2. Modified point-vortex model for unequal vortices

The basic behaviour of two well-separated vortices is similar to that of two point vortices (Trieling et al. Reference Trieling, Dam and van Heijst2010), even in viscous flow (Folz & Nomura Reference Folz and Nomura2014). Kimura & Hasimoto (Reference Kimura and Hasimoto1985) studied the motion of equal point vortices in shear, and Ryzhov, Koshel & Carton (Reference Ryzhov, Koshel and Carton2012) studied the motion of unequal point vortices in an arbitrary deformation flow. Here, the motion of two unequal point vortices in uniform background shear is examined, and the boundary between the major flow regimes is identified.

Two point vortices having circulation ratio $\varLambda \equiv {\varGamma _1}/{\varGamma _2}$ are located initially at $x_1 = -b_0/2$ and $x_2 = b_0/2$ and $y_0 = 0$, where $b_0$ is the initial peak–peak distance, and $x$ and $y$ are the flow and shear directions, respectively. The linear background shear has constant strength $\alpha \equiv {\rm d}U/{{\rm d}y}$. The vortices’ motion is described by a system of equations (Kimura & Hasimoto Reference Kimura and Hasimoto1985)

(2.1)$$\begin{gather} \frac{{\rm d}\kern0.06em x_1}{{\rm d}t} ={-}\frac{\varGamma_2}{2 {\rm \pi}}\,\frac{y_1 - y_2}{b^2} + \alpha y_1, \end{gather}$$

(2.2)$$\begin{gather}\frac{{\rm d}y_1}{{\rm d}t} = \frac{\varGamma_2}{2 {\rm \pi}}\,\frac{x_1 - x_2}{b^2}, \end{gather}$$

(2.3)$$\begin{gather}\frac{{\rm d}\kern0.06em x_2}{{\rm d}t} ={-}\frac{\varGamma_1}{2 {\rm \pi}}\,\frac{y_2 - y_1}{b^2} + \alpha y_2, \end{gather}$$

(2.4)$$\begin{gather}\frac{{\rm d}y_2}{{\rm d}t} = \frac{\varGamma_1}{2 {\rm \pi}}\,\frac{x_2 - x_1}{b^2}, \end{gather}$$

where $(x_i,y_i)$ are the coordinates of vortex $i=1,2$, and the instantaneous peak–peak distance $b = \sqrt {(x_1-x_2)^2 + (y_1-y_2)^2}$ may vary in time. The $y$-coordinate of the centre of rotation of the system is $Y = ({\varGamma _1 y_1 + \varGamma _2 y_2})/({\varGamma _1 + \varGamma _2})$, and is always at $0$.

These equations can be integrated to find the vortex trajectories. In symmetric flow, these trajectories are either closed or open, with a critical stationary case separating these regimes (Kimura & Hasimoto Reference Kimura and Hasimoto1985). Trieling et al. (Reference Trieling, Dam and van Heijst2010) characterized these regimes using a non-dimensional shear strength parameter $\mu _{s} = \alpha b_0^2/\varGamma$ for symmetric vortices. Closed trajectories occur in favourable ($\mu _s < 0$) to weakly adverse ($\mu _{s,sep} > \mu _s > 0$) shear, where $\mu _{s,sep}$ is the critical shear strength associated with the stationary case. In this regime, from their initial position, the vortices follow overlapping elliptical trajectories where an extremum of $b$ occurs when the vortices are aligned vertically, i.e. along the shear direction. This extremum is minimum $b$ in favourable shear, and maximum $b$ in adverse shear, respectively, with circular trajectories occurring in the no-shear case ($\mu _s = 0$). Open trajectories occur in strongly adverse shear ($\mu _s > \mu _{s,sep} > 0$), and the vortices simply move apart continuously; this behaviour is termed separation.

For the case of unequal vortices, a similar parameter is constructed:

(2.5)\begin{equation} \mu = \alpha b_0^2/{\varGamma_2} \end{equation}

based on the stronger vortex, here taken to be $2$. Figure 1 shows example trajectories for a pair having $\varLambda = 0.70$ in various $\mu$. Similar regimes are observed, with the vortices following closed concentric elliptical trajectories for favourable ($\mu < 0$) or weakly adverse ($\mu _{sep} > \mu > 0$) shear, and open trajectories for strongly adverse shear ($\mu > \mu _{sep}$), distinguished by a critical separation shear strength $\mu _{sep}$.

Figure 1. Trajectories of point vortices of pairs (red for vortex 1, blue for vortex 2), having circulation ratio $\varLambda = \varGamma _1/\varGamma _2 = 0.70$, within shear of various strengths $\mu = \alpha b_0^2 /\varGamma _2$. For this case, separation shear strength is $\mu _{sep} = 0.0995$. The filled circles indicate the starting positions for the integrated trajectories (2.1)–(2.4), with (a) $\mu = -0.0992$, (b) $\mu = 0$, (c) $\mu = 0.0992$, and (d) $\mu = 0.128$.

In the symmetric case, an analytical expression for the critical shear strength associated with the stationary case, similar to $\mu _{s,sep}$, was derived by Kimura & Hasimoto (Reference Kimura and Hasimoto1985) using the Hamiltonian of the system (2.1)–(2.4),

(2.6)\begin{equation} H = \frac{-\varGamma_1 \varGamma_2}{4 {\rm \pi}} \ln[((x_1 - x_2)^2 + (y_1 - y_2)^2 )] + \frac{\alpha}{2}(\varGamma_1 y_1^2 + \varGamma_2 y_2^2 ). \end{equation}

When the vortices are unequal, many of their simplifying assumptions cannot be made, but an analytical expression for $\mu _{sep}$ may nevertheless be found in the following manner.

Rearranging (2.6) gives

(2.7)\begin{equation} H - \frac{\alpha}{2}(\varGamma_1 y_1^2 + \varGamma_2 y_2^2 ) = \frac{-\varGamma_1 \varGamma_2}{4 {\rm \pi}} \ln[((x_1 - x_2)^2 + (y_1 - y_2)^2 )]. \end{equation}

Then

(2.8)\begin{equation} \exp\left(H\,\frac{-4 {\rm \pi}}{\varGamma_1 \varGamma_2} - \frac{-4 {\rm \pi}}{\varGamma_1 \varGamma_2}\,\frac{\alpha}{2}(\varGamma_1 y_1^2 + \varGamma_2 y_2^2) \right) = ((x_1 - x_2)^2 + (y_1 - y_2)^2), \end{equation}

and noting that $H$ is constant,

(2.9)\begin{equation} C \exp\left(\frac{4 {\rm \pi}}{\varGamma_1 \varGamma_2}\,\frac{\alpha}{2}(\varGamma_1 y_1^2 + \varGamma_2 y_2^2 ) \right) = ((x_1 - x_2)^2 + (y_1 - y_2)^2) = \xi^2 + \eta^2, \end{equation}

where $\xi \equiv x_1 - x_2$ and $\eta \equiv y_1 - y_2$ (using the nomenclature of Kimura & Hasimoto Reference Kimura and Hasimoto1985), and $C = \xi _0^2$ since $\eta = 0$ in the initial condition.

When the vortices are oriented vertically, the coordinates $y_1$ and $y_2$ are equal to the distances of the weaker and stronger vortices from the centre of rotation, $r_1$ and $r_2$ respectively, since that centre remains fixed in space. Additionally, $\eta = b$ in the vertical orientation. Therefore, at the critical time,

(2.10a,b)\begin{equation} y_1 = r_1 \equiv \frac{1}{1+\varLambda}\,\eta_v,\quad y_2 = r_2 \equiv \frac{\varLambda}{1+\varLambda}\,\eta_v, \end{equation}

where $\eta _v$ is $\eta$ when the vortices are aligned vertically, and $\varLambda \equiv \varGamma _1/\varGamma _2$. So

(2.11) \begin{gather} \xi_0^2 \exp\left(\frac{2 {\rm \pi}\alpha}{\varGamma_1 \varGamma_2}(\varGamma_1 r_1^2 + \varGamma_2 r_2^2) \right) = \eta_v^2,\end{gather}

(2.12) \begin{gather} \xi_0^2 \exp\left(\frac{2 {\rm \pi} \alpha}{\varLambda \varGamma_2} \left(\varLambda\, \frac{\eta_v^2}{(1+\varLambda)^2} + \frac{\varLambda^2 \eta_v^2}{(1+\varLambda)^2} \right) \right) = \eta_v^2, \end{gather}

and ultimately

(2.13)\begin{equation} \xi_0^2 \exp\left(\frac{2 {\rm \pi}\alpha}{\varGamma_2 (1+\varLambda)}\,\eta_v^2\right) = \eta_v^2. \end{equation}

Normalizing (2.13) by the initial peak–peak distance $\xi _0 = b_0$,

(2.14)\begin{equation} \exp\left(\frac{2 {\rm \pi}\alpha b_0^2}{\varGamma_2 (1+\varLambda)} \left(\frac{\eta_v}{b_0} \right)^2\right) = \left(\frac{\eta_v}{b_0}\right)^2. \end{equation}

For this equation to have a finite solution, it must be true that

(2.15)\begin{equation} \frac{2 {\rm \pi}\alpha b_0^2}{\varGamma_2 (1+\varLambda)} \leqslant \frac{1}{\rm e}, \end{equation}

so the critical criterion in terms of $\mu = \alpha b_0^2/\varGamma _2$ is

(2.16)\begin{equation} \mu_{sep} = \frac{(1+\varLambda)}{2 {\rm \pi}{\rm e}}. \end{equation}

Note that for symmetric vortices, this is identical to the criterion found by Trieling et al. (Reference Trieling, Dam and van Heijst2010).

This criterion can be modified to apply for well-separated finite area vortices. Recalling that each vortex's circulation remains constant in this case, $\varGamma _{2} = \varGamma _{2,0} = {\rm \pi}a_{2,0}^2 \omega _{2,0}$ may be substituted into (2.16), where $a_{2,0}$ and $\omega _{2,0}$ are the stronger vortex's initial characteristic radius and peak vorticity, respectively:

(2.17)\begin{equation} \mu_{sep} = (\alpha b_0^2/{{\rm \pi} a_{2,0}^2 \omega_{2,0}})_{sep} = \frac{(1+\varLambda)}{2 {\rm \pi}{\rm e}}. \end{equation}

Then

(2.18)\begin{equation} \left(\frac{-\alpha}{\omega_{2,0}} \right)_{sep} \equiv \zeta_{sep,p} = \frac{-(1+\varLambda)}{2{\rm e}} \left(\frac{a_{2,0}}{b_0} \right)^2, \end{equation}

where $\zeta _{sep,p}$ is a critical shear strength for the separation of finite-area vortices as predicted by this modified point-vortex model (an equivalent expression can be found by normalizing using vortex $1$). It is seen that there is direct dependence between the magnitude of $\zeta _{sep,p}$ and both $\varLambda$ and $(a_{2,0}/b_0)$: a more disparate pair (i.e. having lower $\varLambda$) will separate for weaker adverse shear than a more similar pair, although there is a minimum adverse shear strength required for any separation to occur ($\lim _{\varLambda \to 0} \zeta _{sep,p}(\varLambda ) = (a_{2,0}/b_0)^2/(2\textrm {e}) \neq 0$). Likewise, a given pair (i.e. having a given $\varLambda$) will separate for weaker shear the smaller their initial aspect ratio $a_{2,0}/b_0$ is (i.e. the larger their initial normalized peak–peak distance). Values for $\zeta _{sep,p}$ for various pairs are shown in table 1. In § 4.1, these values are compared to empirical results from simulations of finite-area pairs.

Table 1. Predicted $\zeta _{sep,p}$ from (2.18), with corresponding observed $\zeta _{sep} = -\alpha /\omega _{2,0}$ from numerical simulation of various starting pairs having $a_{2,0}/b_0 = 0.157$ (see § 4.1). The margin of error on the empirical data corresponds to the bracketing values used to determine $\zeta _{sep}$.

3. Set-up and numerical simulations

The initial flow configuration is shown in figure 2: for each case, two finite-area, co-rotating vortices are initially oriented along the flow direction of linear background shear, separated by an initial peak–peak distance $b_0$. Each vortex $i=1,2$ is initially circular with a Gaussian vorticity distribution, and has an initial peak vorticity $\omega _{i,0}$ and characteristic radius $a_{i,0}$, giving an initial circulation ratio

(3.1)\begin{equation} \varLambda_0 \equiv \frac{\varGamma_{1,0}}{\varGamma_{2,0}} = \frac{\omega_{1,0} a_{1,0}^2}{\omega_{2,0} a_{2,0}^2} \leqslant 1. \end{equation}

This study focuses primarily on pairs having either $\omega _{1,0}/\omega _{2,0} < 1$, $a_{1,0}^2/a_{2,0}^2 = 1$ (termed ‘UPEA’ for ‘unequal peaks, equal areas’), or $\omega _{1,0}/\omega _{2,0} = 1$, $a_{1,0}^2/a_{2,0}^2 < 1$ (termed ‘EPUA’ for ‘equal peaks, unequal areas’), as well as symmetric pairs (i.e. those having $\omega _{1,0}/\omega _{2,0} = a_{1,0}^2/a_{2,0}^2 = 1$). A handful of pairs having $\omega _{1,0}/\omega _{2,0} \neq 1$, $a_{1,0}^2/a_{2,0}^2 \neq 1$ (termed ‘UPUA’ for ‘unequal peaks, unequal areas’) are also included in § 4.1. The overall range of circulation ratios considered is $0.6 \leqslant \varLambda _0 \leqslant 1$ (a few additional UPEA cases with $\varLambda _0 = 0.50$ were also performed to aid in the analysis of separations). The pair's initial aspect ratio $a_{2,0}/b_0 = 0.157$ and the circulation Reynolds number $Re_{\varGamma } = \varGamma _{2,0}/\nu = 5000$ are maintained at constant values in order to facilitate comparison with the no-shear case ($\zeta _0 = 0$) examined previously in Folz & Nomura (Reference Folz and Nomura2014, Reference Folz and Nomura2017). This also allows each vortex of the initially well-separated pair to adjust to the combined influence of the other vortex and the shear prior to interacting. This methodology helps to ensure that observed differences between the present results and the no-shear case are attributable primarily to the effects of shear.

Figure 2. Initial flow configuration: two co-rotating vortices, $i=1,2$ (peak vorticity $\omega _{i,0}$, characteristic radius $a_{i,0}$, Gaussian vorticity distribution), whose peak–peak axis is oriented orthogonally to the direction of linear background shear (strength $\alpha = \textrm {d}U/{\textrm {d}y}$). The case shown has $\zeta _0 > 0$.

The numerical simulations are performed using a hybrid finite-difference/pseudo-spectral code with periodic boundary conditions in the flow direction and shear-periodic boundary conditions in the shear direction (see Gerz, Schumann & Elghobashi (Reference Gerz, Schumann and Elghobashi1989) for details of the method). The pair is initially positioned at the centre of a square domain of size $L \times L$ with $2048^2$ grid points, and $b_0 = 1/24 L$. This gives a resolution of approximately $38$ points across the larger core. In comparison with a $1024^2$ grid for cases spanning the considered parameter range, differences in computed vortex quantities were found to be small (e.g. core circulation, an integrated quantity, differed by at most $2\,\%$, and the starting relative straining $(S/\omega )_1$, a pointwise quantity, differed by at most $5\,\%$), the quantitative assessments were similar (e.g. the merging efficiency $\eta$ differed by at most five percentage points, and typically less than two; these quantities are discussed in § 5), and qualitatively all notable phenomena were observed for both resolutions in each case. Domain size independence was also tested using an initial separation distance $b_0 = 1/12 L$ in the favourable case and $b_0 = 1/18 L$ in the adverse case, with similarly small observed differences (e.g. $\eta$ differed by at most six percentage points). The higher $2048^2$ resolution and smaller $b_0 = 1/24 L$ were utilized in all cases in order to minimize spurious variation in core quantities employing the threshold (see § 5), and to maintain maximum fidelity in general (this resolution was also previously found to be sufficient for the no-shear case; see Brandt & Nomura Reference Brandt and Nomura2007, Reference Brandt and Nomura2010).

The relative strength of the constant shear, $\alpha \equiv \textrm {d}U/{\textrm {d}y}$, is characterized in terms of its vorticity $\omega _s$ relative to that of the stronger vortex in the initial condition $\omega _{2,0}$:

(3.2)\begin{equation} \zeta_0 \equiv \frac{\omega_s}{\omega_{2,0}} = \frac{-\alpha}{\omega_{2,0}}. \end{equation}

In order to allow the vortices to adjust to the shear prior to the start of the main convective interaction, the range of $|\zeta _0|$ considered is limited to $|\zeta _0| < \zeta _{adj}$, where $\zeta _{adj}$ is a value chosen such that the viscous increase of $|\zeta |$ ($\sim 1/\omega (t)$; see § 1) would not be sufficient to induce detrainment prior to the end of the adjustment period of the vortices to each other, $t^*_{adj}$. This value ensures that $|\zeta _1(t^*_{adj})|, |\zeta _2(t^*_{adj})| < |\zeta _{cr,s}|$, where the critical shear strength associated with detrainment, $\zeta _{cr,s} \approx -0.063$, was found through simulations of a single vortex in shear (see the supplementary material). An estimate for $t^*_{adj}$ is made using results from Le Dizès & Verga (Reference Le Dizès and Verga2002) for a Gaussian vortex having $Re_{\varGamma } = 2000$ (the lower $Re_{\varGamma }$ result is used to ensure a conservative estimate), which indicate an adjustment period of approximately $t \nu /({\rm \pi} a_{1,0}^2) = 0.05$ using their nomenclature. Using analytical results for a single isolated Gaussian vortex (see § 1) gives a requirement that $|\zeta _{i,0}| \leqslant 0.0387$, which in turn gives a value $|\zeta _{2,0}| \leqslant \zeta _{adj} \equiv 0.0387(\omega _{1,0}/\omega _{2,0})$.

In all cases, temporal results are presented on a convective time scale $t^* = t/T_0$, where $T_0 = (4 {\rm \pi}b_0^2)/(\varGamma _{1,0} + \varGamma _{2,0})$ is the period of revolution of a pair of point vortices having the same $\varLambda _0$ and $\zeta _0 = 0$.

4. General flow behaviour

First, observations of the general flow behaviour are made, and the major interaction regimes identified, for two unequal vortices interacting in the presence of shear (with finite viscosity). Figures 3 and 4 show vorticity contours of example cases demonstrating the major trends with respect to shear strength $\zeta _0$ and vortex circulation ratio $\varLambda _0$. Observations for UPEA and EPUA cases are qualitatively similar, so only UPEA cases are shown. The motion and full flow development of two illustrative cases, $\zeta _0 = 0.0167$, $\varLambda _0 = 0.90$ and $\zeta _0 = -0.0073$, $\varLambda _0 = 0.70$, can be seen in supplementary movies 1 and 2, with useful related information presented in figure 5.

Figure 3. Vorticity contour plots showing time evolution of flows for UPEA pairs having $Re_{\varGamma } = 5000$ and $\varLambda _0 = 0.90$, with varying shear strength $\zeta _0$: (a) $\zeta _0 = 0.1$, (b) $\zeta _0 = 0.0167$, (c) $\zeta _0 = 0.0045$, (d) $\zeta _0 = 0$ (no shear), (e) $\zeta _0 = -0.0045$, ( f) $\zeta _0 = -0.0091$, and (g) $\zeta _0 = -0.1$. For $\varLambda _0 = 0.90$, $\zeta _{sep} = -0.0089$ (see table 1) and $\zeta _{adj} = 0.0348$; cases (b–e) fall within the vortex-dominated regime, $\zeta _{sep} < \zeta _0 < \zeta _{adj}$. For these cases, each column corresponds to an equivalent stage of flow development (see §§ 4 and 5.2): column 1, initial condition; column 2, first quarter-turn; column 3, oriented approximately $45^{\circ }$ above the positive $x$-axis; column 4, start of core detrainment, $t^* = t^*_{start}$; column 5, end of core detrainment and start of mutual entrainment, $t^* = t^*_{det}$; and column 6, end of mutual entrainment, $t^* = t^*_{ent}$. For cases (a, f,g), the column images have been chosen to illustrate the general flow development. The vortices rotate clockwise in favourable shear (counter-clockwise in adverse shear). The contour interval is $10\,\%$, and $t^*$ for each plot is indicated at the lower left. The data in (d) were presented previously in Folz & Nomura (Reference Folz and Nomura2017).

Figure 4. Vorticity contour plots showing time evolution of flows for UPEA pairs having $Re_{\varGamma } = 5000$ and $|\zeta _0| = 0.0073$, with varying $\varLambda _0$: (a) $\varLambda_0 = 0.7$, (b) $\varLambda_0 = 0.9$, (c) $\varLambda_0 = 1.0$ having favourable shear ($\zeta _0 = 0.0073$); and (d) $\varLambda_0 = 1.0$, (e) $\varLambda_0 = 0.9$, ( f) $\varLambda_0 = 0.7$, (g) $\varLambda_0 = 0.5$ having adverse shear ($\zeta _0 = -0.0073$). The columns have meanings equivalent to those in figure 3, but the sixth column of the straining out cases has been omitted since no entrainment occurs. The vortices rotate clockwise in favourable shear (counter-clockwise in adverse shear). The contour interval is $10\,\%$, and $t^*$ for each plot is indicated at the lower left.

Figure 5. Time development of the vortex pair prior to the end of core detrainment for illustrative cases (solid lines): (a,d) trajectories of vortex peaks (red $\times$ indicates vortex 1, blue $+$ indicates vortex 2); (b,e) normalized separation distance $b/b_0$; and (c, f) $\cos ^2(\theta )$ of the angle $\theta$ between the peak–peak axis of the pair and the principal extensional strain eigenvector of the shear, $\boldsymbol {e}_{\alpha }$, which is oriented $45^{\circ }$ from the flow direction. The Reynolds number is $Re_{\varGamma } = 5000$. Filled circles indicate starting positions, a square indicates the time of the first deformation maximum prior to the start of detrainment, and a diamond indicates the second such maximum that occurs in the $\zeta _0 = 0.0167$, $\varLambda _0 = 0.90$ case. These times are taken at the corresponding local maxima in the relative straining of the weaker vortex in each case, $(S/\omega )_1$, as discussed in § 5.1. In (b,e), the dashed line corresponds to the no-shear case having the same $\varLambda _0$.

For comparison, a no-shear merger case is included ($\zeta _0 = 0$, $\varLambda _0 = 0.90$, figure 3d) and reviewed briefly. As discussed in § 1, when shear is not present, the vortices initially (columns 1–4) revolve along concentric circular trajectories (maintaining constant peak–peak distance $b_0$; see dashed lines in figures 5b,e), growing by viscous diffusion and deforming elliptically along the peak–peak axis due to their intensifying mutually induced strain, leading eventually to detrainment of at least one vortex's core fluid in the vicinity of the centre of rotation (column 4). This is the start of the main convective interaction: if both vortex cores detrain (columns 4–5), then a two-way interaction with mutual entrainment (columns 5–6), i.e. merger, occurs (as in figure 3d); otherwise, one vortex detrains and breaks up while the other remains relatively unaffected, i.e. straining out occurs. Due to the presence of viscosity, all like-signed pair interactions without shear are henditions, i.e. they result in a single vortex.

When shear is present ($\zeta _0 \neq 0$), it alters the motion and deformation of a pair (having a given $\varLambda _0$) in a manner and degree determined by its relative strength (figure 3). When shear is favourable ($\zeta _0 > 0$, figures 3a–c) or weakly adverse ($\zeta _{sep} < \zeta _0 < 0$, figure 3e), the interaction is a hendition; otherwise, when shear is strongly adverse ($\zeta _0 < \zeta _{sep} < 0$, figures 3f,g), the vortices undergo separation and move apart continuously. The separation case is discussed in § 4.1.

During shear-influenced henditions (such as those shown in the illustrative cases, i.e. figures 3(b) and 4( f) and supplementary movies 1 and 2), the vortices follow concentric elliptical trajectories in which the peak–peak distance $b$ reaches a minimum (when $\zeta _0 > 0$) or maximum (when $\zeta _0 < 0$) when the peak–peak axis is aligned with the shear direction, akin to the point-vortex case discussed in § 2 (figures 5a,d,b,e). The shear causes periodic amplification of the deformation of the vortices, but, notably, this deformation is greatest when the peak–peak axis is oriented through the first and third quadrants, i.e. approximately along the direction of principal extensional strain of the shear, and not when $b$ is minimal (figures 3a–c,e and 4a–f, columns 1–3, and figure 5). When shear is strongly favourable (figure 3a), it reduces $b$ and amplifies deformation so substantially that it is the predominant cause of the hendition. Otherwise, when shear is weakly favourable or adverse, it is the viscous growth of the vortices and concomitant intensification of mutual strain that eventually cause detrainment to initiate and the main interaction to occur, similar to the no-shear case.

For a given shear strength $\zeta _0$, the circulation ratio of the pair, $\varLambda _0$, influences whether the interaction is a hendition or a separation, and the type of hendition should one occur (figure 4). For a symmetric pair in weakly favourable or adverse shear (figures 4c,d), assuming $\zeta _0 > \zeta _{sep}(\varLambda _0 = 1.0)$ (recall from § 2 that $\zeta _{sep} = \zeta _{sep}(\varLambda _0)$), the interaction is a merger, similar to the no-shear case, but with the vortices experiencing equal shear-induced periodic amplification of deformation. For increasing asymmetry of the pairs (figures 4b–a,e–f), the variation of $b$ increases, the deformation amplification becomes increasingly unequal (greater for the weaker), and during the main convective interaction, the mutual entrainment process becomes increasingly one-sided (e.g. the $\zeta _0 = 0.167$, $\varLambda _0 = 0.90$ illustrative case). When the pair is sufficiently disparate (figures 4a, f), the interaction is entirely one-sided and the weaker vortex is simply destroyed, leaving the stronger one essentially unaffected, i.e. straining out occurs (e.g. the $\zeta _0 = -0.0073$, $\varLambda _0 = 0.70$ illustrative case). In adverse shear, sufficiently small $\varLambda _0$ may result in $\zeta _0 < \zeta _{sep}(\varLambda _0)$ (unless $\zeta _0 > \zeta _{sep}(\varLambda _0 = 0)$), and separation may occur instead, i.e. asymmetry may in a sense ‘cause’ separation in certain circumstances.

Additionally, merger may occur in certain cases with higher $|\zeta _0|$ that for lower or zero $|\zeta _0|$ are straining out. An example is shown in figure 6, an EPUA case having ${\zeta _0 = 0.033}$, $\varLambda _0 = 0.70$ (several lower-$|\zeta _0|$ EPUA $\varLambda _0 = 0.70$ cases are straining out, as is the UPEA case with the same $\zeta _0$ and $\varLambda _0$; this will be seen in § 5.1). In these cases, the influence of the shear enables the second vortex to detrain, and thereby allows for mutual entrainment to occur. It can therefore be said that, in general, the outcome of a given interaction is a function of $\zeta _0$, $\varLambda _0$, and whether the pair is UPEA or EPUA.

Figure 6. Vorticity contours for ${\zeta _0 = 0.033}$, $\varLambda _0 = a_1^2/a_2^2 = 0.70$, EPUA case (contour interval is $10\,\%$, and $Re_{\varGamma } = 5000$). This case is a merger, whereas the equivalent no-shear case is a straining out (discussed further in § 5.1).

In all cases, the resulting vortex or vortices continue(s) to evolve through viscous diffusion akin to a single vortex under the influence of shear (not shown). In adverse shear, this inevitably leads to filamentation (i.e. detrainment) and, ultimately, breakup. See § 1 and references for general discussion of this case.

5. Analysis and characterization of vortex-dominated interactions

When two like-signed vortices interact under the influence of external shear, i.e. when the pair's interaction is vortex-dominated, in viscous flow, hendition always occurs. In these cases, the shear affects primarily the occurrence and timing of core detrainment, which can have a significant effect on the ensuing processes and resulting vortex. As seen in § 4, it can even, in some cases, enable entrainment and merger to occur when, in its absence, they would not. In order to examine the shear's net influence on vortex-dominated henditions, their outcomes must be assessed quantitatively. These results can then be related to basic pair parameters, and, ultimately, a general characterization of vortex-dominated henditions developed.

5.1. Quantitative assessment of interaction outcomes in shear

The outcome of any hendition can be assessed quantitatively in terms of an enhancement factor

(5.1)\begin{equation} \varepsilon \equiv \frac{\varGamma_{end}}{\varGamma_{2,start}} \end{equation}

and a corresponding merging efficiency

(5.2)\begin{equation} \eta \equiv \frac{\varGamma_{end}}{\varGamma_{tot,start}}, \end{equation}

based on the ratio of the circulation of the resulting vortex, $\varGamma _{end}$, at the end of the main convective interaction, $t^*_{end}$, to that of the stronger vortex, $\varGamma _{2,start}$, and of the pair combined, $\varGamma _{tot,start}$, respectively, at the start of the main convective interaction, $t^*_{start}$. (Note that the symbol $\eta$ has a different meaning here than in § 2.) Mergers correspond to $\varepsilon > 1$, while strainings out correspond to $\varepsilon \approx 1$. Note that $\eta$ is not a meaningful quantity for strainings out.

In order to identify the appropriate $\varGamma$, $t^*_{start}$ and $t^*_{end}$ in the course of the flow development, which proceeds continuously due to the viscosity, a method is used that is similar to that developed for the no-shear case in Folz & Nomura (Reference Folz and Nomura2017). The vortex cores are identified via a threshold based on the second invariant of the velocity gradient tensor: $II_t^* = II/II_{peak} = 0.10$, where $II = 1/2 (\omega ^2/2 - S^2)$ is the second invariant at a given location and time, $\omega$ is the local vorticity, $S$ is the local strain rate magnitude, and $II_{peak}$ refers to the value of $II$ at the location of the vorticity peak within a contiguous $II > II_t$ region (see Folz & Nomura (Reference Folz and Nomura2017) for discussion of this choice of $II^*_t$ value).

Aggregate properties of the vortex cores are then computed for the entire flow region meeting the $II > II_t$ criterion, which allows the entire flow development to be monitored continuously, including the transition from two vortices to one:

(5.3)\begin{equation} A_{II} = \displaystyle\int_{II > II_t} {\rm d}A \end{equation}

and

(5.4)\begin{equation} \varGamma_{II} = \displaystyle\int_{II > II_t} \omega \,{\rm d}A, \end{equation}

where $A_{II}$ is the aggregate core area, $\varGamma _{II}$ is the aggregate core circulation, and $\textrm {d}A$ refers to an area element of fluid (similar properties of each individual vortex $i=1,2$ are also computed using a separate $II^*_{t,i}$ based on the peak of each, not shown). The time development of the relative straining of each vortex, $(S/\omega )_i$, is also monitored.

The time development of these quantities is presented for the illustrative cases in figure 7; see the supplementary material for the complete flow development of all the cases shown in figures 3 and 4. The salient features (which are observed in every case considered) are similar to those observed in the no-shear case (Folz & Nomura Reference Folz and Nomura2017). First, there is a period dominated by growth of $A_{II}$ and simultaneous slight decline of $\varGamma _{II}$, corresponding to the initial revolving and viscous growth of the pair. This is followed by diminishing growth (until a local maximum is reached) then decline of $A_{II}$ and simultaneous more rapid decline of $\varGamma _{II}$, until each reaches a local minimum, corresponding to core detrainment. In some cases (such as that in figure 7a), these local minima are followed by rapid rises to local maxima, i.e. spikes, corresponding to mutual entrainment.

Figure 7. Time development of key quantities for illustrative cases, scaled for better visualization:(a) $\zeta _0 = 0.0167$, $\varLambda _0 = 0.90$; (b) $\zeta _0 = -0.0073$, $\varLambda _0 = 0.70$. Left-hand axis: normalized core area $A_{II}/A_{II,0}$ (solid line); and normalized core circulation $\varGamma _{II}/\varGamma _{II,0}$ (dashed line), scaled by a factor of 2 in (a) and 3 in (b). Right-hand axis: relative straining of weaker vortex $(S/\omega )_1$ (through $t^*_{det}$; thick dotted line), with critical $(S/\omega )_{cr}$ also indicated (thin horizontal dotted line). The $+$ signs indicate the start and end times of a supercritical peak of $(S/\omega )_{cr}$ and corresponding troughs in $A_{II}/A_{II,0}$ and $\varGamma _{II}/\varGamma _{II,0}$. The $\unicode{x25CF}$ and $\blacktriangledown$ symbols indicate the start and end of core detrainment, $t^*_{start}$ and $t^*_{det}$, respectively. The $\blacktriangle$ symbols indicate the end of mutual entrainment, $t^*_{ent}$ (occurs in (a) only). See text for definitions of the times.

The most significant effect of shear is the introduction of additional local minima in the $\varGamma _{II}$ development (with corresponding variations in the $A_{II}$ development, e.g. beginning at $t^* = 0.50$ in figure 7(a), and at $t^* = 1.1$ in figure 7(b)). It is seen that these ‘troughs’ correspond to local maxima, or ‘bumps’, in the time development of $(S/\omega )_1$ (which exhibits net growth due to viscous diffusion intensifying $(S/\omega )_1$), which in turn correspond to the periodic amplified deformation, associated with the vortices’ orientation, observed in § 4. (The squares and diamonds in figure 5 correspond to the peaks of bumps in $(S/\omega )_1$ in figure 7; corresponding bumps also occur in $(S/\omega )_2$, not shown here.) Notably, $(S/\omega )_i$ of either vortex may temporarily exceed the critical $(S/\omega )_{cr} = 0.135$ (associated with core detrainment) one or more times, with negligible evident detrainment, before $(S/\omega )_1$ (and in some cases, $(S/\omega )_2$) ultimately surpasses it terminally as $A_{II}$ growth diminishes and $\varGamma _{II}$ decline accelerates.

Due to these nonlinearities, $t^*_{start}$ is taken to be the earlier time at which one vortex achieves – and thereafter maintains – $(S/\omega )_i > (S/\omega )_{cr} = 0.135$ through to the end of detrainment. (In the no-shear case, $t^*_{start}$ had been identified as the time of deviation of $A_{II}$ from linear growth – see Folz & Nomura (Reference Folz and Nomura2017); $t^*_{start}$ as identified in the present manner produces similar results in those cases, not shown.) This time is indicated for the illustrative cases in figure 7 (in figures 3(b–e) and 4(a–f), column 4 corresponds to $t^*_{start}$). Maintaining the critical $(S/\omega)_{cr} = 0.135$ value has been seen to correspond to detrainment both here and in the no-shear case, and is also consistent with the value observed in simulations of a single Gaussian vortex in adverse shear (e.g. Mariotti et al. Reference Mariotti, Legras and Dritschel1994), suggesting that this is a general critical value for Gaussian vortices subject to external strain. In this study, the first detraining vortex is always vortex $1$. This identification of $t^*_{start}$ with the maintaining of $(S/\omega )_1 > (S/\omega )_{cr}$ is consistent with the observation in Trieling et al. (Reference Trieling, Dam and van Heijst2010) that symmetric pairs in shear always merge when their peak–peak distance remains within the critical merging criterion for symmetric pairs without shear; it is noted, though, that they do observe merger to occur in cases (typically having favourable shear) in which the vortices only temporarily surpass the critical criterion. The choice of $t^*_{start}$ here therefore likely constitutes a conservative estimate for the start of the main convective interaction.

The end of the interaction, $t^*_{end}$, is taken to be the time of the first peak (i.e. spike) immediately following the minimum if there is one (i.e. the end of entrainment, $t^*_{ent}$), or the time of the local minimum otherwise (i.e. the end of detrainment, $t^*_{det}$). These times are indicated for the illustrative cases in figure 7 (and in figures 3(b–e) and 4(a–f), column 5 corresponds to $t^*_{det}$, and column 6 to $t^*_{ent}$). In all cases considered, $t^*_{end}$ corresponds to the existence of only a single vortical structure meeting the $II > II_t$ criterion. These are identical to the criteria used in the no-shear case (Folz & Nomura Reference Folz and Nomura2017).

5.2. The influence of shear on the timing and duration of detrainment

The shear has a significant influence on the timing of detrainment, as seen in figures 8(a,b), which show $t^*_{start}$ and $\varDelta _d t^* = t^*_{det} - t^*_{start}$, the duration of the detrainment-dominated portion of the main convective interaction, respectively. The large degree of variation of $t^*_{start}$ with $\zeta _0$ reflects the influence of shear on $b$, and thereby the overall growth of $(S/\omega )_i$ leading to detrainment: lowering and increasing $t^*_{start}$, i.e. accelerating or delaying the onset of detrainment, derives from shear reducing or increasing $b$ from $b_0$, respectively (here associated with favourable and adverse shear; this is discussed further below). The variation of $t^*_{start}$ with $\varLambda _0$ (and whether the pair is UPEA or EPUA) is consistent with that observed in the no-shear case: $t^*_{start}$ decreases for increasing pair disparity (lower $\varLambda _0$; Folz & Nomura Reference Folz and Nomura2017). It is also noted that as $\zeta _0$ increases from $0$, the variation of $t^*_{start}$ with $\varLambda _0$ is reduced, consistent with the (increasingly strong, favourable) shear playing an increasingly significant role in the initiation of detrainment. The observed variation of $\varDelta _d t^*$ likewise reflects the influence of shear on $b$, with reduction and increase likewise generally shortening and prolonging the duration of detrainment, although the effect here is not strictly monotonic. This is attributed to the complexity of the detrainment process, as well as the method of assigning $t^*_{start}$ and $t^*_{end}$. It is also seen that $\varDelta _d t^*$ increases with decreasing $\varLambda _0$ (consistent with Folz & Nomura Reference Folz and Nomura2017), and that in some cases this can be quite significant, especially for $\zeta _0 < 0$.

Figure 8. For vortex-dominated henditions: (a) time of interaction start, $t^*_{start}$, and (b) the duration of detrainment, $\varDelta _d t^* \equiv t^*_{det} - t^*_{start}$, as functions of $\zeta _0$ and $\varLambda _0$ (both UPEA and EPUA). The meanings of symbol shapes and colours are indicated in (a,b), respectively; a black outline indicates UPEA cases (and symmetric), and a white outline indicates EPUA cases ($Re_{\varGamma } = 5000$).

It should be understood that the net influence of the shear on $\varDelta _d t^*$ derives not only from the time variation of $b$, but also from directional effects that inhibit core detrainment. In order to examine this influence, the angle $\phi _i$ is computed for each vortex $i=1,2$, where $\phi _i$ is the angle between the peak–peak axis and $\boldsymbol {e}_{pk,i}$, the unit vector corresponding to the direction of principal extensional strain evaluated at the vorticity peak. The angle $\phi _i$ serves as an instantaneous indicator of the vortex's response to the net directional influence of the strain rate fields induced by the shear and the other vortex as their relative prevalence and direction vary in time. The time variations of $\cos ^2(\phi _i)$ for each vortex for no-shear cases having $\varLambda _0 = 0.90$ and $\varLambda _0 = 0.70$ are presented in figures 9(a,b), and it is seen that $\cos ^2(\phi _i) \approx 0.5$ is maintained (i.e. $\phi _i \approx 45^{\circ }$, after the initial adjustment period) until detrainment begins, after which $\cos ^2(\phi _i)$ decreases (i.e. $\phi _i$ increases); this occurs for both vortices in the $\varLambda _0 = 0.90$ case – a merger – and only for the weaker vortex in the $\varLambda _0 = 0.70$ case – a straining out. It can therefore be said that $\cos ^2(\phi _i) < 0.5$ is associated with core detrainment. When shear is present (shown for the illustrative cases in figures 9c,d), its principal extensional strain rate remains fixed in the Eulerian frame (oriented $45^{\circ }$ from the background flow direction), which initially causes $\cos ^2(\phi _i)$ of both vortices to oscillate about $0.5$ as they revolve; this occurs such that the bumps in $(S/\omega )_i$ correspond to maximum $\cos ^2(\phi _i)$, i.e. when the directional effects are unfavourable for detrainment. This may explain why little evident detrainment occurs even when $(S/\omega )_i > (S/\omega )_{cr}$ during a bump. After $t^*_{start}$, one vortex (at least) maintains $(S/\omega )_i > (S/\omega )_{cr}$ and $\cos ^2(\phi _i) < 0.5$ simultaneously, thereby undergoing detrainment and causing hendition to occur. These observations suggest that in order for core detrainment to occur, the vortex must maintain both sufficient relative straining ($(S/\omega )_i > (S/\omega )_{cr}$) and conducive directionality ($\cos ^2(\phi _i) < 0.5$) of its straining response.

Figure 9. Time development of $\cos ^2(\phi _i)$, where $\phi _i$ is the angle between the peak–peak axis and the principal extensional strain eigenvector at each peak, $\boldsymbol {e}_{pk,i}$ (left-hand axis; solid red line shows vortex 1, dashed blue line shows vortex 2), for no-shear cases having (a) $\varLambda _0 = 0.90$ and (b) $\varLambda _0 = 0.70$; and the illustrative shear UPEA cases having (c) $\zeta _0 = 0.0167$, $\varLambda _0 = 0.90$ and (d) $\zeta _0 = -0.0073$, $\varLambda _0 = 0.70$. In all cases, $Re_{\varGamma } = 5000$. In (c,d), the time development of $b/b_0$ is also included for reference (right-hand axis; solid black line), and $\blacksquare$ (and $\blacklozenge$) indicate times of first (and, in (c), second) local $S/\omega$ peak (see figures 5 and 7).

It is critical to note that accelerating/delaying $t^*_{start}$ and shortening/prolonging $\varDelta _d t^*$ are not necessarily associated with ‘favourable’ or ‘adverse’ shear. If the vortices are initially oriented along the shear direction, then the favourable shear delays/prolongs while adverse shear accelerates/shortens (since, in this orientation, the vortices are initially located at minimum $b$ in the favourable case and maximum $b$ in the adverse). This can be seen in table 2, which shows $t^*_{start}$ and $\varDelta _d t^*$ for equivalent cases having each initial orientation (i.e. maintaining all pair parameters described in § 3). A full exploration of the relationship between the initial orientation, the flow development, and outcomes is beyond the scope of this study, but it can be concluded that the observed effects of shear on timing derive primarily from the relative motion of the vortices that it engenders, and not, strictly speaking, the shear's relative sense.

Table 2. Time of start of core detrainment, $t^*_{start}$, and duration of detrainment, $\varDelta _d t^* \equiv t^*_{det} - t^*_{start}$, for equivalent UPEA cases initially oriented horizontally (Horiz) and vertically (Vert) ($Re_{\varGamma } = 5000$).

5.3. Results for $\varepsilon$ and $\eta$

It has been seen that shear affects the timing and duration of core detrainment, and it is known that the outcome of an asymmetric pair interaction derives from the relative timing of detrainment and destruction of the vortices (Brandt & Nomura Reference Brandt and Nomura2010). The influence of the shear on interaction outcomes is therefore most significant when the outcome of the case is particularly sensitive to changes in this timing. To examine this, and the influence of shear on interaction outcomes overall, $\varepsilon$ and $\eta$ are computed for every case using (5.1) and (5.2).

Figure 10 shows each hendition case, with $\varepsilon$ indicated via colour, for both UPEA and EPUA cases (similar trends are observed for $\eta$). Linear interpolation has also been employed to estimate $\varepsilon$ values between the simulated cases, in order to better visualize the overall trends. In the broadest sense, the variation of $\varepsilon$ is similar to that of the no-shear case: $\varepsilon$ is maximal ($\varepsilon \approx 2$) for symmetric pairs, and minimal ($\varepsilon \approx 1$) for highly disparate ones (approximately $\varLambda _0 < 0.70$ for UPEA, and $\varLambda _0 < 0.60$ for EPUA). The $\varLambda _{0,cr}$ value at which this minimum $\varepsilon$ is reached is seen to vary with $\zeta _0$, particularly in the transitional range (approximately $0.80 \geqslant \varLambda _0 \geqslant 0.70$ for UPEA, and $0.80 \geqslant \varLambda _0 \geqslant 0.60$ for EPUA), being generally lower for higher $|\zeta _0|$ (with the notable exception of the $\zeta _0 \leqslant 0$, $0.75 \geqslant \varLambda _0 \geqslant 0.70$ subregion). More generally, the variation of $\varepsilon$ with $\zeta _0$ is not always monotonic, for $\varLambda _0 > \varLambda _{0,cr}$.

Figure 10. Interaction outcomes for vortex-dominated henditions for (a) UPEA and (b) EPUA cases. Symbols indicate the outcomes of the cases included in figure 8, categorized: $\Box$ indicates merger; $+$ indicates straining out (see § 5.1). Colours indicate enhancement factor $\varepsilon$: values for each denoted case are exact, while those in between are produced by linear interpolation (both plots use same colour map, indicated in (b)). The solid black line indicates $\zeta _0 = 0$, the dashed line indicates $\zeta _{sep,p}$ (see § 2), and the dash-dotted line indicates $\zeta _{adj}$ (see §§ 3 and 4). The data for the no-shear ($\zeta _0 = 0$) cases were presented previously in Folz & Nomura (Reference Folz and Nomura2017). Note that $\varepsilon$ inherently declines with decreasing $\varLambda _0$, since the theoretical maximum is $\varepsilon _{max} = 1 + \varLambda _0$. In all cases, $Re_{\varGamma } = 5000$.

Also indicated in figure 10, via the symbols, is the occurrence of mutual entrainment (i.e. merger) or not (i.e. straining out) in each case (as ascertained from the presence of one or more spikes post-$t^*_{det}$ in the $\varGamma _{II}$ time development; see § 5.1). It is seen that the presence of shear can engender entrainment in some cases where it does not occur in the equivalent no-shear case, and that this typically occurs for more disparate pairs when shear is stronger (again there are notable exceptions, particularly in the EPUA $\zeta _0 \leqslant 0$, $0.75 \geqslant \varLambda _0 \geqslant 0.70$ subregion). This promotion of merger is attributed to the shear – of either sense – altering the rate of increase of $(S/\omega )_i$ of each vortex, making detrainment of the second vortex more likely prior to the destruction of the first.

The variations of $\varepsilon$ and $\eta$ with $\zeta _0$ and $\varLambda _0$ for all hendition cases are presented in figure 11. It is seen that, within the transition region, cases with $\zeta _0 \neq 0$ typically have higher $\varepsilon$ and $\eta$ than the no-shear case with the same $\varLambda _0$, and that increasingly favourable shear generally (though not universally) results in greater enhancement and more efficient merger for EPUA cases. (Note that the variation for the symmetric case results largely from the use of $II_t$ in conjunction with its uniquely reciprocal mutual strain.) The handful of cases with lower $\varepsilon$ and $\eta$ than in the no-shear case have $\zeta _0 < 0$: the weaker $\zeta _0 = -0.0045$ cases always produce $\varepsilon < \varepsilon (\zeta _0 = 0)$ for a given $\varLambda _0$, while the stronger $\zeta _0 = -0.0073$ begins to produce $\varepsilon > \varepsilon (\zeta _0 = 0)$ for more disparate pairs ($\varLambda _0 \leqslant 0.75$ for EPUA) only until the straining out regime is reached, i.e. stronger adverse shear better enables entrainment to occur. Similar trends are observed in the $\varepsilon {-}\varLambda$ and $\eta{-}\varLambda$ variations, where $\varLambda$ is evaluated at $t^*_{start}$ (not shown). It is noted that the initially vertical cases considered in § 5.2 produce $\varepsilon$ and $\eta$ similar to those for their initially horizontal counterparts (table 2; not otherwise included in the present results or discussion).

Figure 11. For vortex-dominated henditions: (a) $\varepsilon$ and (b) $\eta$, as functions of $\varLambda _0$. See figure 8 for meanings of symbol shapes, colours and outlines. In all cases, $Re_{\varGamma } = 5000$.

An example of shear of either sense promoting merger is presented in figures 12(a,b,c), which show the time development of $(S/\omega )_i$ for EPUA $\varLambda _0 = 0.70$ cases having ${\zeta _0 = 0.033}$, no shear and $\zeta _0 = -0.0073$, respectively. In these cases, entrainment does not occur for the no-shear case, but it does occur for ${\zeta _0 = 0.033}$ ($\varepsilon = 1.46$; vorticity contours for this case are shown in figure 6), and to a small degree for $\zeta = -0.0073$ ($\varepsilon = 1.11$; vorticity contours not shown). It is seen that the influence of the shear causes $(S/\omega )_2$ to surpass $(S/\omega )_{cr}$ in both $\zeta _0 \neq 0$ cases, allowing detrainment of the second vortex to occur. For the ${\zeta _0 = 0.033}$ case, this is done primarily by accelerating the rate of increase of $(S/\omega )_2$ (i.e. reducing $t^*_{start}$); for the $\zeta = -0.0073$ case, this is done primarily by prolonging the detrainment of vortex $1$ (i.e. increasing $\varDelta _d t^*$), allowing $(S/\omega )_2$ to increase sufficiently to surpass $(S/\omega )_{cr}$ prior to the end of the interaction.

Figure 12. Time development of $(S/\omega )_1$ and $(S/\omega )_2$ (red solid and blue dashed lines, respectively; right-hand axis), along with $b/b_0$ (thick dotted line; left-hand axis), for EPUA cases having $Re_{\varGamma } = 5000$ and $\varLambda _0 = 0.70$, with (a) $\zeta _0 = 0.0167$, (b) $\zeta _0 = 0$ (no shear), and (c) $\zeta _0 = -0.0073$. In all plots, the horizontal thin dotted line indicates $(S/\omega )_{cr} = 0.135$. The $\zeta _0 \neq 0$ cases are mergers ($\varepsilon > 1$), whereas the no-shear case is a straining out ($\varepsilon \approx 1$).

5.3.1. Mutuality and the fundamental characterization of henditions in shear

Fundamentally, the outcome of the interaction of two like-signed vortices derives from the degree of mutuality of the interaction. In the no-shear case, Folz & Nomura (Reference Folz and Nomura2017) introduced a mutuality parameter $MP = (S/\omega )_1/(S/\omega )_2$, which compares the relative straining of each vortex at $t^*_{start}$, and thereby captures the degree of mutuality of the interaction. When the relative straining of both vortices is similar at $t^*_{start}$ (approximately $1 \leqslant MP < 1.8$ for $Re_{\varGamma } = 5000$), the second vortex can begin to detrain before the first is destroyed, enabling mutual entrainment (i.e. merger occurs); when the disparity is greater (approximately $MP > 1.8$), it cannot, and the first detraining vortex is simply destroyed (i.e. straining out occurs). In the vortex-dominated regime, $MP$ is computed for the shear cases in the same manner.

Figure 13 shows the variations of $\varepsilon$ and $\eta$ with $MP$. A generally monotonic relationship is observed between $\varepsilon$ and $MP$ (likewise $\eta$ and $MP$), with $\varepsilon \approx 2$ (and $\eta \approx 1$) near $MP = 1$, then generally declining as $MP$ increases until $MP \approx 2$, at which point $\varepsilon \approx 1$ is reached and thereafter maintained (recall that $\eta$ ceases to be a meaningful quantity for strainings out). However, significantly more scatter is observed in the shear case (as compared with figure 9 in Folz & Nomura Reference Folz and Nomura2017). This is attributed to $MP$ being a pointwise quantity, and therefore sensitive to significant and rapid variations as the vortices revolve.

Figure 13. For vortex-dominated henditions: (a) $\varepsilon$ and (b) $\eta$ as functions of $MP = (S/\omega )_1/(S/\omega )_2$. See figure 8 for meanings of symbol shapes, colours and outlines. The dotted line indicates $MP = 1$. In all cases, $Re_{\varGamma } = 5000$.

However, $MP$ is related closely to the vortex enstrophy ratio, an integrated quantity less susceptible to fluctuations:

(5.5)\begin{equation} \frac{Z_2}{Z_1} = \frac{\displaystyle\int\nolimits_{II > II_{t,2}} \omega_2^2 \,{\rm d}A_2}{ \displaystyle\int\nolimits_{II >II_{t,1}} \omega_1^2 \,{\rm d}A_1}, \end{equation}

where the integral is evaluated for each vortex at $t^*_{start}$. This quantity was also seen to effectively characterize the variation of $\varepsilon$ and $\eta$ in the no-shear case (Folz & Nomura Reference Folz and Nomura2017).

Figure 14 shows the variations of $\varepsilon$ and $\eta$ with $Z_2/Z_1$. It is seen that the scatter is greatly reduced (even for high $|\zeta _0|$), such that the variation is essentially monotonic outside of a transition region, reached at $1.65 \pm 0.01$ and extending to $1.91 \pm 0.02$ (where the margins of error for each are found from the extremal midpoint between a merger and a straining out case). In this region, cases having similar $Z_2/Z_1$ but different $\zeta _0$, $\varLambda _0$ and/or UPEA/EPUA status can produce significantly different $\varepsilon$ and $\eta$, with higher $|\zeta _0|$ generally (though not universally) corresponding to higher $\varepsilon$ and $\eta$. The consistently lesser enhancement and less efficient merging when weak adverse shear is present ($\zeta _0 = -0.0045$) are attributed to greater dissipation of detrained fluid due to the increase of $b$, while the greater enhancement and more efficient merging produced by stronger adverse shear ($\zeta _0 = -0.0073$) are attributed to the interaction being sufficiently prolonged to enable entrainment. The lower bound of the transition region is comparable to the critical $(Z_2/Z_1)_{cr} \approx 1.63 \pm 0.03$ for straining out observed in the no-shear case, and merger cases having greater $Z_2/Z_1$ correspond to higher $|\zeta _0|$.

Figure 14. For vortex-dominated henditions: (a) $\varepsilon$ and (b) $\eta$ as functions of $Z_2/Z_1$. See figure 8 for meanings of symbol shapes, colours and outlines. The dotted line indicates $Z_2/Z_1 = 1$. In all cases, $Re_{\varGamma } = 5000$.

As such, it can be concluded that favourable and sufficiently strong adverse shear (approximately $\zeta _0 < -0.0045$) promote enhancement and merger for interactions that are moderately disparate (i.e. having approximately $1.65 < Z_2/Z_1 < 1.9$). Efficient merger occurs in interactions with a high degree of mutuality, regardless of shear strength (typically $\eta > 0.85$ for $Z_2/Z_1 < 1.65$), while more disparate ones always result in straining out of the weaker vortex ($\varepsilon \approx 1$ for $Z_2/Z_1 > 1.9$).

The fact that the variations of $\varepsilon$ and $\eta$ with $Z_2/Z_1$ are generally monotonic is taken as evidence that the choice of $t^*_{start}$ ($(S/\omega )_1 > (S/\omega )_{cr} = 0.135$ terminally) does effectively characterize the start of convective detrainment (see § 5.2). These trends are also observed when only the relative vorticity is considered (not shown), indicating that they reflect the shear's effect on the physical mechanisms of vortex interaction and not spurious consequences of the use of $II_t$.

6. Summary and discussion

This study considers the interaction of a pair of unequal vortices in background shear with finite viscosity. In these cases, the flow development is determined by the relative significance of the vortices’ mutual influence and that of the shear. Sufficiently adverse shear causes the vortices to separate; the critical adverse shear strength for separation, $\zeta _{sep}$, varies with the pair's circulation ratio $\varLambda _0$ (and aspect ratio), and its empirical values are well-predicted by point-vortex analysis. Otherwise, the interaction between the vortices is a hendition (§ 1.1), resulting in a single vortex.

In henditions occurring in background shear, the flow development is essentially governed by three constituent external influences occurring simultaneously, whose relative significance varies in time: the shear causes the peak–peak distance $b$ between the vortices to vary as they revolve; the vortices influence each other through their mutually induced strain, which depends on $b$; and the strain induced by the constant shear acts directly on each vortex. When the shear is strongly favourable, it causes such significant reduction of $b$ and amplified deformation of the vortices that it is the principal cause of hendition; otherwise, when the shear is weakly favourable or weakly adverse, the flow is vortex-dominated.

In vortex-dominated henditions in shear, viscous diffusion causes the vortices’ mutually induced strain to become predominant, which enables sufficient persistence of straining for one vortex to begin detraining core fluid: the relative straining remains sufficiently strong in magnitude ($(S/\omega )_i > (S/\omega )_{cr} = 0.135$ for detraining vortex $i$), and the directionality of the vortex's response to the external strain field maintains a conducive relative orientation ($\cos ^2(\phi _i) < 0.5$) for sustained detrainment despite the continuing influence of shear on $b$ and on each vortex directly. The flow development then proceeds similarly to the no-shear case: if the second vortex is induced to detrain before the first is destroyed, then a two-way interaction leads to a mutual entrainment process that produces an enhanced resulting vortex i.e. merger occurs; otherwise, the interaction is essentially one-way and the detraining vortex is broken up, leaving the other largely unaffected, i.e. straining out occurs.

The post-interaction vortex is assessed quantitatively in terms of an enhancement factor $\varepsilon$ and a merging efficiency $\eta$, using a method adapted from one utilized in the no-shear case (Folz & Nomura Reference Folz and Nomura2017). It is found that all vortex-dominated hendition outcomes across the parameter range considered are effectively characterized by the pair's core enstrophy ratio at the start of detrainment, $Z_2/Z_1$, which encapsulates the mutuality of the interaction similarly to $MP = (S/\omega )_1/(S/\omega )_2$ (utilized previously in the no-shear case) but is less sensitive to the time variation caused by the shear. For $Z_2/Z_1$ near unity, merger essentially conserves circulation, while mergers between more disparate vortices become less efficient until straining out occurs. Within the transition region, approximately $1.65 < Z_2/Z_1 < 1.9$, weak adverse shear reduces enhancement; otherwise, the presence of shear of either sense generally promotes enhancement and merger relative to the no-shear case (in which straining out occurs for $Z_2/Z_1 > (Z_2/Z_1)_{cr} \approx 1.63$ for $Re_{\varGamma } = 5000$, similar to the value at which it first occurs with shear present). In favourable shear, this results from more rapid increase of $(S/\omega )_2$ due to reduction of $b$, while in adverse shear, this results from prolonging the interaction due to a combination of increase of $b$ and periodic detrainment-inhibiting orientation effects allowing $(S/\omega )_2$ to reach $(S/\omega )_{cr}$ (with simultaneous $\cos ^2(\phi _i) < 0.5$) before the first vortex is destroyed.

Additional study must examine vortex pair parameters beyond those considered here. In particular, it is possible that the boundary values of the $Z_2/Z_1$ transition region may vary with parameters such as the Reynolds number or initial aspect ratio, either of which would be expected to affect the rate of increase of $(S/\omega )_i$ and therefore promote or inhibit detrainment of the second vortex. It has also been seen that the initial orientation of the vortices relative to the shear is significant: certain effects can be associated with the opposite sense of shear in the orthogonal initial orientation (e.g. when the vortices are initially oriented along the shear direction, favourable shear increases $b$ and prolongs the interaction, while adverse shear reduces and shortens). Moreover, it would be desirable to consider a time-varying background flow, which might better reflect that experienced by a vortex pair in turbulence (the concept of persistence of straining leading to detrainment may be a particularly significant concept in such flows). It is hoped that the current study may provide a step towards a more all-encompassing characterization of vortex interactions in background flow that incorporates these additional parameters, and others (the influence of an opposite-signed vortex would be another priority for consideration). Such studies remain for future work.