3.1. A case study at $(\textit {Re},\textit {Fr})=(3200,0.1875)$

We use one of the new runs at $(\textit {Re},\textit {Fr})=(3200,0.1875)$ as a case study to demonstrate how a freely evolving columnar TG vortex array can reach the LAST regime through a shear-driven transition to turbulence. Figure 2 visualises the flow at $t=52$ , by which time the primary zigzag instability has saturated and secondary structures have emerged. This case sits near the high- $\textit {Re}$ , low- $\textit {Fr}$ corner of the parameter space explored in table 1, with a peak buoyancy Reynolds number $\textit {Re}_b^\star \approx 2$ and the corresponding horizontal Froude number $\textit {Fr}_h^\star \approx 0.02$ . Panel (a) shows a three-dimensional rendering of the vertical vorticity $\omega _z$ (red/blue volume) overlaid with iso-surfaces of the streamwise vorticity $\omega _x$ (tan/cyan). The $\omega _x$ structures are organised into thin, horizontally elongated sheets that span the gaps between adjacent vortex columns and stack vertically within the computational domain. Panel (b), a projection along $+x$ onto the ( $y$ $,z$ ) plane, shows $\omega _x$ concentrated in quasi-horizontal sheets of alternating sign, interleaved with the $\omega _z$ signature of the (now deformed) columnar vortices. The red frame in panel (a) marks the $x=\pi$ transect analysed in figure 3.

Figure 2.

Instantaneous vorticity field at $t=52$ for $(\textit {Re},\textit {Fr})=(3200,0.1875)$ . (a) Three-dimensional view: volume rendering of the vertical vorticity $\omega _z$ (red/blue) overlaid with iso-surfaces of the streamwise vorticity $\omega _x$ (tan/cyan). The red frame marks the $x=\pi$ transect visualised in figure 3. (b) Projection of the same field along the $+x$ axis onto the ( $y$ , $z$ ) plane.

Figure 3.

Time evolution of the streamwise vorticity $\omega _x$ on the ( $y$ , $z$ ) plane at $x=\pi$ for $(\textit {Re},\textit {Fr})=(3200,0.1875)$ (red frame in figure 2 a). Snapshots in (a), at $t=48,52,54\, \text{and}\,56$ , show the development from faint quasi-horizontal striations to chevron-like roll-up. Snapshots in (b), at $t=58,60,62\, \text{and}\,66$ , show Kelvin–Helmholtz-like roll-up extending across the $y$ -direction, followed by relaminarisation.

Figure 4.

Horizontal-velocity reversal and vertical shear for $(\textit {Re},\textit {Fr})=(3200,0.1875)$ at $t=48$ . (a) Vertical vorticity $\omega _z$ on the $x=\pi /2$ plane. (b, c) Horizontal streamlines sampled at $z=2\pi /5$ and $z=\pi /5$ , corresponding to the red and blue dotted lines in (a); the grey band in each panel marks the region near $y=\pi$ where the sign of $v$ reverses between the two heights. (d)–(f) Squared vertical shear $(\partial _z u)^2+(\partial _z v)^2$ , $(\partial _z u)^2$ and $(\partial _z v)^2$ , sampled at the green dotted line in (a), midway between the two streamline levels.

The time evolution of the $\omega _x$ structures on the $x=\pi$ plane is shown in figure 3 for $t\in [48,66]$ . At $t=48$ , $\omega _x$ appears as a faint pattern of quasi-horizontal striations with vertical spacing consistent with the saturating zigzag mode. By $t=52$ , a localised patch of stronger $\omega _x$ emerges around $y=\pi$ , with clearly alternating sign in $z$ . Between $t=54$ and $t=56$ , this patch concentrates around $y=\pi$ and develops into a chevron-like array of tilted sheets that roll up into a train of billow-like structures reminiscent of Kelvin–Helmholtz instability. From $t=58$ onward the roll-up spreads to fill the entire $y$ -extent, and the flow develops progressively finer scales. The flow eventually relaminarises by $t=66$ . This sequence differs qualitatively from the convection-driven transition seen in G25 at larger $\textit {Fr}$ , where the overturning billows appear in pairs localised between adjacent vortex columns (see G25’s figure 4) rather than as an extended train that progressively fills the horizontal extent of the domain and cascades to small scales.

The horizontal velocity field during the transition to turbulence is examined in figure 4 at $t=48$ , an instant chosen to be among the last few snapshots at which the dynamics remains recognisably linear; for $t\gtrsim 48$ the perturbations grow rapidly and the base flow is no longer clearly identifiable. Panels (b,c) show streamlines sampled at two nearby heights, $z=2\pi /5$ and $z=\pi /5$ (red and blue dotted lines in panel (a), respectively). The streamline patterns at the two levels are clearly out of phase. In particular, comparing panels (b) and (c) shows a region near $y=\pi$ where the sign of $v$ reverses between the two heights, reflecting a combination of horizontal shifting and deformation of the columnar TG cells that varies considerably with $z$ , so that the in-plane flow direction is partially reversed between adjacent horizontal planes. This small vertical shear is the kinematic origin of the stacked shear layers seen in figure 3: as panels (d)–(f) show, the squared vertical shear of the horizontal velocity is concentrated in thin bands at intermediate heights between the two streamline levels, and, in the horizontal plane, these bands are strongly localised, concentrated near the hyperbolic stagnation points of the base flow at the peripheries of the deformed cells. The decomposition into $(\partial _z u)^2$ and $(\partial _z v)^2$ further reveals that $(\partial _z v)^2$ dominates while $(\partial _z u)^2$ contributes only weakly, consistent with the $v$ -reversal highlighted in the grey zone of panels (b,c). Although small in amplitude at $t=48$ , it is this vertical shear that grows into the secondary shear instabilities driving the transition to LAST.

Figure 5.

Vortex-stretching diagnostic for $(\textit {Re},\textit {Fr})=(3200,0.1875)$ at $t=48$ : (a) streamwise vorticity $\omega _x$ and (b) normal strain $\partial u/\partial x$ . Intense $\omega _x$ sheets coincide with regions of $\partial u/\partial x\gt 0$ , indicating amplification of $\omega _x$ by streamwise stretching in the base flow.

Figure 5. Long description

Panel A: A 3D visualization of streamwise vorticity. The axes are labeled x and y. The color gradient ranges from -2.0 to 2.0, with lighter colors indicating lower values and darker colors indicating higher values. The visualization shows regions of varying vorticity intensity. Panel B: A 3D visualization of normal strain. The axes are labeled similarly. The color gradient ranges from -1.5 to 1.5, with blue indicating lower values and red indicating higher values. The visualization shows regions of varying strain intensity.

Figure 5 identifies the amplification mechanism of the $\omega _x$ layers. Panels (a) and (b) show $\omega _x$ and $\partial u/\partial x$ respectively at the same instant. The most intense $\omega _x$ sheets are spatially coincident with regions where $\partial u/\partial x\gt 0$ , indicating that normal strain in the base flow near the hyperbolic stagnation points is actively stretching horizontal vortex tubes oriented along $x$ , thereby amplifying $\omega _x$ . This is directly analogous to the stretching of $\omega _y$ by $\partial v/\partial y$ identified in G25 (their figure 7), but with the roles of $x$ and $y$ interchanged. Beyond this stretching by the normal strain, the same base-flow geometry also accounts for the evolution of the $\omega _x$ sheets seen in figure 3 between $t=52$ and $t=56$ , and offers a mechanism for the lateral spreading of the turbulent activity across the $y$ -extent of the domain. At early times, two sheets of the same sign on either side of the hyperbolic stagnation point are separated by a clear gap (see figure 2 b); between $t=52$ and $t=56$ , these sheets are convectively transported towards each other by the base-flow velocity $v$ , which converges towards the hyperbolic stagnation point at $(x,y)=(\pi ,\pi )$ (figures 4 b and 4 c), and merge near $y=\pi$ , where Kelvin–Helmholtz-like billows subsequently roll up and intensify. The roll-up then extends outward along $y$ as the convergent base flow continues to advect newly amplified vorticity into the merger region, offering a plausible advective mechanism for the lateral spreading of the turbulent activity seen in figure 3.

The flow structure and mechanism inferred from figures 2, 3, 4 and 5 place the present case squarely in a layered anisotropic regime and reveal a transition to turbulence that is predominantly shear-driven, in sharp contrast to the moderately stratified cases of G25. At $\textit {Fr}\geqslant 0.5$ , G25 showed that the saturation of the zigzag instability overturns the isopycnals between adjacent vortex columns, producing compact billows of horizontal vorticity that appear in localised pairs and break down convectively without ever developing a laterally extended secondary structure. The same paired, convection-driven breakdown was documented by Waite & Smolarkiewicz (Reference Waite and Smolarkiewicz2008) for a Lamb–Chaplygin vortex pair. In the present case, no such paired overturns are observed. Instead, the $\omega _x$ field organises into horizontally extended sheets, each of which rolls up into a horizontal train of Kelvin–Helmholtz-like billows along $y$ , with successive sheets stacked on top of one another in $z$ ; these subsequently break down to progressively finer scales before relaminarising. This resembles the shear-driven secondary-instability mode identified by Deloncle et al. (Reference Deloncle, Billant and Chomaz2008) and Augier & Billant (Reference Augier and Billant2011) for a bent vortex pair, and, in a different configuration, by Lewin & Caulfield (Reference Lewin and Caulfield2024) for horizontally sheared stratified flow. The resulting flow, consisting of quasi-horizontal layers that move largely independently, separated by thin sheets of intense vertical shear, is the structure that strongly stratified theory (Billant & Chomaz Reference Billant and Chomaz2001; Brethouwer et al. Reference Brethouwer, Billant, Lindborg and Chomaz2007) associates with the SST (or LAST) regime.

3.2. Effects of $\textit {Re}$ and $\textit {Fr}$

Figure 6.

Spanwise vorticity on a representative vertical transect at peak dissipation, for $\textit {Re}\in \{800,1600,3200\}$ (columns) and $\textit {Fr}\in \{0.125,0.25,0.5,1.0,2.0\}$ (rows). In each panel, the parenthesised pair at the upper left is $(\textit {Re}_b^\star ,\,2\pi \textit {Fr}_v^\star )$ , which will be discussed in § 4, and the symbol at the lower right indicates the flow regime: $\triangle$ weakly stratified turbulence (WST), $\bigcirc$ LAST, $\Diamond$ LAVAF, $\square$ viscously controlled flow (VCF). The horizontal axis spans $x\in [0,2\pi ]$ and the vertical axis $z\in [0,\pi ]$ .

Figure 7.

As figure 6, but focused on the strongly stratified corner: $\textit {Re}\in \{1600,2400,3200\}$ (columns) and $\textit {Fr}\in \{0.125,0.1875,0.25\}$ (rows). The $\textit {Fr}=0.1875$ row and the $\textit {Re}=2400$ column consist of runs new to this study, and resolve the transition between the LAVAF ( $\Diamond$ ) and LAST ( $\bigcirc$ ) regimes.

Having established in § 3.1 that the run at $(\textit {Re},\textit {Fr})=(3200,0.1875)$ reaches the LAST regime through a shear-driven transition, we now survey the entire parameter set of table 1 to examine how the flow structure depends on $\textit {Re}$ and $\textit {Fr}$ . Figure 6 shows the spanwise vorticity on a representative vertical transect at peak dissipation for $\textit {Re}\in \{800,1600,3200\}$ and $\textit {Fr}\in \{0.125,0.25,0.5,1.0,2.0\}$ , and figure 7 focuses on the strongly stratified corner with the new $\textit {Re}=2400$ column and $\textit {Fr}=0.1875$ row included. The $(\textit {Re}_b^\star ,\,2\pi \textit {Fr}_v^\star )$ pair annotated on each panel will be discussed later.

Four qualitatively distinct flow regimes emerge across figures 6 and 7, labelled by the symbols listed in table 1.

1. (i) Weakly stratified turbulence ( $\triangle$ ) occurs at the larger values of $\textit {Fr}$ , specifically $\textit {Fr}\geqslant 0.5$ : the spanwise vorticity field is densely populated by small-scale, nearly isotropic turbulent structures with no visible layering, and this featureless turbulence persists across all three Reynolds numbers surveyed.

2. (ii) Layered anisotropic stratified turbulence ( $\bigcirc$ ) occurs at smaller $\textit {Fr}$ and sufficiently high $\textit {Re}$ : the flow organises into a stack of alternating-sign horizontal layers of spanwise vorticity that have clearly rolled up into billow-like structures and subsequently cascaded to fine scales, displaying the largest dynamic range of spatial scales among the layered cases, with sharp vertical gradients between layers and turbulent fluctuations within them.

3. (iii) Layered anisotropic viscously affected flow ( $\Diamond$ ) retains the stacked-layer structure and the layers still undulate and roll up, but the fine-scale turbulence visible in the LAST panels is absent; the billows are smooth, without secondary instabilities or a turbulent cascade, so that the flow is layered and anisotropic but strongly affected by viscosity.

4. (iv) Viscously controlled flow ( $\square$ ) has layers that are noticeably thicker for the same $\textit {Fr}$ , with the spanwise vorticity smooth throughout and no sign of roll-up: viscosity appears to control the flow, and no secondary instability develops within the layers.

The trends in $(\textit {Re},\textit {Fr})$ can now be inferred from figures 6 and 7. Following a column of fixed $\textit {Re}$ from high to low $\textit {Fr}$ , the flow leaves WST and enters the layered regime; the layers thin visibly as $\textit {Fr}$ decreases, consistent with the strongly stratified scaling $\ell _v\sim \textit {Fr} L$ (Billant & Chomaz Reference Billant and Chomaz2001; Brethouwer et al. Reference Brethouwer, Billant, Lindborg and Chomaz2007). Following a row of fixed $\textit {Fr}\leqslant 0.25$ from low to high $\textit {Re}$ , the layered flow evolves from smooth and viscously controlled (VCF) through undulating but non-turbulent (LAVAF) to fully turbulent within layers (LAST). Figure 7 resolves the LAVAF–LAST boundary with the new $\textit {Re}=2400$ column: at $\textit {Fr}=0.1875$ , the flow is LAVAF at $\textit {Re}=1600$ and LAST at $\textit {Re}=2400$ and $\textit {Re}=3200$ , while at $\textit {Re}=3200$ it is LAVAF at $\textit {Fr}=0.125$ and LAST at $\textit {Fr}=0.1875$ and $\textit {Fr}=0.25$ . The LAST regime therefore occupies a semi-infinite region of $(\textit {Re},\textit {Fr})$ space: if $\textit {Fr}$ is too large, the layering is lost and the flow becomes WST, while if either $\textit {Re}$ or $\textit {Fr}$ is too small, the cascade within the layers is suppressed and the flow becomes LAVAF or VCF. A quantitative placement of these boundaries in terms of $\textit {Re}$ and $\textit {Fr}$ will be revisited in the following analysis.

4.1. Time series at $\textit {Re}=3200$

We begin by examining how a set of turbulence diagnostics evolves in time as the columnar TG array breaks down and relaminarises. Here, we focus on the three runs at $\textit {Re}=3200$ and $\textit {Fr}\in \{0.125,0.1875,0.25\}$ , which span the parameter range over which the flow undergoes a shear-driven transition to turbulence (figure 7). The most fully developed turbulent state in each run is reached at the time of peak volume-averaged dissipation $\varepsilon _k=\langle \epsilon _k\rangle$ , where $\langle . \rangle$ denotes a volume average over the entire domain and $\epsilon _k$ is the point-wise kinetic-energy dissipation rate. We mark this instant with a star symbol on every curve and revisit it in the remainder of this section.

The vertical and horizontal length scales of the turbulence are diagnosed from the kinetic-energy spectrum. Following Zhou & Diamessis (Reference Zhou and Diamessis2019), who used the integral scale $\ell _v$ to characterise the vertical scale associated with spontaneously formed shear layers in stratified turbulent wakes, we define

(4.1) \begin{equation} \ell _{v}(t)\equiv 2\pi \int _{k_{\textit{min}}}^{k_{\textit{nyq}}} k^{-1}\tilde {E}_{k}(k,t)\,\mathrm{d}k\,\Big /\int _{k_{\textit{min}}}^{k_{\textit{nyq}}} \tilde {E}_{k}(k,t)\,\mathrm{d}k, \end{equation}

where $k_{\textit{min}}=1.0$ is the minimum wavenumber, $k_{\textit{nyq}}$ is the Nyquist wavenumber and $\tilde {E}_k(k,t)$ is the one-dimensional vertical kinetic-energy spectrum. The horizontal integral length scale $\ell _h$ is defined analogously from the horizontal spectrum. The horizontal Froude number $\textit {Fr}_h=U_h/(N\ell _h)$ then combines $\ell _h$ with the root-mean-square horizontal velocity $U_h=\langle u^2+v^2\rangle ^{1/2}$ and $N$ to gauge how stratified the dynamics is in the horizontal, while a vertical Froude number $\textit {Fr}_v=U_h/(N\ell _v)=(2\pi )^{-1}(\ell _b/\ell _v)$ measures the ratio of the buoyancy scale $\ell _b$ to the emergent vertical scale $\ell _v$ (§ 4.3).

Figure 8 shows the time evolution of the large-scale quantities. Up to approximately $t=40$ the three runs are nearly indistinguishable in terms of the horizontal velocity and integral length scales. The horizontal integral scale $\ell _h$ sits near $2\pi$ , as expected for the columnar base flow, and $U_h$ decays viscously. The vertical scale $\ell _v$ locks onto the wavelength of the saturated zigzag mode (G24) and stays relatively constant thereafter. The value of $\ell _v$ depends strongly on $\textit {Fr}$ , but once rescaled by the buoyancy scaling $U_h/N$ the three runs yield a narrow range of $\textit {Fr}_v$ values. The horizontal Froude number $\textit {Fr}_h$ tracks the initial $\textit {Fr}$ at this stage and remains relatively constant, because $\ell _h$ and $U_h$ are both decaying.

Around $t\approx 40$ the curves separate. The $\textit {Fr}=0.25$ run (green) is the first to break down into turbulence: $U_h$ decays sharply as the turbulent transition occurs, and $\ell _h$ decreases as turbulent scales develop, reaching its minimum near the star and recovering thereafter. The $\textit {Fr}=0.1875$ and $\textit {Fr}=0.125$ runs follow with progressively later excursions. The star, marking the peak of $\varepsilon _k$ , sits near the bottom of the $\ell _h$ dip and on the steep portion of $U_h$ in every case, confirming that the turbulence is most fully developed when the horizontal integral scale is most strongly perturbed away from its base-flow value. After the star, $\textit {Fr}_h$ decays rapidly as $\ell _h$ recovers while $U_h$ continues to fall due to turbulent dissipation. It is interesting to note that the $\textit {Fr}$ dependence of the two length scales runs in opposite directions (figure 8 b): $\ell _h^\star$ grows weakly as $\textit {Fr}$ decreases, reflecting the inhibited forward cascade at strong stratification, while $\ell _v^\star$ decreases with $\textit {Fr}$ as it locks onto the buoyancy scale $U_h/N$ (§ 4.3).

Figure 9 shows the corresponding small-scale and mixing diagnostics. The Kolmogorov scale $\ell _K$ is roughly constant at early times and dips to a similar minimum across the three runs near the star, since $\varepsilon _k^\star$ is largely insensitive to $\textit {Fr}$ . The Thorpe scale $\ell _T$ is essentially zero before $t\approx 45$ and spikes sharply around the star, confirming that density overturns appear only during the active turbulent burst and disappear once the flow relaminarises. The peak $\ell _T$ is comparable for the two highest- $\textit {Fr}$ runs, but for $\textit {Fr}=0.125$ it is roughly an order of magnitude smaller, falling below the corresponding $\ell _K$ . This indicates that no resolved overturns are present and that the flow at $\textit {Fr}=0.125$ never enters a state of active density overturning. The ratio $\varepsilon _p/\varepsilon _k$ rises from near zero, peaks close to the star at values of $O(0.5)$ , and decays back towards zero as the turbulence subsides. Together, these diagnostics show that each run passes through a short window of active turbulence and scalar mixing centred on the star, bracketed by long phases in which the flow is either still transitioning out of the base state or already relaminarising.

Figure 8.

Time series of $U_h$ , the integral length scales $\ell _h$ (upper, unshaded) and $\ell _v$ (lower, shaded), $\textit {Fr}_h$ and $\textit {Fr}_v$ , for the cases with $\textit {Re}=3200$ and $\textit {Fr}=0.25$ (green dashed), $0.1875$ (red dotted) and $0.125$ (blue solid). The star on each curve marks the time of peak volume-averaged dissipation $\varepsilon _k$ .

Figure 9.

Time series of $\ell _K$ , $\ell _T$ and $\varepsilon _p/\varepsilon _k$ for the same cases as in figure 8. Here, $\varepsilon _p=(\textit {Re}\,\textit {Pr}\,\textit {Fr}^2)^{-1}\,\langle |\boldsymbol{\nabla }\rho |^2\rangle$ is the potential-energy dissipation rate. The star on each curve marks the time of peak volume-averaged dissipation $\varepsilon _k$ .

4.2. Phase-plane trajectories

Figure 10 shows the trajectories of the runs with $\textit {Re}\geqslant 1600$ in the $(\textit {Re}_h,1/\textit {Fr}_h)$ plane, the regime diagram of Brethouwer et al. (Reference Brethouwer, Billant, Lindborg and Chomaz2007). The three panels correspond to $\textit {Re}=1600$ , $2400$ and $3200$ , and within each panel the four colours mark $\textit {Fr}=0.5$ , $0.25$ , $0.1875$ , $0.125$ . The trajectories within each panel are separated by $\textit {Fr}$ , since the base flow has the same velocity and length scales across runs and only $N$ varies, so the initial $\textit {Fr}_h$ is set directly by $\textit {Fr}$ . Increasing $\textit {Re}$ shifts each curve to the right and raises the value of $\textit {Re}_h^\star \textit {Fr}_h^{\star 2}$ at peak dissipation; the perpendicular distance from a trajectory to the dotted line $\textit {Re}_h\textit {Fr}_h^2=1$ is a visual indicator of the magnitude of $\textit {Re}_h\textit {Fr}_h^2$ at that instant. Across the database, $\textit {Re}_h^\star$ depends strongly on $\textit {Re}$ and only weakly on $\textit {Fr}$ , whereas $\textit {Fr}_h^\star$ depends almost entirely on $\textit {Fr}$ , an observation we will use in § 4.5. Region I, with $1/\textit {Fr}_h$ large and $\textit {Re}_h\textit {Fr}_h^2\gt 1$ , is where one expects SST; region III, with $1/\textit {Fr}_h$ small, is where one expects WST; region II, with $\textit {Re}_h\textit {Fr}_h^2\lt 1$ , is the viscously affected stratified regime. The grey band $0.031\lt \textit {Fr}_h\lt 0.044$ runs between the largest $\textit {Fr}_h^\star$ recorded for any LAST run in table 1 and the smallest $\textit {Fr}_h^\star$ recorded for any WST run, and identifies a potential $\textit {Fr}_h$ cutoff between WST and the strongly stratified regimes (§ 4.3). The diagram was originally intended for statistically stationary flows, for which the location of the initial condition on the diagram is not a concern. In the transient setting studied here, however, the initial condition for the low- $\textit {Fr}$ runs ( $\textit {Fr}\leqslant 0.25$ ) already lies well inside region I, i.e. SST, even though the laminar columnar vortex array has yet to break down, so caution should be exercised when using this regime diagram as a diagnostic for a transient flow.

Figure 10.

Trajectories of the runs in the $(\textit {Re}_h,\,1/\textit {Fr}_h)$ plane, for $\textit {Re}=1600$ , $2400$ and $3200$ (panels left to right) and $\textit {Fr}=0.5$ , $0.25$ , $0.1875$ , $0.125$ (purple, green, red, blue). The vertical tick on each curve marks $t=0$ and the star marks peak dissipation $\varepsilon _k$ . The dotted line is a visual guide of constant $\textit {Re}_h\textit {Fr}_h^{2}=1.0$ . The grey band $0.031\lt \textit {Fr}_h\lt 0.044$ is bounded below by the maximum $\textit {Fr}_h^\star$ over all LAST runs in table 1, attained at $(\textit {Re},\textit {Fr})=(3200,0.25)$ , and above by the minimum $\textit {Fr}_h^\star$ over all WST runs, attained at $(\textit {Re},\textit {Fr})=(800,0.5)$ . Roman numerals label the three regions of the $(\textit {Re}_h,1/\textit {Fr}_h)$ plane in the regime classification of Brethouwer et al. (Reference Brethouwer, Billant, Lindborg and Chomaz2007): I. SST; II. viscously affected stratified flow; III. WST.

An alternative phase plane, $(\textit {Re}_b,2\pi \textit {Fr}_h)$ , is shown in figure 11, after de Bruyn Kops & Riley (Reference de Bruyn Kops and Riley2019). The buoyancy Reynolds number $\textit {Re}_b=\varepsilon _k/(\nu N^2)$ quantifies the scale separation between the Ozmidov scale $\ell _O$ and the Kolmogorov scale $\ell _K$ . When $\textit {Re}_b$ is sufficiently large, one can follow the argument in the appendix of Riley & Lindborg (Reference Riley and Lindborg2008) to show that $\ell _O$ is indeed the scale at which inertial and buoyancy forces are comparable, relying on the existence of an inertial range in the turbulence spectrum; when $\textit {Re}_b^\star =O(1)$ , this separation is marginal and the corresponding interpretation of $\ell _O$ is less clear cut due to the lack of an inertial range. The bottom-right corner of each panel, with $\textit {Re}_b\gt 1$ and $\textit {Fr}_h$ small, is where one expects LAST; to the left of $\textit {Re}_b=1$ the flow is viscously affected; at large $\textit {Fr}_h$ the flow is weakly stratified. Each trajectory begins at low $\textit {Re}_b$ and moderate $\textit {Fr}_h$ , reflecting the initially laminar base flow. As the columnar vortex array breaks down, $\textit {Re}_b$ climbs and the curve sweeps to the right while $\textit {Fr}_h$ stays nearly constant. Once turbulence peaks, marked by the star at maximum $\textit {Re}_b$ , the curve turns back: the turbulence weakens and $\textit {Fr}_h$ falls rapidly as $U_h$ decays and $\ell _h$ grows (figure 8). de Bruyn Kops & Riley (Reference de Bruyn Kops and Riley2019) used $2\pi \textit {Fr}_h=0.6$ as the cutoff below which the flow qualifies as ‘stratified turbulence,’ equivalent to LAST or SST. By that threshold every run in figure 11 accesses the strongly stratified regime. We instead propose a different cutoff in $\textit {Fr}_h$ , highlighted in grey, that separates the layered anisotropic regimes (LAST and LAVAF) from the weakly stratified regime; this new cutoff is motivated by the vertical length-scale scaling examined in the next subsection.

Figure 11.

Trajectories of the same runs as in figure 10, replotted in the $(\textit {Re}_b,\,2\pi \textit {Fr}_h)$ plane. The vertical tick marks $t=0$ on each curve and the star marks peak dissipation. The vertical dashed line indicates $\textit {Re}_b=1$ . The grey band, $0.031\lt \textit {Fr}_h \lt 0.044$ , is the same as in figure 10.

4.3. Scaling of the vertical length scale

The hallmark of strongly stratified, layered anisotropic regimes is the buoyancy-controlled vertical scale $\ell _v\propto U_h/N$ (Billant & Chomaz Reference Billant and Chomaz2001; Brethouwer et al. Reference Brethouwer, Billant, Lindborg and Chomaz2007), equivalent to the vertical Froude number $\textit {Fr}_v=U_h/(N\ell _v)$ being approximately constant. In this subsection we test the criterion at peak dissipation across all simulations in table 1, and assess how strongly stratified, as measured by an appropriate version of the Froude number, the flow has to be for the buoyancy-driven scaling to hold.

Figure 12.

Value of $2\pi \textit {Fr}_v^\star$ at peak dissipation versus $\textit {Fr}$ , for all runs. Symbols denote regime, as defined in § 3.2 and listed in table 1: triangles ( $\triangle$ ) WST, circles ( $\bigcirc$ ) LAST, diamonds ( $\Diamond$ ) LAVAF, squares ( $\square$ ) VCF. The horizontal dashed line marks the buoyancy-controlled plateau $2\pi \textit {Fr}_v^\star \approx 0.74$ followed by the LAST and LAVAF runs, with $0.74$ the mean and $0.08$ the standard deviation across those data points; the dotted diagonal line marks the high- $\textit {Fr}$ isotropic scaling $2\pi \textit {Fr}_v^\star \propto \textit {Fr}$ followed by the WST runs.

Figure 13.

Testing two vertical-length scalings at peak dissipation for all runs. (a) Value of $\ell _v^\star /(2\pi )$ versus $U_h^\star /N$ ; the dotted line is the buoyancy-driven scaling $\ell _v^\star \propto U_h^\star /N$ (Billant & Chomaz Reference Billant and Chomaz2001; Brethouwer et al. Reference Brethouwer, Billant, Lindborg and Chomaz2007) and the dashed line is the high- $\textit {Fr}$ , isotropic limit $\ell _v^\star =\text{const}$ . (b) Value if $2\pi \textit {Fr}_v^\star$ versus $\textit {Fr}_h^\star$ ; the dotted line is the buoyancy-driven plateau $\textit {Fr}_v^\star =\text{const}$ and the dashed line is the high- $\textit {Fr}_h$ scaling $\textit {Fr}_v^\star \propto \textit {Fr}_h^\star$ . Symbols are as in figure 12 and colour reflects the magnitude of $\log _{10}\textit {Re}_b^\star$ .

Two scalings for the vertical length scale are possible. In the strongly stratified, layered anisotropic limit (Billant & Chomaz Reference Billant and Chomaz2001; Brethouwer et al. Reference Brethouwer, Billant, Lindborg and Chomaz2007), the vertical scale is set by the buoyancy length, $\ell _v\propto U_h/N$ ; the vertical Froude number $\textit {Fr}_v$ approaches a constant and $\ell _v/\ell _h\sim \textit {Fr}_h\ll 1$ , reflecting anisotropic flow structures. In the weakly stratified, nearly isotropic limit, $\ell _v$ is set by the geometry of the flow rather than by stratification, $\ell _v\propto \ell _h\sim L$ ; the vertical Froude number then tracks the base-flow value, $\textit {Fr}_v=U_h/(N\ell _v)\sim \textit {Fr}$ , and $\ell _v/\ell _h\sim 1$ .

Figure 12 plots $2\pi \textit {Fr}_v^\star$ , which can be interpreted as the inverted ratio of the vertical scale $\ell _v^\star$ to the buoyancy scale $2\pi U_h^\star /N$ , against the base-flow Froude number $\textit {Fr}$ . A clear transition is visible as $\textit {Fr}$ is reduced from $0.5$ to $0.25$ . For $\textit {Fr}\geqslant 0.5$ only WST runs (triangles) are present and $2\pi \textit {Fr}_v^\star$ scales approximately linearly with $\textit {Fr}$ , i.e. the isotropic scaling. For $\textit {Fr}\leqslant 0.25$ the remaining three regimes are present (circles for LAST, diamonds for LAVAF, squares for VCF), and a subset of the points (circles and diamonds) collapse onto a plateau near $2\pi \textit {Fr}_v^\star \approx 0.74$ that is essentially independent of $\textit {Fr}$ , i.e. the buoyancy-controlled scaling.

Figure 13 displays the same scaling in two ways, with colour now reflecting $\log _{10}\textit {Re}_b^\star$ . Panel (a) shows $\ell _v^\star /(2\pi )$ against $U_h^\star /N$ . The LAST and LAVAF points (red and brown, intermediate $\textit {Re}_b^\star$ ) collapse onto the dotted line $\ell _v^\star \propto U_h^\star /N$ , i.e. the buoyancy-controlled scaling, while the WST points (green, large $\textit {Re}_b^\star$ ) saturate near the dashed line $\ell _v^\star =\text{const}$ . Panel (b) shows $2\pi \textit {Fr}_v^\star$ against $\textit {Fr}_h^\star$ and reflects the same scalings: the buoyancy-driven plateau $\textit {Fr}_v^\star =\text{const}$ is followed largely by the red and brown circles and diamonds, while the green WST triangles lie on the dashed line $\textit {Fr}_v^\star \propto \textit {Fr}_h^\star$ . Panel (b) also locates a cutoff in $\textit {Fr}_h^\star$ between WST and the layered anisotropic regimes. The largest $\textit {Fr}_h^\star$ recorded for any LAST run is $0.031$ and the smallest for any WST run is $0.044$ , so the regime cutoff is expected to lie somewhere in this gap $0.031\lt \textit {Fr}_h\lt 0.044$ . It is well below the value $2\pi \textit {Fr}_h\approx 0.6$ proposed by de Bruyn Kops & Riley (Reference de Bruyn Kops and Riley2019), and of the same order as, though slightly larger than, the criterion $\textit {Fr}_h\approx 0.02$ estimated by Lindborg (Reference Lindborg2006) from scaling arguments for atmospheric flows.

The outliers from both scalings in figure 13 are the VCF points (blue and purple, small $\textit {Re}_b^\star$ ). These cases are strongly stratified by the $\textit {Fr}$ criterion, but the turbulence is too weak for the buoyancy scaling to establish itself: viscosity holds $\ell _v^\star$ above the buoyancy scaling (panel (a) of figure 13). This points to a second requirement for the layered anisotropic regimes (LAST and LAVAF) beyond strong stratification, namely that the turbulence becomes vigorous enough for the buoyancy-controlled vertical scale to be realised. From table 1, the smallest $\textit {Re}_b^\star$ recorded for any LAVAF run is $0.58$ and the largest for any VCF run is $0.45$ , placing the cutoff in $\textit {Re}_b^\star$ between LAVAF and VCF in this gap $0.45\lt \textit {Re}_b\lt 0.58$ .

4.4. Turbulence diagnostics

4.4.1. Thorpe scale

Having located the layered anisotropic regimes in the phase planes of § 4.2 and tested the buoyancy-controlled vertical scale in § 4.3, we now turn to two further diagnostics, the Thorpe scale and the cumulative mixing efficiency, that sharpen the distinction between LAST and the neighbouring regimes. Figure 14(a) plots $\ell _T^\star /\ell _O^\star$ against $\textit {Fr}$ , where $\ell _O^\star =\textit {Fr}^{3/2}\,\varepsilon _k^{\star 1/2}$ is the Ozmidov scale at peak dissipation. The WST (triangles) and LAST (circles) points all sit close to $\ell _T^\star /\ell _O^\star \sim 1$ , with the Thorpe scale lying above the Ozmidov scale in most turbulent cases, except for a few WST runs at the highest $\textit {Fr}=2.0$ . The LAVAF points (diamonds) are scattered, with one case near unity and another well below; the VCF point (square) lies more than an order of magnitude below. One LAVAF case, $(\textit {Re},\textit {Fr})=(2400,0.125)$ , and three VCF cases, $(800,0.1875)$ , $(800,0.125)$ and $(1600,0.125)$ , show no resolved overturns ( $\ell _T\approx 0$ ) and are absent from the logarithmic plot. Panel (b) plots $\ell _T^\star /\ell _K^\star$ against $\textit {Fr}$ and the regimes separate cleanly. The WST and LAST points all lie above the Kolmogorov scale with $\ell _T^\star /\ell _K^\star \gt 1$ , so the overturns evolve in a spatial range where viscosity is not in control. The LAVAF and VCF points fall below unity, indicating overturns (if present) smaller than the viscous scale and so completely controlled by viscosity. This separation aligns with the visualisations of § 3.2: the LAST and LAVAF cases at $\textit {Fr}=0.1875$ in figure 7 (middle row) share the same large-scale layered structure, but the LAVAF case shows smooth undulations rather than fine-scale turbulence, the visual signature of a viscosity-controlled dynamics lacking a dynamic range of scales.

Figure 14.

Thorpe-scale ratios at peak dissipation versus $\textit {Fr}$ , for all runs. (a) $\ell _T^\star /\ell _O^\star$ , where $\ell _O^\star =(\varepsilon _k^\star \,\textit {Fr}^3)^{1/2}$ is the Ozmidov scale; (b) $\ell _T^\star /\ell _K^\star$ . Symbols and colour as in figure 13. Cases with $\ell _T^\star \simeq 0$ (no resolved overturns) are not shown.

Figure 15.

Value of $\ell _T^\star /\ell _K^\star$ versus $\textit {Re}_b^\star$ for all runs. The trend crosses unity near $\textit {Re}_b^\star \approx 1$ , identifying the lower- $\textit {Re}_b^\star$ cutoff below which Thorpe-scale overturns are damped to within the Kolmogorov scale. Symbols as in figure 12.

Figure 15 replots $\ell _T^\star /\ell _K^\star$ against $\textit {Re}_b^\star$ and the regimes collapse onto a single trend across four orders of magnitude in $\ell _T^\star /\ell _K^\star$ , with the curve crossing $\ell _T^\star /\ell _K^\star =1$ at $\textit {Re}_b^\star \approx 1$ . Cases with $\textit {Re}_b^\star \gtrsim 1$ have $\ell _T^\star \gt \ell _K^\star$ and overturn more freely, being less damped by viscosity; cases with $\textit {Re}_b^\star \lesssim 1$ have $\ell _T^\star \lt \ell _K^\star$ and are viscously controlled. This identifies $\textit {Re}_b^\star \approx 1$ as the lower cutoff in $\textit {Re}_b^\star$ separating LAST from LAVAF, and provides the second of the two regime boundaries for LAST, revisited in § 4.5.

4.4.2. Mixing efficiency

The cumulative mixing efficiency $\varGamma$ , defined here as the ratio of the time-averaged potential-energy dissipation to time-averaged kinetic-energy dissipation, sets the rate at which stratified turbulence converts kinetic energy into potential energy, and remains the central parameter used to relate small-scale mixing to large-scale stratification budgets in the ocean and atmosphere (Caulfield Reference Caulfield2021). For the LAST regime in particular, most reported values come from forced simulations (e.g. Maffioli et al. Reference Maffioli, Brethouwer and Lindborg2016), in which forcing and dissipation reach a statistical balance. Decaying configurations that also access LAST are less common; Lewin & Caulfield (Reference Lewin and Caulfield2024) and Zhou (Reference Zhou2024) are recent examples based on horizontal shear layers and a stratified wake, respectively. The freely evolving columnar TG arrays studied here add to this dataset: the LAST regime is reached through an instability of a laminar base flow, and we report the cumulative $\varGamma$ over the resulting turbulent event.

Figure 16.

Cumulative mixing efficiency $\varGamma =\overline {\varepsilon _p}/\overline {\varepsilon _k}$ for all runs: (a) $\varGamma$ versus $\textit {Fr}$ , (b) $\varGamma$ versus $\ell _O^\star /\ell _T^\star$ ; cases with $\ell _T^\star =0$ are not shown, hence the smaller vertical range. Symbols and colour as in figure 13. Here, the overline denotes a time average over the simulation duration, i.e. $\overline {\varepsilon _p}=(1/T) \int _0^T \varepsilon _p \mathrm{d}t$ , $\overline {\varepsilon _k}=(1/T) \int _0^T \varepsilon _k \mathrm{d}t$ , where $T=100$ .

Figure 17.

Cumulative mixing efficiency $\varGamma$ versus the buoyancy Reynolds number $\textit {Re}_b^\star$ at peak dissipation, for all runs in table 1. Symbols as in figure 16.

Figure 16(a) shows $\varGamma$ against $\textit {Fr}$ and highlights the range of $\varGamma$ attained by each regime. Across the LAST runs, $\varGamma$ falls in the range $0.24$ – $0.49$ (table 1). These values are consistent with the high-efficiency mixing reported by Riley & de Bruyn Kops (Reference Riley and de Bruyn Kops2003) for decaying turbulence generated by three-dimensional (not columnar) TG vortices, who found $\varGamma \approx 0.4$ ; with the SST phase identified by Zhou (Reference Zhou2024) in stratified wakes, where $\varGamma$ plateaus at $0.45$ – $0.49$ before dropping once viscosity dominates; and with the LAST values $\varGamma \approx 0.3$ – $0.4$ reported by Lewin & Caulfield (Reference Lewin and Caulfield2024) in their freely evolving horizontal shear layer. Figure 17 plots $\varGamma$ against $\textit {Re}_b^\star$ for every run. For $\textit {Re}_b^\star \lesssim 1$ the VCF squares rise steeply from $\varGamma \approx 0.057$ at the smallest $\textit {Re}_b^\star$ to approximately $0.16$ – $0.18$ near the LAVAF threshold. Above $\textit {Re}_b^\star \approx 1$ the LAST circles and WST triangles populate a plateau at $\varGamma \approx 0.3$ – $0.5$ , with a slight decrease at the largest $\textit {Re}_b^\star$ . This behaviour, a rise at low $\textit {Re}_b^\star$ followed by a plateau and mild decline, is reminiscent of Shih et al. (Reference Shih, Koseff, Ivey and Ferziger2005), who used forced sheared simulations. We attribute the high efficiency of LAST relative to VCF or LAVAF to the fact that overturns at the Thorpe scale are less damped by viscosity in LAST: at $\textit {Re}_b^\star \gtrsim 1$ the Thorpe and Ozmidov scales both lie above the Kolmogorov scale (§ 4.4.1), so the buoyancy-driven overturns that produce $\varepsilon _p$ can develop without being smeared by viscous dissipation, whereas in the regimes of $\textit {Re}_b^{\star }\lesssim 1$ the overturning Thorpe scale is pushed down to the viscous range and overturns are suppressed before they can convert kinetic energy into potential energy.

Figure 16(b) shows $\varGamma$ against the scale ratio $\ell _O/\ell _T$ at peak dissipation. The LAST runs cluster near and below $\ell _O/\ell _T=1$ , and $\varGamma$ decreases as $\ell _O/\ell _T$ grows past unity towards the WST runs. Mashayek et al. (Reference Mashayek, Caulfield and Peltier2017) argued, on the basis of decaying Kelvin–Helmholtz-driven mixing layers, that mixing is most efficient when the largest overturns are comparable to the Ozmidov scale, $\ell _O/\ell _T\sim 1$ : the overturning scale holding the available potential energy is then small enough to escape suppression by ambient stratification, yet large enough to sustain a broad inertial subrange that supports irreversible mixing. Mashayek et al. (Reference Mashayek, Caulfield and Alford2021) extended this Goldilocks picture to oceanic shear-induced overturns. Zhou (Reference Zhou2024) reported peak $\varGamma$ at the somewhat smaller ratio $\ell _O/\ell _T\in [0.37,0.52]$ for stratified wakes, with high mixing efficiency $\varGamma \in [0.45,0.49]$ . Taken together with the $\varGamma$ values reported specifically for LAST (e.g. Maffioli et al. Reference Maffioli, Brethouwer and Lindborg2016; de Bruyn Kops & Riley Reference de Bruyn Kops and Riley2019; Lewin & Caulfield Reference Lewin and Caulfield2024), a consensus is emerging that the LAST regime mixes efficiently, with $\varGamma$ in the range $0.3$ – $0.5$ across forced and decaying configurations. Based on the results here and in § 4.4.1, we conjecture that this high efficiency in LAST is tied to overturns that are large compared with the Kolmogorov scale, hence free of excessive viscous damping, and comparable to the Ozmidov scale, hence in the Goldilocks regime of Mashayek et al. (Reference Mashayek, Caulfield and Peltier2017).

4.5. Regime boundaries

We now place the four regimes on the base-flow parameter plane $(\textit {Re},1/\textit {Fr})$ in figure 18. Two questions raised in the introduction return here for the LAST regime: how stratified is strongly stratified, and how turbulent is turbulent enough to access this regime? These questions correspond to two distinct transitions rather than a single boundary. The first was addressed in § 4.3: the onset of buoyancy-controlled vertical scaling $\ell _v\propto U_h/N$ occurs for $\textit {Fr} \lt 0.5$ , and the associated cutoff in the turbulence-based horizontal Froude number lies in the gap $0.031\lt \textit {Fr}_h^\star \lt 0.044$ . The second was addressed in § 4.4.1: the lower cutoff in $\textit {Re}_b^\star$ , which separates LAST from LAVAF, lies near $\textit {Re}_b^\star =1$ , where the Thorpe scale crosses the Kolmogorov scale (figure 15), and thus marks the onset of fine-scale turbulence within the layers. These two criteria are therefore not competing estimates of a single regime boundary, but complementary diagnostics of stratification strength and turbulent activity. Taken together, they define the LAST regime as the region of $(\textit {Re},1/\textit {Fr})$ space in which the flow is both sufficiently strongly stratified to obey the buoyancy-controlled vertical scaling and sufficiently turbulent to sustain overturns. The remainder of this subsection quantifies both thresholds in terms of $\textit {Re}$ and $\textit {Fr}$ .

Figure 18.

Regime diagram in the $(\textit {Re},1/\textit {Fr})$ plane. Symbols denote the four regimes WST ( $\triangle$ ), LAST ( $\bigcirc$ ), LAVAF ( $\Diamond$ ) and VCF ( $\square$ ), as defined in § 3.2; colour reflects $\log _{10}\textit {Re}_b^\star$ . The dotted line delineates $\textit {Re}_b^\star =1$ , equivalent to $0.0090\,\textit {Re}\textit {Fr}^{5/3}=1$ from the fit in figure 21, and marks the lower- $\textit {Re}_b$ cutoff below which the flow can no longer sustain LAST. The grey band $0.26\leqslant \textit {Fr}\leqslant 0.37$ marks the upper- $\textit {Fr}$ cutoff separating LAST from WST. Together, the two boundaries delineate the LAST regime as the wedge between the band and the dotted line.

Figure 19.

Empirical scalings of the turbulence-based parameters with the base-flow Froude number $\textit {Fr}$ , at peak dissipation. (a) Value of $\textit {Re}_h^\star /\textit {Re}$ versus $\textit {Fr}$ ; the dashed line is $\textit {Re}_h^\star /\textit {Re}\propto \textit {Fr}^{-1/3}$ . (b) Value of $\textit {Fr}_h^\star$ versus $\textit {Fr}$ ; the dashed line is $\textit {Fr}_h^\star \propto \textit {Fr}$ . Symbols as in figure 18.

Figure 20.

Value of $\textit {Re}_b^\star$ versus $\textit {Re}_h^\star \textit {Fr}_h^{\star 2}$ for all runs. The dotted line is the one-to-one relation $\textit {Re}_b^\star =\textit {Re}_h^\star \textit {Fr}_h^{\star 2}$ expected from Taylor’s dissipation scaling $\varepsilon _k\sim U_h^3/\ell _h$ (e.g. Brethouwer et al. Reference Brethouwer, Billant, Lindborg and Chomaz2007; Maffioli & Davidson Reference Maffioli and Davidson2016). Symbols as in figure 18.

Figure 21.

Value of $\textit {Re}_b^\star$ versus $\textit {Re}\textit {Fr}^{5/3}$ for all runs. The dotted line, $\textit {Re}_b^\star =0.0090\,\textit {Re}\textit {Fr}^{5/3}$ , is a least-squares fit to only the LAST and LAVAF cases. (Note that a different scaling was proposed in G25 based on the mostly weakly stratified data available then.) Symbols as in figure 18.

Figure 19(b) plots $\textit {Fr}_h^\star$ against $\textit {Fr}$ for all runs and shows that the two are approximately linearly related. An empirical fit gives

(4.2) \begin{equation} \textit {Fr}_h^\star \approx 0.118\,\textit {Fr}, \end{equation}

which holds across all 24 runs. Inverting this relation, the gap $0.031\lt \textit {Fr}_h^\star \lt 0.044$ maps to the base-flow Froude number gap $0.26\lt \textit {Fr}\lt 0.37$ , plotted as the grey band in figure 18 below which only WST triangles appear.

The condition $\textit {Re}_b^\star \gtrsim 1$ from § 4.4.1 must also be expressed in terms of the base-flow parameters $(\textit {Re},1/\textit {Fr})$ if it is to be drawn on figure 18. The natural starting point is the dissipation scaling $\varepsilon _k\sim U_h^3/\ell _h$ , which, combined with the definitions of $\textit {Re}_b^\star$ , $\textit {Re}_h^\star$ and $\textit {Fr}_h^\star$ , yields $\textit {Re}_b^\star \simeq \textit {Re}_h^\star \textit {Fr}_h^{\star 2}$ (e.g. Brethouwer et al. Reference Brethouwer, Billant, Lindborg and Chomaz2007). Figure 20 tests this relation directly. Evidently, it does not hold one to one: the points cluster near the dotted reference line for $\textit {Re}_b^\star$ of order unity and below, but are systematically above it at higher $\textit {Re}_b^\star$ . The relation $\textit {Re}_b^\star \simeq \textit {Re}_h^\star \textit {Fr}_h^{\star 2}$ is therefore not exact, as expected, since it is based on a scaling argument. But it remains a reasonable approximation in the range $\textit {Re}_b^\star \sim 1$ relevant to the LAST–LAVAF cutoff, reminiscent of the decaying turbulence DNSs of Maffioli & Davidson (Reference Maffioli and Davidson2016). We use this approximate relation, together with the empirical scalings $\textit {Re}_h^\star /\textit {Re}\propto \textit {Fr}^{-1/3}$ in figure 19(a) and $\textit {Fr}_h^\star \propto \textit {Fr}$ in panel (b), both reflecting the trends discussed in § 4.2, to predict $\textit {Re}_b^\star$ in terms of the base-flow parameters: $\textit {Re}_b^\star \sim \textit {Re}_h^\star \textit {Fr}_h^{\star 2}\propto \textit {Re}\textit {Fr}^{-1/3}\textit {Fr}^2=\textit {Re}\textit {Fr}^{5/3}$ . Figure 21 shows that the data follow this scaling well, and a least-squares fit to the LAST and LAVAF cases gives

(4.3) \begin{equation} \textit {Re}_b^\star \approx 0.0090\,\textit {Re}\textit {Fr}^{5/3}. \end{equation}

The same strategy of empirical power-law fits for regime cutoffs was used by Zhou & Diamessis (Reference Zhou and Diamessis2019) for wakes in the SST regime. Setting $\textit {Re}_b^\star =1$ in (4.3) gives the lower- $\textit {Re}_b^\star$ cutoff in the $(\textit {Re},1/\textit {Fr})$ plane, i.e. $\textit {Re}\textit {Fr}^{5/3}\approx 1.1\times 10^2$ , plotted as the red dotted line in figure 18. Together, the two boundaries identify the LAST regime as the wedge-like region in figure 18 between the grey band on the upper- $\textit {Fr}$ side and the red dotted line on the lower- $\textit {Re}_b^\star$ side.