=0$ , i.e. when the inflexion point and the density jump coincide, is a non-trivial special case that is not captured by this theory and needs to be dealt with numerically. We see in § 5 that asymmetric configurations away from this special case can be significant on the flow’s absolute instability properties.

For the case of symmetric confinement when $h_1=h_2=h$ (3.9) becomes

(3.11) \begin{eqnarray} &&SI(h)-I(-h)+\frac {1}{f'(0)^2}\left (\frac {1-S}{d}+\frac {(1+S)}{h}\right ) \nonumber \\ &&-J_0\left [I(h)+\frac {1}{f'(0)^2}\left (\frac {1}{h}-\frac {1}{d}\right )\right ]\left [I(-h)-\frac {1}{f'(0)^2}\left (\frac {1}{h}+\frac {1}{d}\right )\right ]=0. \end{eqnarray}

For a homogeneous shear layer, Healey (Reference Healey2009) showed that this scenario is a local minimum of the plate separation, i.e. that the symmetric confinement case is locally the most unstable flow configuration. In the case of the KH mode in an inhomogeneous shear layer this is not always the case, as we now discuss.

Consider an almost symmetric configuration with $h_2$ close to $h$ with $d$ , $S$ and $J_0$ fixed, then we can solve (3.9) for $h_1$ along the neutral curve. Therefore, by writing

(3.12) \begin{align} h_2 &= h+\delta ,\end{align}

(3.13) \begin{align} h_1 &= h+h_{11}\delta +h_{12}\delta ^2+O(\delta ^3),\ \end{align}

where $|\delta |\ll 1$ is a small parameter (not to be confused with $\widehat {\delta }$ used in (2.2)–(2.4)), we can substitute this into (3.9) and equate like powers of $\delta$ . Substituting into (3.9), using the asymptotic result

(3.14) \begin{eqnarray} \int _{a+b}^a g(t)\,{\textrm {d}} t&=&\int _{a+b}^a\left (g(a)+g'(a)(t-a)+\frac {1}{2}g''(a)(t-a)^2+O\left ((t-a)^3\right )\right )\,{\textrm {d}} t,\nonumber \\ &=&-g(a)b+\frac {1}{2}g'(a)b^2-\frac {1}{6}g''(a)b^3+O(b^4) \end{eqnarray}

for $|b|\ll 1$ and noting that as $f(t)$ is an odd function then $g(t)$ is an even function such that $g(-t)=g(t)$ and $g'(-t)=-g'(t)$ , we derive (3.11) at $O(1)$ ,

(3.15) \begin{align} -\frac {\left(1+S h_{11}\right)}{f(h)^2}-J_0 f(h)^2\!\left [\!I(h)\!+\!\frac {1}{f'(0)^2}\!\left (\frac {1}{h}-\frac {1}{d}\right )\!\right ]\!+h_{11}\!\left [\!I(-h)-\frac {1}{f'(0)^2}\!\left (\frac {1}{h}+\frac {1}{d}\right )\!\right ]=0 \end{align}

at $O(\delta )$ and

(3.16) \begin{eqnarray} &&\left [-S+J_0\left [I(-h)-\frac {1}{f'(0)^2}\left (\frac {1}{h}+\frac {1}{d}\right )\right ]\right ] h_{12} \nonumber \\ &&+\left [S\frac {f'(h)}{f(h)}-J_0\left [I(-h)-\frac {1}{f'(0)^2}\left (\frac {1}{h}+\frac {1}{d}\right )\right ]\right ]h_{11}^2 \nonumber \\ &&+\frac {J_0}{f(h)^2}+\frac {f'(h)}{f(h)}+\frac {f'(h)}{f(h)}J_0\left [I(h)+\frac {1}{f'(0)^2}\left (\frac {1}{h}-\frac {1}{d}\right )\right ]=0 \end{eqnarray}

at $O(\delta ^2)$ .

At leading order the symmetrically confined case (3.11) is satisfied, while at higher orders these equations can be solved for $h_{11}$ and $h_{12}$ . In order to highlight the qualitative behaviour of this analysis we consider the zero local Richardson number, $J_0=0$ case, with the $J_0\neq 0$ , $|J_0|\ll 1$ case having qualitatively similar behaviour, as we demonstrate for the $\tanh$ mixing layer in § 3.2. With $J_0=0$ we find that, at $O(\delta )$ ,

(3.17) \begin{align} h_{11}=-\frac {1}{S}, \end{align}

while at $O(\delta ^2)$ ,

(3.18) \begin{align} h_{12}=\frac {(1+S)}{S^2}\frac {f'(h)}{f(h)}. \end{align}

Therefore,

(3.19) \begin{equation} h_{1}=h-\frac {\delta }{S}+\frac {(1+S)}{S^2}\frac {f'(h)}{f(h)}\delta ^2+O(\delta ^3). \end{equation}

This means that the plate separation distance for neutral waves for slightly asymmetrically confined flows is

(3.20) \begin{equation} h_1+h_2=2h+\frac {S-1}{S}\delta +\frac {1+S}{S^2}\frac {f'(h)}{f(h)}\delta ^2. \end{equation}

For the homogeneous case $(S=1)$ from Healey (Reference Healey2009), this quantity is always larger than the symmetric case that showed that the asymmetric case was always less unstable than the corresponding symmetric case for a monotonically increasing $f$ .

For the inhomogeneous case here, there is an additional term at $O(\delta )$ . The significance of this term is that if $\delta \gt 0$ , i.e. moving both plates apart (while remaining on the neutral curve), then the symmetrically confined case is more temporally unstable than the asymmetric case. However, if $\delta \lt 0$ (and sufficiently small), i.e. both plates are moved together (while remaining on the neutral curve), then the asymmetrically confined flow can be more temporally unstable than the symmetrically confined counterpart. The interesting feature of this result is that it is independent of the position of the fixed density jump $y=d$ .

Note that when $\delta \lt 0$ , the $O(\delta )$ term in (3.20) is negative, but the $O(\delta ^2)$ term is positive. This combination of terms ultimately becomes positive for $\delta \lt \delta _c$ where

(3.21) \begin{align} \delta _c=S\frac {1-S}{1+S}\frac {f(h)}{f'(h)}. \end{align}

Thus, for particularly small confinement asymmetries, the flow is more unstable than the symmetric case, but for large confinement asymmetries, the flow is ultimately more stable than the symmetric configuration. This is shown numerically for the tanh shear layer in § 3.2.

For the case of semi-confined flows, we cannot simply let $h_1$ or $h_2$ tend to $\infty$ , as the integrals are not convergent. Also, in the unbounded half of the domain, we expect the behaviour $v_1\sim \exp (-\alpha y)$ or $v_2\sim \exp (\alpha y)$ , respectively, but we expect these forms in the limit of small $\alpha$ ; hence, we expect $v_1\sim 1-\alpha y$ or $v_2\sim 1+\alpha y$ and we use these to fix the arbitrary constants $A_1, B_1$ and $A_2, B_2$ , respectively. Note, here we are considering the limit $\alpha \to 0$ , whereas in (3.6) and (3.7) we were able to consider $\alpha =0$ directly. This then gives the expressions

(3.22) \begin{align} v_1(y)&= f(y)\left [1-\alpha y-\alpha \int _{\infty }^y\left (g(t)-1\right )\,{\textrm {d}} t+\frac {\alpha }{f'(0)^2 y}\right ], \end{align}

(3.23) \begin{align} v_2(y)&= f(y)\left [1+\alpha y+\alpha \int _{-\infty }^y\left (g(t)-1\right )\,{\textrm {d}} t-\frac {\alpha }{f'(0)^2 y}\right ], \end{align}

for an unbounded region above the shear layer and an unbounded region below the shear layer, respectively. Thus, the semi-confined relations can be found by matching (3.22) with (3.7) and matching (3.23) with (3.6), which lead to the expressions

(3.24) \begin{align} &\alpha \left [I(-h_2)-\frac {1}{f'(0)^2}\left (\frac {1}{h_2}+\frac {1}{d}\right )\right ] \nonumber \\ &\quad = -\left [\!S\!+\!J_0\!\left [\!I(-h_2)-\frac {1}{f'(0)^2}\!\left (\frac {1}{h_2}\!+\!\frac {1}{d}\right )\!\right ]\right ]\!\left [\!1-\alpha d-\alpha\! \int _{\infty }^d\left (g(t)-1\right )\,{\textrm {d}} t\!+\!\frac {\alpha }{f'(0)^2 d}\!\right ],\end{align}

(3.25) \begin{align} &\alpha S\left [I(h_1)+\frac {1}{f'(0)^2}\left (\frac {1}{h_1}-\frac {1}{d}\right )\right ] \nonumber \\ &\quad =\left [\!1\!+\!J_0\!\left [\!I(h_1)+\frac {1}{f'(0)^2}\left (\frac {1}{h_1}-\frac {1}{d}\right )\!\right ]\right ]\!\left [1+\alpha d+\alpha\! \int _{-\infty }^d\left (g(t)-1\right )\,{\textrm {d}} t-\frac {\alpha }{f'(0)^2 d}\!\right ],\end{align}

for flows bounded below and above, respectively. To find the asymptotic forms of these expressions, we write $\alpha =\alpha _0+h_2\alpha _1$ in (3.24) and $\alpha =\alpha _0+h_1\alpha _1$ (3.25) and consider the limits $h_2\to 0$ and $h_1\to 0$ , respectively. These lead to

(3.26) \begin{align} \alpha & =\frac {J_0}{1+J_0 D_+}\!+\!\frac {f'(0)^2}{(1+J_0 D_+)^2}\!\left [S(1\!+\!(J_0-1)D_+)-(J_0-1)(J_0 D_+ + 1)C\right ]h_2+O(h_2^2),\nonumber \\ \alpha &=\frac {J_0}{S-J_0 D_-}\!+\frac {f'(0)^2}{(S-J_0 D_-)^2}\!\left [S-(J_0-1)(J_0D_- -S)C-(J_0-1)D_-\right ]h_1+O(h_1^2), \end{align}

where

(3.27) \begin{align} C=\int _0^d g(t)\,{\textrm {d}} t-\frac {1}{f'(0)^2 d},\,\,\,D_{\pm }=d+\int _{\pm \infty }^d(g(t)-1)\,{\textrm {d}} t-\frac {1}{f'(0)^2 d}. \end{align}

We again highlight the asymptotic forms of these eigenrelations for the case $J_0=0$ , which simplify to

(3.28) \begin{equation} \alpha \sim S f'(0)^2 h_2 \end{equation}

and

(3.29) \begin{equation} \alpha \sim \frac {1}{S} f'(0)^2 h_1, \end{equation}

respectively. Both of these expressions are in a form similar to that found for the homogeneous shear layer in Healey (Reference Healey2009).

3.2. Confinement effect on the tanh shear layer for KH modes

The general long-wave results of the last section will be illustrated in this section for the smooth tanh shear layer with $f(y)$ given in (2.14). This flow field provides a convenient flow model on which to examine both temporal and absolute instability properties due to the fact that some analytic solutions can be derived. The profile has been used in the study by Huerre & Monkewitz (Reference Huerre and Monkewitz1985) to examine the absolute instability of an unconfined shear layer, in Healey (Reference Healey2009) to examine the absolute instability of asymmetrically confined homogeneous shear layers and in Carpenter et al. (Reference Carpenter, Balmforth and Lawrence2010) to examine the instability of stratified shear layers to name a few.

With the form of $f(y)$ given by (2.14), Michalke (Reference Michalke1964) showed that on the neutral curve, (3.1) exhibits an exact solution. Therefore, in our two layers, $j=1,2$ , we can write the general solution for the eigenmode on the neutral curve as

(3.30) \begin{align} v_j(y)=A_j\!\left [2\alpha \cosh (\alpha y)-\sinh (\alpha y)\tanh \left (\frac {y}{2}\right )\!\right ]\!+\!B_j\!\left [2\alpha \sinh (\alpha y)-\cosh (\alpha y)\tanh \left (\frac {y}{2}\right )\!\right ], \end{align}

where $A_j,B_j$ give four unknown constants that are fixed by the two boundary conditions and the two matching conditions at the density jump. The boundary conditions at the confining plates are again $v(h_1)=v(-h_2)=0$ while if the flow is unconfined then the condition becomes (2.11), which is equivalent to $v(\pm \infty )=0$ .

Unlike the homogeneous case in Healey (Reference Healey2009), there are no simple closed forms for the relations that link $\alpha , h_1, h_2, d, S$ and $J_0$ . However, we can succinctly write down the forms of the neutral curve eigenconditions by introducing the following functions:

(3.31) \begin{align} T_1(\xi )&= 2\alpha -\tanh (\alpha \xi )\tanh \left (\frac {\xi }{2}\right ),\quad T_2(\xi )=2\alpha \tanh (\alpha \xi )-\tanh \left (\frac {\xi }{2}\right ),\nonumber \\ T_3(\xi )&=\left (2\alpha ^2\tanh (\alpha \xi )\tanh \left (\frac {\xi }{2}\right )-\alpha \right )\tanh \left (\frac {\xi }{2}\right ), \nonumber \\ T_4(\xi )&=\left (2\alpha ^2\tanh \left (\frac {\xi }{2}\right )-\alpha \tanh (\alpha \xi )\right )\tanh \left (\frac {\xi }{2}\right ),\nonumber \\ T_5(\xi )&= 2\alpha +\tanh \left (\frac {\xi }{2}\right ),\quad \quad T_6(\xi )=2\alpha -\tanh \left (\frac {\xi }{2}\right ),\nonumber \\ T_7(\xi )&=\left (\alpha +2\alpha ^2\tanh \left (\frac {\xi }{2}\right )\right )\tanh \left (\frac {\xi }{2}\right ),\quad T_8(\xi )=\left (\alpha -2\alpha ^2\tanh \left (\frac {\xi }{2}\right )\right )\tanh \left (\frac {\xi }{2}\right ). \end{align}

Then the neutral curve relations can be given by

(3.32) \begin{align} ST_5(d)T_8(d)-T_6(d)T_7(d)-J_0T_5(d)T_6(d)= 0\,\,\,\textrm {unconfined}& ,\end{align}

(3.33) \begin{align} T_1(h)\left [ST_4(d)T_5(d)+T_2(d)T_7(d)\right ]+T_2(h)\left [ST_3(d)T_5(d)+T_1(d)T_7(d)\right ]& \nonumber \\ +J_0T_5(d)\left [T_1(d)T_2(h)+T_1(h)T_2(d)\right ]= 0, \quad h_2=h,\quad h_1=\infty,\end{align}

(3.34) \begin{align} T_1(h)\left [ST_2(d)T_8(d)+T_4(d)T_6(d)\right ]-T_2(h)\left [ST_1(d)T_8(d)+T_3(d)T_6(d)\right ]& \nonumber \\ +J_0T_6(d)\left [T_1(d)T_2(h)-T_1(h)T_2(d)\right ]= 0,\quad h_1=h, \quad h_2=\infty,\end{align}

(3.35) \begin{align} (S-1)\left [T_1(h)^2T_2(d)T_4(d)-T_2(h)^2T_1(d)T_3(d)\right ] \nonumber \\ +(1+S)T_1(h)T_2(h)\left [T_2(d)T_3(d)-T_1(d)T_4(d)\right ] \nonumber \\ -J_0\left [T_1(h)^2T_2(d)^2-T_1(d)^2T_2(h)^2\right ]= 0 ,\quad h_{1,2}=h,\end{align}

(3.36) \begin{align} (S-1)\left [T_1(h_1)T_1(h_2)T_2(d)T_4(d)-T_2(h_1)T_2(h_2)T_1(d)T_3(d)\right ]& \nonumber \\ +T_2(d)T_3(d)\left [T_2(h_1)T_1(h_2)+ST_1(h_1)T_2(h_2)\right ] \nonumber \\ -T_1(d)T_4(d)\left [T_1(h_1)T_2(h_2)+ST_2(h_1)T_1(h_2)\right ] \nonumber \\ +J_0\left [T_1(h_1)T_2(d)-T_1(d)T_2(h_1)\right ]\left [T_1(h_2)T_2(d)+T_1(d)T_2(h_2)\right ]=0,\quad h_1\neq h_2.\end{align}

The unconfined condition (3.32) can be expanded and written as a cubic polynomial in $\alpha$ with coefficients depending on $d$ , $S$ and $J_0$ , i.e.

(3.37) \begin{eqnarray} 4(S+1)\tanh ^2\left (\frac {d}{2}\right )\alpha ^3-2\left [(S-1)\tanh \left (\frac {d}{2}\right ){\textrm {sech}}^2\left (\frac {d}{2}\right )+2J_0\right ]\alpha ^2&& \nonumber \\ -(S+1)\tanh ^2\left (\frac {d}{2}\right )+J_0\tanh ^2\left (\frac {d}{2}\right )&=&0,\\[6pt]\nonumber \end{eqnarray}

which has a positive root that, for $J_0=0$ , is

(3.38) \begin{equation} \alpha =\frac {(S-1){\textrm {sech}}^2\left ({d}/{2}\right )+\left [(S-1)^2{\textrm {sech}}^4\left ({d}/{2}\right )+4(S+1)^2\tanh ^2\left ({d}/{2}\right )\right ]^{1/2}}{4(S+1)\tanh \left ({d}/{2}\right )}. \end{equation}

This value is a constant for a given $d\neq 0$ and $S$ and corresponds to the $h_{1,2}\to \infty$ limit of the other cases in (3.33)–(3.36). When $S=1$ , this reduces to $\alpha =1/2$ and all the other relations reduce to the equivalent results from Healey (Reference Healey2009), as shown in § B of the supplementary material.

Figure 3.

Plot of the temporal neutral curve of the KH modes for the cases $S=5$ , $d=1$ with (a) $J_0=0$ , $h_2=3h_1/2$ ; (b) $J_0=0$ , $h_2=9h_1/10$ ; and (c) $J_0=0.3$ , $h_2=3h_1/2$ . The horizontal dashed line is the neutral curve for the unconfined shear layer (3.38), while the other results are neutral curves for (i) a semi-confined shear layer with $h_2=h$ (3.33), (ii) a semi-confined shear layer with $h_1=h$ (3.34), (iii) a symmetrically confined shear layer with $h_1=h_2=h$ (3.35), and (iv) a asymmetrically confined shear layer with the above plate position ratios and $(h_1+h_2)/2=h$ (3.35). The dotted lines represent the small $\alpha$ asymptotic result (3.28) in (a) and (b). In each panel, result 2 terminates at $h=1$ because this is the point at which the upper plate meets the density jump producing a homogeneous shear layer.

The results considered here in (3.32)–(3.36) with $d\neq 0$ can be used to define the neutral curves, we plot examples of these neutral curves in figure 3 for $S=5$ , $d=1$ with $J_0=0$ , $h_2=3h_1/2$ (panel a), $J_0=0$ , $h_2=9h_1/10$ (panel b) and $J_0=0.3$ , $h_2=3h_1/2$ (panel c). Here the flow is unstable under (and to the right of) the solid lines. As the curves lie progressively further to the right (larger values of $h$ ), then the flows become progressively more stable as we move in this direction. For example, in panel (a) the symmetrically confined result (result 3) is more stable than the two semi-confined results (results 1 and 2), but the asymmetric result with $h_2=3h_1/2$ (result 4) is more stable than this symmetric result. In panel (b) we have a case with $\delta \lt 0$ in (3.20) ( $h_2=9h_1/10$ ) and in this case we showed analytically in § 3.1 that the asymmetric case can be more unstable than the symmetric case, which is confirmed here, with the positions of results 3 and 4 reversed, although the neutral curves lie close to each other. In panels (a) and (b) the dotted line gives the result of the asymptotic result (3.28), while result 2 terminates at $h=1$ because at this point the upper plate would meet the density interface, changing the problem to a homogeneous problem. In panel (c) we consider the same case as in panel (a) except with $J_0=0.3$ . Here we find that the significant difference with the $J_0=0$ case is that each result is more unstable than the corresponding $J_0=0$ result, with unstable modes existing for a larger range of $\alpha$ values.

Figure 4.

Temporal instability growth rates of the KH modes for the shear layer (2.14) with $r=1$ for the cases given in (a) figure 3(a) and (b) figure 3(c), confined by symmetric plates at $y=\pm h$ . The curve labelled $h=\infty$ corresponds to the unconfined result.

Figure 3 shows that, for $h\geq 5$ , the effect of confinement on the neutral curve for symmetric confinement is small, but the stabilising effect on the growth rate $\textrm {Im}(\omega )=\textrm {Im}(\alpha c)$ for $h\leq 5$ is significant; see figure 4. These growth rates need to be calculated numerically by solving (2.9) directly for the (now) complex eigenvalue $c$ and, unlike on the neutral curve, these results depend on the shear rate $r$ , which we set to $r=1$ here. These plots are essentially vertical slices through the results in figures 3(a) and 3(c).

4. Long-wavelength dispersion relations

Before we examine the absolute instability properties of these stratified shear flows, we first consider the long-wavelength disturbances for absolute instability. When we consider disturbances with wavelengths longer than the thickness of the shear layer, then a suitable approximation for the velocity function (1.1) is to consider a step function where $f(y)=-1$ for $y\lt 0$ and $f(y)=1$ for $y\gt 0$ . This profile allows for an analytic form of the dispersion relation to be derived, which is fast and straight forward to solve numerically. This is helpful for plotting contours of $\omega$ in the complex $\alpha$ plane, which is an essential part of identifying the existence of pinch point saddles, as described in § 5. For the homogeneous shear layer, Healey (Reference Healey2009) identified that it is these long-wavelength disturbances that are significant to the form of the neutral stability curve, especially at stronger confinements. The same observation was also made by Juniper (Reference Juniper2006) for confined jets and wakes. In anticipation that long-wave solutions will be significant in the confinement of asymmetric stratified shear flows, we construct the long-wavelength dispersion relations here.

If we let $0\lt \epsilon \ll 1$ be a small parameter that characterises the size of the wavenumber, then if the plate separation is of $O(\epsilon ^{-1})$ , i.e. of order of the wavelength of the disturbance, and we let the position of the density jump also be $O(\epsilon ^{-1})$ away from the shear layer, then both these effects enter the problem at leading order. Healey (Reference Healey2009) showed that the scaling on the phase speed $c=\omega /\alpha$ is the same as for the KH instability, namely

(4.1) \begin{equation} \alpha =\epsilon \alpha _0,\,\,\,\,\,\omega =\epsilon c_0\alpha _0+O(\epsilon ^2), \end{equation}

with the boundary conditions

(4.2) \begin{align} v(h_1/\epsilon )=0,\,\,\,v(-h_2/\epsilon )=0, \end{align}

and with the density jump at $y=d/\epsilon$ . Healey (Reference Healey2009) also showed that $v=O(\epsilon )$ in these regions.

Inside the shear layer and the density-jump layer the variable $y=O(1)$ , but outside these layers we introduce a slow spatial variable $Y=\epsilon y$ , where $Y=O(1)$ . Therefore, the shear and density layers are significantly thinner than the rest of the flow domain, and so the consequence of this is that we can determine the long-wave dispersion relations by solving the piecewise-linear form of the problem, as depicted in figure 5, for the case $r\gt 0$ and $d\gt 0$ .

Figure 5.

Schematic of the piecewise-linear, long-wave form of the problem for the case $d\gt 0$ and $r\gt 0$ .

Here, in the three layers, the base flow velocity and density profiles are constants, and so the solutions satisfy

(4.3) \begin{equation} \frac {{\textrm {d}}^2v_{j}}{{\textrm {d}} Y^2}-\alpha _0^2v_j=0, \end{equation}

to be solved with the jump conditions (2.13), which now need to be solved at $Y=0$ as well as $Y=d$ .

For the case in figure 5, we solve (4.3) in each layer with the respective boundary conditions at $Y=h_1,\,-h_2$ to give

(4.4) \begin{align} v_1&= A\sinh \alpha _0(Y-h_1), \end{align}

(4.5) \begin{align} v_2 &= B\cosh \alpha _0Y+C\sinh \alpha _0Y,\end{align}

(4.6) \begin{align} v_3 &= D\sinh \alpha _0(Y+h_2). \end{align}

Matching at $Y=0$ and $Y=d$ leads to

(4.7) \begin{align} B &=\frac {1+r-c_0}{1-r-c_0}D\sinh \alpha _0 h_2, \end{align}

(4.8) \begin{align} C &=\frac {1-r-c_0}{1+r-c_0}D\cosh \alpha _0 h_2, \end{align}

(4.9) \begin{align} A\sinh \alpha _0(d-h_1)&= B\cosh \alpha _0 d+C\sinh \alpha _0 d,\end{align}

(4.10) \begin{align} A\cosh \alpha _0(d-h_1)-S\left [B\sinh \alpha _0d+C\cosh \alpha _0d\right ]&=\frac {(1-S)F^{-2}}{\alpha _0(1+r-c_0)^2}A\sinh \alpha _0(d-h_1).\end{align}

The dispersion relation for $d\gt 0$ is then found via a non-trivial solution to this system of equations. The full set of long-wave dispersion relations used for both finite and infinite $h_1$ and $h_2$ can be found in § D of the supplementary material.

5. Absolute instability of symmetrically confined shear layers

To determine the absolute instability properties of a flow, we consider a spatio-temporal stability analysis where both $\alpha$ and $\omega$ are allowed to be complex. We are interested in seeking whether instability wavepackets have growth, $\textrm {Im}(\omega )=\omega _i\gt 0$ , along the characteristic

(5.1) \begin{align} \frac {x}{t}=0, \end{align}

i.e. at the point at which the instability disturbance is generated. Spatio-temporal theory has been documented in many works such as Briggs (Reference Briggs1964), Huerre & Monkewitz (Reference Huerre and Monkewitz1990), Healey (Reference Healey2006a , Reference Healey2007), Juniper (Reference Juniper2006, Reference Juniper2007), Garrett & Peake (Reference Garrett and Peake2007), Brambley (Reference Brambley2010), Pier & Peake (Reference Pier and Peake2015) and Poole & Turner (Reference Poole and Turner2023, Reference Poole and Turner2024) to name a few, and the reader is directed to these works and references therein for more information. Determining the absolute instability characteristics of the flow can be shown to amount to seeking the value of $\omega _i$ at special saddle points of contours of $\omega _i$ in the complex $\alpha$ plane. These special saddle points are known as pinch points and can be determined via the Briggs criterion (Briggs Reference Briggs1964), which effectively states that the Fourier inversion contour, which runs along the real $\alpha$ axis, can be deformed onto the saddle, while remaining in the valleys of the saddle in the rest frame (Healey Reference Healey2006b ). For the results which follow in §§ 5 and 6.1, we check whether our saddle points are pinch points in the following two ways: firstly, we plot contours of $\omega _i=\textrm {constant}$ , such as those in figure 6, for various confinement and asymmetry parameters and determine by eye whether the Fourier inversion contour can be deformed off the real $\alpha$ axis and through the saddle, while remaining in the valleys of the saddle; secondly, in ambiguous cases where a pinch point cannot be determined by eye, we explicitly use Briggs criterion to determine whether the saddle is one which pinches upstream and downstream propagating branches of solutions.

For a given set of parameters $h_1,h_2,S,d,r$ , we find the four unknowns $\alpha _r,\alpha _i,\omega _r$ and $\omega _i$ (where $\alpha =\alpha _r+\textrm {i}\alpha _i$ , $\omega =\omega _r+\textrm {i}\omega _i$ ) by solving

(5.2) \begin{equation} \mathbb{D}(\alpha ,\omega )=\frac {\partial \mathbb{D}}{\partial \alpha }(\alpha ,\omega )=0 \end{equation}

simultaneously using Newton’s method. From an initial guess $(\alpha ^0,\omega ^0)$ the $n$ th iterate of $(\alpha ^n,\omega ^n)$ is updated via

(5.3) \begin{align} \left (\begin{array}{c} \alpha ^{n+1} \\ \omega ^{n+1}\end{array}\right )=\left (\begin{array}{c} \alpha ^{n} \\ \omega ^{n}\end{array}\right )-\left (\begin{array}{cc} \mathbb{D}_\alpha (\alpha ^n,\omega ^n) & \mathbb{D}_\omega (\alpha ^n,\omega ^n) \\ \mathbb{D}_{\alpha \alpha }(\alpha ^n,\omega ^n) & \mathbb{D}_{\alpha \omega }(\alpha ^n,\omega ^n)\end{array}\right )^{-1}\left (\begin{array}{c} \mathbb{D}(\alpha ^n,\omega ^n) \\ \mathbb{D}_\alpha (\alpha ^n,\omega ^n) \end{array}\right ), \end{align}

where the subscripts denote partial derivatives. We continue iterations until $||(\alpha ^{n+1},\omega ^{n+1})-(\alpha ^{n},\omega ^{n})||_2\lt 10^{-8}$ , at which point we consider the solution to be converged.

Our main interest is in determining the position of the neutral stability curve for a given set of system parameters, and so in this case our vector of unknowns is $(\alpha _r,\alpha _i,\omega _r,r)^T$ , as $\omega _i=0$ on this curve. We again find these values using Newton’s method but this time we split the two complex equations in (5.2) into four real equations for numerical convenience.

Despite our interest being on how asymmetry of the system affects the absolute instability properties, we begin first by considering a symmetrically confined shear layer with $h_1=h_2=h$ and, in particular, with $d=0$ . In this case we are restricted to numerical solutions of the full system only, because the long-wavelength theory in (D-1) and (D-2) of the supplementary material both simplify to

(5.4) \begin{align} \left [(c_0+r-1)^2S+(c_0-r-1)^2\right ]\tanh (\alpha _0h)+(S-1)F^{-2}\frac {\tanh ^2(\alpha _0h)}{\alpha _0}=0. \end{align}

When $F^{-2}=0$ , this has no dispersive terms, i.e. no saddle points and, thus, no absolute instability properties of the shear layer. In this case the long-wave theory breaks down as dispersion is a higher-order effect. However, when $d\neq 0$ , the long-wave theory does not breakdown for symmetric confinement, as we will see later, and dispersion returns as a leading-order effect.

Figure 6.

Plot of $\omega _i$ contours in the $\alpha$ plane for $S=2$ , $r=1.09$ , $d=F^{-2}=0$ for symmetric confinement with (a) $h=50$ and (b) $h=20$ . The white circle and square denote the dominant pinch points for $h=50$ and $h=20$ , respectively, the thick black contour signifies the value $\omega _i=0$ , the black and white lines denote branch cuts and the red contour signifies an example path for the Fourier inversion contour. In (a) the flow is convectively unstable with $\omega _i=-0.00451$ (circle saddle) and in (b) the flow is absolutely unstable with $\omega _i=0.00579$ (square saddle).

In figure 6 we consider the form of the $\omega _i=\,$ constant contours for the cases $S=2$ , $r=1.09$ , $d=F^{-2}=0$ with $h=50$ (panel a) and $h=20$ (panel b). In panel (a) the flow is convectively unstable, as the dominant pinch point saddle at $\alpha =0.0723-0.2578\textrm {i}$ has $\omega _i=-0.00451$ . The saddle at $\alpha =0.0129-0.0609\textrm {i}$ also lies on the Fourier inversion contour (an example of this contour is denoted by the red contour in each panel), but has $\omega _i=-0.00934$ and so is subdominant to the other saddle. In this case, determining whether saddle points lie on the Fourier inversion contour is a straight forward task by eye, by determining how the Fourier inversion contour is deformed off the $\alpha _r$ axis while remaining in the valleys of the saddle points. The red inversion contour example for $h=50$ comes along the $\alpha _r$ axis from $\alpha _r=-\infty$ (not shown) and then passes down the left-hand side of the figure, passing through the string of saddles at $0.0036{-}0.0298\textrm {i}$ , $0.0129{-}0.0609\textrm {i}$ , $0.0092{-}0.0917\textrm {i}$ , $0.0254{-}0.1258\textrm {i}$ , $0.0116{-}0.1548\textrm {i}$ and $0.0390{-}0.1924\textrm {i}$ before finally passing through the saddle point at $0.0723{-}0.2578\textrm {i}$ and back up to the $\alpha _r$ axis where it extends to $\alpha _r=\infty$ . The dominant saddle in this case is the same saddle point that exists in the unconfined $(h\to \infty )$ limit and so would determine the absolute instability properties of the flow if the plates were not present. This unconfined saddle usually occurs with $\alpha _r=O(1)$ , and thus, is not determined by the long-wave theory. The array of saddles close to the imaginary axis are due to the inclusion of the bounding plates, which puts infinitely many poles along the imaginary axis at $\alpha =\alpha _p$ , where $\alpha _p$ satisfies

(5.5) \begin{equation} S\tanh (\alpha _p(d-h))-\tanh (\alpha _p(d+h))=0; \end{equation}

see § E of the supplementary material. When $S=1$ or $d=0$ , this reduces to

(5.6) \begin{align} \alpha _p=\pm \frac {n\pi }{2h}\textrm {i}, \end{align}

as found in Healey (Reference Healey2009) for the homogeneous case.

As the flow is further confined to $h=20$ in figure 6(b), the saddle points move in the complex $\alpha$ plane and we find scenarios where the confinement saddles dominate over the unconfined saddle. In this case the flow is absolutely unstable as the pinch point saddle at $\alpha =0.0619-0.1502\textrm {i}$ has the largest $\omega _i$ value with $\omega _i=0.00579$ . Note that in both panels of figure 6 the inversion contour eventually returns to the real $\alpha$ axis at large values of $\alpha _r$ .

Figure 7.

(a,b) Plot of the neutral curve for absolute instability for symmetric confinement in the $(h,r)$ plane for the case $d=F^{-2}=0$ and $S=1, 1.1, 2, 5, 20$ and $100$ . The solid lines represent cases with $r\gt 0$ and the dashed lines have $r\lt 0$ . Above each neutral curve the flow is absolutely unstable and convectively unstable below. Panel (b) is a close-up of (a). In (c,d) we consider the cases $S=2$ and $S=100$ , respectively, with $d=0$ . The arrows show increasing Richardson numbers with $F^{-2}=0.0,0.01,0.025,0.05,0.1$ in (c) and $F^{-2}=0.0,0.0001,0.0002,0.0004,0.0006$ in (d).

In figures 7(a) and 7(b) we consider the neutral absolute instability curves for the symmetrically confined shear layer with $F^{-2}=d=0$ . The flows are absolutely unstable above the contours and convectively unstable below, and results with $r\lt 0$ are given by the dashed lines while $r\gt 0$ results are given by the solid lines. For the case $S=2$ with $r=1.09$ depicted in figure 6, we see for large $h$ , the neutral curve is approximately constant and this is because here the unconfined saddle is dominant and varying the plate position does not alter its stability properties significantly, while for $h\lesssim 29$ , the confinement saddle switches to being the dominant saddle. Here we find that the neutral curve moves to a smaller value of $|r|$ , giving an absolute instability for a larger set of parameters, ultimately making the flow more absolutely unstable. This change in saddle dominance is signified by the gradient discontinuities of the neutral curve.

Figure 7(a) shows that, for $r\lt 0$ (i.e. having the faster stream in the denser fluid), the unconfined saddle is dominant for a large range of $h$ . Whereas for $r\gt 0$ (i.e. having the slower stream in the denser fluid), the confined saddles become dominant for increasing values of $h$ . The reason for this is that when $r\lt 0$ , the wavelength of the unconfined mode reduces ( $\alpha _r$ increases) for increasing $h$ , and thus, it moves further from the confined modes in the complex $\alpha$ plane. For $r\gt 0$ , the wavelength increases and, thus, the saddle moves towards the imaginary axis, allowing the confinement modes to more readily dominate the absolute instability properties of the flow. In fact, for $S\geq 5$ , only confinement saddles contribute to the neutral curve for $h\leq 250$ given here. These confinement saddles cause the neutral curve to lower and for $S\geq 5$ , the flow has a region of absolute instability with $0\lt r\lt 1$ , which signifies a co-flow shear layer. The minimum $r$ value is $r=0.9402$ , which occurs at $S=8.5778$ , as shown in figure 8.

Figure 8.

Plot of the minimum $r$ value as a function of $S$ for the symmetric confinement neutral curves in figures 7(a) and 7(b) for $F^{-2}=0$ and $r\gt 0$ .

In figures 7(c) and 7(d) we consider the neutral curves for the cases $S=5$ and $S=100$ , respectively, when the Richardson number and, hence, $F^{-2}\neq 0$ . In this work our focus is on high-speed flows, i.e. small Richardson numbers or large Froude numbers, such that the results are a small perturbation from the $F^{-2}=0$ results, but here we consider a range of $F^{-2}$ values such that its effect on the results can be observed. The results in figures 7(c) and 7(d) show that increasing $F^{-2}$ has the effect of stabilising the absolute instability of the flow, i.e. raising the neutral curves, but the effect of the confining modes dominating the absolute instability properties for small and moderate confinements is still evident.

Figure 9.

Plot of the neutral curve for absolute instability in the $(d/h,r)$ plane for the cases (a) $S=2$ , (b) $S=5$ , (c) $S=20$ , and (d) $S=100$ for symmetrically confined flow with $F^{-2}=0$ . The solid lines represent the neutral curve for the full problem, while the dashed lines represent the results from the long-wave analysis. In (d) the dotted line for $d/h\lt 0$ represents the contour where $\alpha _r=0$ . At this point the contribution from this saddle should be discounted (Juniper Reference Juniper2006). Here AU and CU denote regions that are absolutely and convectively unstable, respectively.

In this symmetrically confined scenario we now consider how the neutral stability region is affected by moving the mean density position, $d$ , away from $y=0$ in figure 9 for the case $F^{-2}=0$ with $r\gt 0$ . Here the solid line gives the neutral curve of the shear flow in the $(r,d/h)$ plane with $h=100$ fixed, and the dashed line represents the neutral curve of the long-wave dispersion relation from § 4. All the results show genuine asymmetry, with the minimum value of $r\gt 0$ occurring for $d\lt 0$ in each case. For $S=2$ , the value of $h=100$ is such that for $d=0$ , the confined saddles do not determine the flow stability properties and so as $d$ is varied, the long-wavelength result greatly differs from the actual calculated result. The result also very quickly tends to a constant $r$ for $|d/h|\gtrsim 0.2$ , which is just the result for a tanh shear layer in a single fluid, i.e. the effect of the second fluid stream not containing the shear layer has become less significant to the calculation. For the results $S=5$ and $20$ , the confinement saddles become more significant and we see a range of $d/h$ values where the long-wave results agree well with the actual results. This occurs more predominantly for $d\gt 0$ where the centre of the shear layer lies in the denser of the two fluids. For $S=100$ in panel (d), the results again have a minimum $r$ for $d/h\lt 0$ , but in this case there appears to be a mode for which the flow is unstable with $r=1$ for $d/h\lt 0$ . However, this is a mode for which $\alpha _r\to 0$ at this neutral value. Juniper (Reference Juniper2006) argues that these modes should be discounted on the physical grounds that these modes have a large wavelength, violating the fixed base flow assumption, and as they decay slowly in the $y$ direction, their amplitude should tend to zero to avoid having infinite energy. Applying this reasoning here, the small dotted line close to $d/h=0$ plots the contour for modes that have $\alpha _r=0$ . The flow is then absolutely unstable for $(d/h,r)$ values above the upper solid line for $d/h\lt 0$ , which agrees with the long-wave approximation for a range of values.

Figure 10.

Plot of the neutral curve for absolute instability in the $(h,r)$ plane for the cases (a) $S=2$ and (b) $S=20$ with $F^{-2}=0$ for a symmetrically confined flow. In each panel the dashed line is the $d=0$ result from figure 7, while the solid lines give the equivalent neutral curve with $d=d_{\textit{min}}$ . Above the neutral curve the flow is absolutely unstable and convectively unstable below.

The key result from figure 9 is the existence of a shear rate value, $r$ , smaller than that when $d=0$ , for which the flow is absolutely unstable, essentially showing density asymmetry can further destabilise the flow. In figure 10 we replot the neutral curves for $S=2$ (panel a) and $S=20$ (panel b) with $d=0$ from figure 7 and compare the result to the curve with $d=d_{\textit{min}}$ . The results show that varying $d$ to $d_{\textit{min}}$ can have a sizable effect on the position of the neutral curve, even producing a region of co-flow for $S=2$ that is not present when $d=0$ . The discontinuities in the neutral curve occur at the points where the saddles switch dominance, but these discontinuities would not be observed in an experiment by varying the confinement parameter only, because the value $d_{\textit{min}}$ also varies along these curves. For a fixed value of $d$ , the neutral curve would be continuous but could still lie below the $d=0$ result at specific $h$ values.

Figure 11.

Plot of the neutral curve for absolute instability in the $(d/h,r)$ plane for the cases (a) $S=2$ , (b) $S=20$ for symmetrically confined flow with $F^{-2}\neq 0$ . In each case the arrows show results with increasing $F^{-2}$ with $F^{-2}=0.0,0.05,0.1$ in (a) and $F^{-2}=0.0,0.0005,0.002$ in (b).

In figures 11(a) and 11(b) we consider the $S=2$ and $S=20$ cases from figure 9 except with $F^{-2}\neq 0$ . As seen in figure 7, increasing $F^{-2}$ acts to stabilise the absolute instability by raising the neutral curves. In figure 11(a) there is a small region of $d\lt 0$ , around $r\approx 1.6$ , where the neutral curve is lowered, but overall, the effect of including gravity effects appears to act to smooth out the neutral curve to that value of $r$ at large values of $|d/h|$ . Increasing $F^{-2}$ also has the effect of moving the $d_{\textit{min}}$ value towards $d=0$ , and hence, in an experimental flow scenario, this destabilisation effect maybe harder to observe.

Having established that moving the mean density position away from the centre of the shear layer destabilises the flow, we next investigate the effect of asymmetric confinement.

6. Absolute instability of asymmetrically confined shear layers

6.1. Finite asymmetric confinement

In this section our focus is on the effect of asymmetric confinement of the stratified shear layer, so we fix $d=0$ such that the centre of the shear layer and the mean density position coincide. In this section we also only consider the limiting case when $F^{-2}=0$ . We have seen that the effect of $0\lt F^{-2}\ll 1$ in § 5 acted to stabilise the absolute instability by raising the value of $r$ on the neutral curve, while retaining the qualitative features of the $F^{-2}=0$ result. Here we assume that the same qualitative effect of having $F^{-2}\neq 0$ will be observed and, hence, why we only consider the $F^{-2}=0$ case. We will however reconsider the $F^{-2}\neq 0$ effect when we consider a continuous density profile in § 7. In figure 12 we examine a fixed lower plate at $h_2=100$ and a variable upper plate with $h_1=Hh_2$ , where the asymmetry parameter $H$ is varied to trace out the neutral curve in the $(H,r)$ plane. Here we only consider $r\gt 0$ as these cases see the largest effect due to the confined modes. These results show that asymmetric confinement with $H\lt 1$ and $H\gt 1$ both produce a destabilising effect on the flow away from the symmetric case $(H=1)$ . The most dramatic effect occurs for $H\lt 1$ , which is the case when the less dense side of the shear layer is more strongly confined. The minimum values of $r_{\textit{min}}=r(H_{\textit{min}})$ and the corresponding $H_{\textit{min}}$ values are given in table 1. We observe that the value of $H_{\textit{min}}$ reduces with increasing $S$ , while $r_{\textit{min}}$ has a minimum in $5\leq S\leq 100$ . For $H\gt 1$ , we plot the results from the long-wave theory as the dashed lines for $S\geq 5$ , and these show good qualitative agreement with the general results. For $S=2$ in figure 12(a), the long-wave theory is not a good approximation to the general result because as $H\to \infty$ the dominant saddle is the unconfined saddle, not a saddle related to confinement, and so we do not add this result to the panel. This can be observed in the semi-confined results in § 6.2. The reason for this increase in absolute instability and the multiple minima of the neutral curve has been put down to the constructive interference of modes on either side of the shear layer having similar cross-stream wavelengths (Juniper Reference Juniper2006; Healey Reference Healey2009).

Table 1.

Table of values of $r_{\textit{min}}=r(H_{\textit{min}})$ together with the corresponding $H_{\textit{min}}$ value for the neutral curves in figures 12 and 13, with $d=F^{-2}=0$ .

Figure 12.

Plot of the neutral curve for absolute instability in the $(H,r)$ plane for the cases (a) $S=2$ , (b) $S=5$ , (c) $S=20$ and (d) $S=100$ . The solid lines represent the neutral curve for the full problem, while the dashed lines represent the results from the long-wave analysis. Here the flow is asymmetrically confined with $h_2=100$ and $h_1=Hh_2$ with $d=F^{-2}=0$ .

Figure 13.

Plot of the neutral curve for absolute instability in the $(H,r)$ plane for the cases (a) $S=2$ , (b) $S=5$ , (c) $S=20$ , and (d) $S=100$ . The solid lines represent the neutral curve for the full problem, while the dashed lines represent the results from the long-wave analysis. Here the flow is asymmetrically confined with $h_1=100$ and $h_2=Hh_1$ with $d=F^{-2}=0$ . The horizontal dotted lines in (c) and (d) give the corresponding semi-confined result.

In figure 13 we consider asymmetric confinement again, but here $h_1=100$ is fixed and $h_2=Hh_1$ is varied. The results are similar to those seen in figure 12, with $r_{\textit{min}}$ again occurring when the more dense fluid domain is largest ( $H\gt 1$ in this case). This time the long-wave theory is in good agreement for all $S$ values and, for $S=20$ and $100$ , the semi-confined limit result is given by the horizontal dotted line, which clearly has not been reached by the time $H=100$ . In panel (c) for $H\gt 30$ , there is a frequent changing of the dominant saddle as the large $H$ limit is reached. A similar observation was seen for jets and wakes in Juniper (Reference Juniper2006) (see their figure 15, $S=0.316$ and $S=3.16$ ). This saddle switching occurs when there are a large number of confinement saddles with similar values of $\omega _i$ , closely situated next to the imaginary axis. Then as $H$ increases, the dominant saddle rapidly changes to one higher up the axis. It is worth noting that the asymmetrically confined shear layer in the long-wave limit has the same dispersion relation as a symmetrically confined jet/wake flow and, as a consequence, the results presented in figures 12 and 13 are related to figures 11 and 15 in Juniper (Reference Juniper2006).

Figure 14.

Plot of the neutral curve for the asymmetrically confined tanh shear layer in the $(H,r)$ plane close to the value $H_{\textit{min}}$ for $h_1=100$ , $h_2=Hh_1$ , $S=20$ in figure 13 and $d=0,1,2,-1,-2$ , with $F^{-2}=0$ .

Where there is agreement, the long-wave theory gives excellent agreement with the full theory for $H\gt 1$ , except near $H=1$ where the long-wave theory breaks down with $d=0$ as noted earlier. The robustness of the long-wave theory is investigated in § F of the supplementary material. With this robustness in mind, we use the long-wave theory to examine the qualitative behaviour of the minimum shear value $r_{\textit{min}}$ of the problem when we allow $d\neq 0$ in the asymmetrically confined case for $S=20$ in figure 13. Here, figure 14 shows that, for asymmetric confinements with $H\gt 1$ , the $d=0$ result appears to be the most unstable case. For the four non-zero values of $d$ considered, the neutral curve lies above the $d=0$ neutral curve close to $H_{\textit{min}}$ . A numerical verification of $d\in [-5,5]$ confirms that the $d=0$ case is the most unstable scenario.

The conclusion of the results in this section and the previous section are that, for near symmetric cases, $|H-1|\ll 1$ , moving the mean density position such that $d\lt 0$ makes the flow more unstable (i.e. reduces the value of $r$ on the neutral curve), while for larger values of $H$ , the dominant saddle point passes to a confinement saddle and here asymmetric confinement dominates the absolute instability properties of the flow and, thus, the $d=0$ result is the most unstable result. For the $S=20$ case in figure 13(c), this switching from a $d=0$ saddle to a confinement saddle occurs at $H=1.033$ , hence, there is only a small region of parameter space where having $d\neq 0$ is the dominant asymmetrical flow feature, but its effect can still be significant.

6.2. Semi-confined shear layers (infinite asymmetric confinement)

When the flow is confined by a single plate ( $h_1\to \infty$ or $h_2\to \infty$ ), this is a form of asymmetric confinement that we call semi-confinement. In these cases, (2.9) and the jump conditions (2.13) are solved with one boundary condition from (2.10) and one from (2.11). For the homogeneous shear layer $(S=1)$ , Healey (Reference Healey2009) showed that placing the plate in the faster stream of the shear layer enhanced the absolute instability, while having the plate in the slower stream stabilised the absolute instability.

Figure 15.

Plot of the neutral curves of the tanh shear layer for $S=0.9,1,2,5,20,100$ for the semi-confined shear flows for (a) the unconfined above case ( $h_1\to \infty$ ) and (b) the unconfined below case ( $h_2\to \infty$ ) with $d=F^{-2}=0$ . The solid lines represent cases with $r\gt 0$ and the dashed lines have $r\lt 0$ .

Figure 16.

Schematic diagram of the four semi-confined scenarios considered in this paper. Directional arrow means ‘more unstable than’ and when solid denote that the second flow is always more unstable than the first flow, but when dashed that the second flow is usually more unstable than the first flow, except near $S=1$ .

For the stratified case, our findings show a more complicated situation. In figure 15 we consider the neutral curve in the $(h_{1,2},r)$ plane for various density ratios for $h_1\to \infty$ (panel a) and $h_2\to \infty$ (panel b). The interpretation of these results is given in the schematic diagram in figure 16. This figure shows the four different scenarios under consideration, with directional arrows meaning ‘more unstable than’. We find that placing the plate in the faster and lighter stream is the most unstable configuration, while placing the plate in the slower and lighter stream is the least unstable configuration. The solid directional arrows signify that the second case is always more unstable than the first case, whereas the dashed arrow indicates that this second case is usually more unstable, but not for density ratios close to $S=1$ . This can be seen from the results in figure 15, because as $S$ is increased from $S=1$ with $r\lt 0$ in panel (b) and $S$ is increased from $S=1$ with $r\gt 0$ in panel (a), the results in panel (b) are more unstable initially ( $S$ close to $1$ ), but ultimately the results in panel (a) are more unstable as $S$ is increased further.

6.3. Stability hierarchy mechanism

The semi-confinement hierarchy of results presented in figures 15 and 16 can be inferred from a simple model for the tanh shear layer in a similar, but subtly different, argument to that presented in Healey (Reference Healey2009) for the homogeneous case. Here we consider the piecewise-linear shear layer used by Rayleigh (Reference Rayleigh1896), i.e.

(6.1) \begin{equation} f(y)=\left \{\begin{array}{ccl} 1 & \textrm {for } y\gt 1, \\ y & \textrm {for }-1\lt y\lt 1, \\ -1 & \textrm {for } y\lt 1, \end{array}\right . \end{equation}

together with the density profile (2.12). Substituting this into (2.9) and using the jump conditions (2.13), with $F^{-2}=0$ , one can derive the dispersion relation for the unconfined problem using the piecewise-linear approach as in § 4 (Drazin & Reid Reference Drazin and Reid1981; Drazin Reference Drazin2002). The unconfined dispersion relation is given by the cubic polynomial in the phase speed $c-1$ ,

(6.2) \begin{eqnarray} &&(1+S)(c-1)^3+\frac {r}{\alpha }(1-S)\!\left [1-e^{-2\alpha }\right ]\!(c-1)^2-\frac {r^2}{\alpha ^2}(1+S)\!\left [(2\alpha -1)^2-e^{-4\alpha }\right ]\!(c-1)\nonumber \\ &&\,\,\,\,\,\,\,\,\,\,\,+\frac {r^3}{\alpha ^3}(S-1)\left [(2\alpha -1)+e^{-2\alpha }\right ]^2=0, \end{eqnarray}

whose solutions in the long-wave (small $\alpha$ ) limit are

(6.3) \begin{eqnarray} c_1&=&1+r\frac {S-1+\textrm {i}\sqrt {S}}{S+1}+O(\alpha ),\nonumber \\ c_2&=&1+r\frac {S-1-\textrm {i}\sqrt {S}}{S+1}+O(\alpha ),\nonumber \\ c_3&=&1-r\frac {S-1}{S+1}\alpha +O(\alpha ^2). \end{eqnarray}

The key mode here is $c_3$ , which is not present in the homogeneous case. This mode has dispersion at higher order and, thus, we can determine its group velocity as

(6.4) \begin{equation} \frac {{\textrm {d}}\omega }{{\textrm {d}}\alpha }=1-2r\frac {S-1}{S+1}\alpha +O(\alpha ^2). \end{equation}

Absolute instability is associated with a mode with zero group velocity at some point, and it can be seen from (6.4) that, for $r\gt 0$ , the group velocity of this mode is reduced, while for $r\lt 0$ , the group velocity is increased, suggesting that in the unconfined problem, flows with $r\gt 0$ are more absolutely unstable, which is in line with the results presented in figure 15 in the far field.

But what about when confining plates are considered. If we now consider disturbances to the profile (6.1) that satisfy the boundary conditions $v(1+h_1/\epsilon )=0$ and $v(-1-h_2/\epsilon )=0$ with $\alpha =\epsilon \alpha _0$ and $\epsilon \ll 1$ , then we find the same mode now has the form

(6.5) \begin{align} c_3=1-\epsilon r(S-1)\frac {\coth (\alpha _0h_1)\coth (\alpha _0h_2)}{\coth (\alpha _0h_2)S+\coth (\alpha _0h_1)}\alpha _0+O(\epsilon ^2), \end{align}

which is again dispersive with group velocity

(6.6) \begin{eqnarray} \frac {{\textrm {d}}\omega }{{\textrm {d}}\alpha }&=&1-2\epsilon r(S-1)\frac {\coth (\alpha _0h_1)\coth (\alpha _0h_2)}{\coth (\alpha _0h_2)S+\coth (\alpha _0h_1)}\alpha _0\nonumber \\ &&+\epsilon r(S-1)\frac {h_1\textrm {cosech}^2(\alpha _0h_1)\coth (\alpha _0h_2)+h_2\coth (\alpha _0h_1)\textrm {cosech}^2(\alpha _0h_2)}{\coth (\alpha _0h_2)S+\coth (\alpha _0h_1)}\alpha _0^2 \nonumber \\ &&-\epsilon r(S-1)\frac {\coth (\alpha _0h_1)\coth (\alpha _0h_2)(h_1\textrm {cosech}^2(\alpha _0h_1)+Sh_2\textrm {cosech}^2(\alpha _0h_2))}{(\coth (\alpha _0h_2)S+\coth (\alpha _0h_1))^2}\alpha _0^2\nonumber \\ && +O(\epsilon ^2). \end{eqnarray}

Figure 17.

Plot of the group velocity $ {{\textrm {d}}\omega }/{{\textrm {d}}\alpha }$ from (6.6) for the cases $h_2\to \infty$ with $\alpha _0h_1=1$ (solid line) and $h_1\to \infty$ with $\alpha _0h_2=1$ (dashed line) together with $\epsilon |r|\alpha _0=0.1$ .

In figure 17 we plot (6.6) for the cases $h_2\to \infty$ with $\alpha _0h_1=1$ (solid line) and $h_1\to \infty$ with $\alpha _0h_2=1$ (dashed line) together with $\epsilon |r|\alpha _0=0.1$ . What this figure demonstrates is that, for $r\gt 0$ , the $h_2\to \infty$ mode is slowed more than the $h_1\to {\infty }$ result and so is the more likely to reach a point with zero group velocity first, and hence, is more absolutely unstable. For $r\lt 0$ , the opposite is true, with the $h_2\to \infty$ mode group velocity increased less than the $h_1\to \infty$ mode and, hence, is more stable. While this mechanism is consistent with the results in figures 15 and 16 it does not explain all the observations in our results, such as why the strongest destabilisation was not observed in the semi-confined limit.

7. Effect of a continuous density profile

In this final section we consider the absolute instability asymmetric properties for the tanh shear layer, but with a continuous density profile, in order to investigate whether the absolute instability destabilising effects of asymmetry observed for the step density profile (2.12) are robust and are equally observed for a continuous density profile. A smooth density profile is used to model a flow of two miscible fluids that includes an amount of mixing in between them.

The continuous density profile we consider is

(7.1) \begin{equation} \widehat {S}(y)=S+\frac {1}{2}(1-S)\left [1+\tanh \left (\frac {y-d}{\varDelta }\right )\right ], \end{equation}

where $\varDelta$ is a smoothing parameter. When $\varDelta \to 0$ , we recover the density jump (2.12) considered before, while for $\varDelta =O(1)$ , we have a density profile with the same width as the shear layer. The absolute instability properties of the flow are determined using the same approach as outlined in § 5, but instead of the jump conditions (2.13) being applied, continuity conditions on $v$ and $v'$ are applied at $y=d$ . Unlike for the density-jump case, we find that we need to reduce the integration step size when solving (2.9) for cases with $\varDelta \lt 0.5$ in order to make sure the density profile is fully resolved in the calculation.

In this section the limiting case of zero Richardson number ( $F\to \infty$ ) does not make sense, because in this case the continuous nature of the density profile is then neglected. Hence, in this section we do not consider results with $F^{-2}=0$ , and instead compare results against the results of the step profile in § 5 and § 6 for non-zero $0\lt F^{-2}\ll 1$ . With this in mind, we note that the global temporal eigenvalues for the continuous density profile have the same qualitative structure as seen in figure 2(d). In calculations, not shown, we identify two unstable modes, one linked to the KH instability and one linked to an internal mode due to the existence of the density variation. For the continuous profiles, we find that these internal modes ultimately stabilise for short-wave disturbances more readily than for the density-jump case in § 3. As there are similarities between the mode structures for both the continuous density profile and the density-jump profile, we expect qualitative agreement between the absolute instability results for the two flow models.

Figure 18.

Plot of the neutral curves of the tanh shear layer with the continuous density profile (7.1) with symmetric confinement $h_{1}=h_2=h$ in the $(h,r)$ plane for $d=0$ and (a) $(S,F^{-2})=(2,0.01)$ and $(S,F^{-2})=(20,0.001)$ . The arrows indicate an increasing density smoothing parameter $\varDelta =0,\,0.1,\,0.5,\,1$ and $2$ .

In figure 18 we consider the neutral curve for the tanh shear layer with symmetric confinement $(h_1=h_2=h)$ for $S=2$ with $F^{-2}=0.01$ and $S=20$ with $F^{-2}=0.001$ . The results are presented for a range of smoothness parameters from $\varDelta =0$ (the density-jump result) to $\varDelta =2$ . The results show that, for both values of $S$ , wider density profiles lead to more stable absolute instability flow properties for all confinement values considered. In panel (a) ( $S=2$ ) the absolute instability properties are due to both the unconfined and confinement saddles, and the stabilisation effect observed is similar for both saddle types, i.e. the neutral curve is effectively translated up the $r$ axis. In panel (b) ( $S=20$ ) the neutral curve properties are due to confinement saddles only, and in this case we observe that different confinement saddles are stabilised by a different amount, however, the overall qualitative effect is the same for all saddle points, i.e. they are stabilised as $\varDelta$ increases.

Figure 19.

Plot of the neutral curves of the tanh shear layer with the continuous density profile (7.1) with symmetric confinement $h_{1}=h_2=h$ in the $(d/h,r)$ plane for $h=100$ and (a) $(S,F^{-2})=(2,0.05)$ and $(S,F^{-2})=(20,0.0005)$ . The arrows indicate an increasing density smoothing parameter $\varDelta =0,\,0.1,\,0.5,\,1$ and $2$ .

Therefore, as the result of smoothing out the density jump is to stabilise the absolute instability, then one question to ask is: Do we still observe the region of enhanced absolute instability for $d\lt 0$ with $r\gt 0$ with the continuous density profile? In figure 19 we investigate this question for $S=2$ and $S=20$ . The results show that while the overall magnitude of the variations of the neutral curve reduce as $\varDelta$ is increased, there is also a shift in the position of the minimum $r$ value to a smaller value of $d/h$ . This is most clearly seen in the $S=20$ result in figure 19(b), and in this case the continuous density profile does actually lead to a more absolutely unstable flow at smaller values of $d/h$ as $\varDelta$ is increased. However, it is clear from these results that the main effect of the wider, continuous density profile is to smooth out the neutral curve and reduce the absolute instability for all values of $d/h$ .

Figure 20.

Plot of the neutral curves of the tanh shear layer with the continuous density profile (7.1) with $\varDelta =1$ (dashed lines) with asymmetric confinement in the $(H,r)$ plane for $(S,F^{-2})=(2,0.01)$ and (a) $h_1=Hh_2$ with $h_2=100$ , (b) $h_2=Hh_1$ with $h_1=100$ . The solid lines correspond to the discontinuous density profile with $F^{-2}=0.01$ , and the lower dotted lines are the corresponding $F^{-2}=0$ results from figures 12(a) and 13(a).

For the case where $d=0$ with asymmetric confinement ( $h_1=Hh_2$ and $h_2=Hh_1$ etc.), figure 20 confirms the stabilising effect of a continuous density profile for $S=2$ . The variation is maximal for $H\lt 1$ because in this case the variable plate is moving closer to the shear layer. The continuous profile results follow the discontinuous profile results well at the global minimums of the two neutral curves and, hence, the instability of asymmetric confinement on the absolute instability for inhomogeneous shear flows is still observed. Therefore, the results of this section demonstrate that the absolute instability effects observed due to asymmetry for the discontinuous density profile are robust and are qualitatively similar for continuous density profiles. These results also show that the case $F^{-2}\neq 0$ for the asymmetric confinement case can have a significant effect on the values of minimum $r$ on the neutral curve, particularly for the case when the lower plate position is varied in figure 20(b).