2.1 Zigzag instability of vortex pairs

Billant & Chomaz (Reference Billant and Chomaz2000) and Otheguy, Chomaz & Billant (Reference Otheguy, Chomaz and Billant2006) describe the ‘zigzag instability’ of a vertical cylindrical counter-rotating vortex pair with vertical vorticity, similar to a Lamb–Chaplygin dipole, in a uniformly stratified fluid. The dipole becomes unstable to a three-dimensional instability, periodic in the vertical direction, that ultimately leads to layers. The nonlinear evolution of the instability of the vortex pair, involving Kelvin–Helmholtz and gravitational instabilities, has been investigated numerically by Deloncle, Billant & Chomaz (Reference Deloncle, Billant and Chomaz2008), Waite & Smolarkiewicz (Reference Waite and Smolarkiewicz2008) and Augier and Billant (Reference Augier and Billant2011).

Although small-scale layers are apparent close to a vertical grid bar in the startup stages of some grid-mixing experiments (e.g. Thorpe Reference Thorpe1982, figure 8a), possibly generated by the instability of transient recirculating eddies (§ 2.2), it is not however evident how close-knit vertically coherent pairs of vortices might be produced by moving grids; streets of individual vortices are more likely, at least in type (b) experiments.

2.2 Zigzag instability in wakes

The wake of a vertical cylinder steadily moving horizontally through a uniformly stratified fluid when $40<Re_{0}<200$ and $0.3<Fr_{0}<200$ is described in experiments and a numerical model by Meunier (Reference Meunier2012). (Values of $U$ and $b$ in the definitions of $Re_{0}$ in (1.1) and $Fr_{0}$ in (1.2) are taken as the cylinder speed and diameter, respectively.) Until the onset of motions in directions parallel to the cylinder axis at $Re_{0}\sim 190$ , the wakes in unstratified and stratified fluids are similar. At small $Re_{0}$ wakes are steady; shear layers detach from the boundary layers on each side of the cylinder, producing a pair of recirculating eddies in the wake. When $45<Re_{0}\gtrsim 180$ , an array of vortices is produced in the cylinder’s wake, the well-known Kármán vortex street of an unstratified fluid. The vortex array consists of two rows of vortices approximately $2b$ apart and with vortices of one row staggered in relation to those of the other. The Strouhal number (equal to $b$ divided by the horizontal separation, $B$ , of vortices in a row) is approximately $0.019Re_{0}^{1/2}$ . Holford & Linden (Reference Holford and Linden1999b ) describe the wakes of square vertical bars moving horizontally through a uniformly stratified fluid when $1<Fr_{0}<1.83$ . The staggered double row of vortices observed in the wake of a cylinder appears for $40<Re_{0}<180$ but, as $Re_{0}$ increases, slowly develops perturbations away from the vertical. Vertical motion develops at higher $Re_{0}$ , wakes becoming more fully turbulent at $Re_{0}\sim 700$ . Holford & Linden (Reference Holford and Linden1999a ) show that even when $Re_{0}=1030$ vertical vortices can be identified in the wake.

Although ‘zigzag instability’ was first used to describe the instability of single vertical vortex pairs in stratified fluids, Deloncle, Billant & Chomaz (Reference Deloncle, Billant and Chomaz2011) adopt the term to describe the instability of steady vortex arrays in uniformly stratified fluids, including the vortex street shed by a horizontally moving vertical bar. Deloncle et al. examine the stability of a vortex sheet but not how it is generated by a moving grid. For a horizontal Froude number, $Fr_{h}\equiv |{\it\Gamma}|/(2{\rm\pi}R^{2}N)\leqslant 1$ , where ${\it\Gamma}$ is the circulation of the vortices of radius $R$ , and when the street spacing ratio, ${\it\kappa}$ , equal to the distance between the vortex rows divided by $B$ , is less than 0.41, they show that the vortex street is most unstable to a three-dimensional instability of the zigzag type that vertically bends the vortices. (As Deloncle et al. remark, the spacing ratio, ${\it\kappa}$ , of vortex sheets behind two-dimensional buff bodies is usually less than 0.41.) Numerical and theoretical calculations are made with Reynolds number, $|{\it\Gamma}|/(2{\rm\pi}{\it\nu})$ , equal to $5\times 10^{4}$ (corresponding to an effectively inviscid flow). The vertical wavelengths, ${\it\lambda}_{z}$ , of the fastest growing disturbance in a Kármán vortex street can be derived from Deloncle et al.’s figures 6 and 10. Their non-dimensional measures, $BFr_{h}(2{\rm\pi}/{\it\lambda}_{z})=8.5$ and 5.2 when ${\it\kappa}=0.2$ and 0.4, respectively, and $Fr_{h}=0.1$ , have corresponding ${\it\lambda}_{z}/b=0.37$ and 0.60, respectively, supposing that (as commonly found) $B\approx 5b$ , where $b$ is the width of the vertical bar forming the vortex street. Larger values, ${\it\lambda}_{z}/b=4.9$ and 8.8, are found when ${\it\kappa}=0.2$ , and $Fr_{h}=1$ and 1.2, respectively.

The estimates of ${\it\lambda}_{z}$ do not appear to have been verified experimentally or related to the thickness of layers observed in grid mixing experiments. Bosco & Meunier (Reference Bosco and Meunier2014) combine laboratory and numerical experiments to show that instability, and consequent layers, with a vertical wavelength of approximately $(4\pm 1)b$ occurs in the wake of a steadily moving vertical cylinder of diameter $b$ when $1.0<Fr_{0}<5.2$ and $180<Re_{0}<190$ (a range much lower than that at which Deloncle et al. estimate ${\it\lambda}_{z}$ ). The instability is similar to the classical mode A instability in the unstratified case described by Williamson (Reference Williamson1996). Bosco & Meunier attempted to visualize the zigzag instability but it was not observed, possibly because of the limited dimensions of their laboratory tank. Turning for comparison, to the type (b) experiments: when $Fr_{0}=1.77$ and $Re_{0}=210$ , Park et al. (Reference Park, Whitehead and Gnanadeskian1994, run number 2) find layers of height $h\approx 5.11b$ when a single cylindrical bar moves repeatedly back and forth along the length of a tank. This estimate of $h$ is slightly above Bosco & Meunier’s range of ${\it\lambda}_{z}$ . (Park et al.’s other experiments are at greater $Re_{0}$ , beyond that of Bosco & Meunier.) In two experiments by Holford & Linden (Reference Holford and Linden1999a ) in which a single vertical 2.8 cm square bar moves repeatedly at constant speed through the length of a tank layers are generated with height, $h\approx 3.6b$ . In these experiments $Re_{0}=746$ and 1380 (higher than the range examined by Bosco & Meunier) and $Fr_{0}=1.2$ . Although this value of $h$ is within Bosco & Meunier’s range of vertical wavelengths, another experiment by Holford & Linden with a single square bar has the smaller height, $h\approx 2.3b$ when $Re_{0}=930,Fr_{0}=0.98$ .

In summary, because of the differing ranges of $Re_{0}$ , there is at best only weak evidence that the onset of zigzag instability quantified numerically by Deloncle et al. relates to that found experimentally either by Bosco & Meunier in the wake of steadily moving cylindrical bars, or to layers in the type (b) experiments with a single cylindrical bar by Park et al. or a square bar by Holford & Linden; it has not been established that the instability may be important in the generation of the layers produced by grid mixing. More experiments are needed to quantify the parameter ranges in which zigzag instability occurs and its scale and development.

2.3 The Phillips/Posmentier instability

This instability does not depend on ‘structured turbulence’, e.g. the presence of vortices in the wake of a grid, and nor does the PPI theory provide a prediction of a vertical scale of layers that it causes. A necessary condition for the occurrence of the PPI mechanism is that the vertical density flux generated by turbulence in a stratified fluid decreases as the modulus of the (negative) upward-vertical density gradient increases, a trend physically consistent with vertical motion being inhibited by increasing stratification or larger $|\text{d}{\it\rho}/\text{d}z|$ . An equivalent condition addressed by Holford & Linden (Reference Holford and Linden1999a ) is that the mixing efficiency, defined as the ratio of the change in the mean potential energy of the turbulent fluid to the change in available (potential and kinetic) energy, decreases as the overall (or ‘bulk’) Richardson number increases.

Layers observed in type (b) experiments (table 1 and figure 3) have sometimes been ascribed to PPI without evidence that the condition necessary for PPI is satisfied in the turbulent conditions of the experiments before layers form. Holford & Linden (Reference Holford and Linden1999a ), however, make careful measurements of the buoyancy flux and find that, even when layers subsequently develop, in the initial state the flux increases with density gradient; layers are therefore not initiated by the PPI mechanism. (An anonymous referee suggests that PPI is not detected in the initial evolution of layers because experiments have not, so far, had sufficient resolution to accurately resolve the fluxes in the initial smooth density profile.) A further important conclusion of Holford & Linden is that the flux condition necessary for the PPI mechanism is satisfied in stratified regions once the layers are formed; a layered structure is reinforced by the mechanism (see § 3.3.2). Martin & Rehmann (Reference Martin and Rehmann2006) reach the same conclusion.

Table 3.

Experiments of type (a) with a continuously oscillating grid in a uniformly stratified fluid. The parameters are $M=$ mesh size, $b=$ grid bar diameter/width, $s=$ stroke (as used in appendix A), and $C$ and ${\it\beta}$ are non-dimensional constants defined in (3.1) and (3.2). The values of $a$ , ${\it\omega}$ , $N$ , $y$ (3.4) and $M/{\it\eta}$ (with ${\it\eta}$ given by (3.5)) are those available in figures or tables in the three publications.

2.4 Other mechanisms

There are other mechanisms, described in later sections, by which layers may be generated. Intrusions caused by the collapse of overturns in the turbulent region generated by an oscillating grid (in type (a) and (c) experiments) are discussed in §§ 3.3.1 and 3.4. Two mechanisms in type (b) experiments are further possible candidates: the distortion of vertical vortices by internal waves (§ 4.4) and mixing from the upper and lower boundaries (§ 4.5).

3.1 Local scales of turbulent motion

Three examples of the oscillating grid experiments of type (a) are listed in table 3. In the region close to an oscillating grid where, in comparison with the effects of stratification, mechanically driven turbulence is strong, motion will approximate to the unstratified turbulence generated by biplanar grids with square bars of width, $b$ , investigated by Hopfinger & Toly (Reference Hopfinger and Toly1976). They derive empirical expressions for the r.m.s. vertical velocity fluctuations, $w$ , and the integral length scale, $l$ :

(3.1) $$\begin{eqnarray}w=2^{1/2}C(a{\it\omega})(aM)^{1/2}/({\rm\pi}x),\end{eqnarray}$$

and

(3.2) $$\begin{eqnarray}l={\it\beta}x,\end{eqnarray}$$

at a horizontal distance, $x$ , from the central grid position, where $a$ is the amplitude of the grid motion, $M$ is the grid mesh length $-5$ or 10 cm in Hopfinger & Toly’s experiments – and ${\it\omega}$ is the grid frequency $(\text{rad}~\text{s}^{-1})$ . Values of the non-dimensional coefficients, $C$ and ${\it\beta}$ , are discussed in appendix A. The eddy time scale, $l/w$ , increases as $x^{2}$ . At some distance $x$ , grid-generated eddies will suffer significant collapse as a result of stratification when the eddy time scale, $l/w$ becomes comparable to the buoyancy time scale, $N^{-1}$ , i.e. when the mean local Froude number, $Fr=w/(Nl)$ , reaches a critical value. Using (3.1) and (3.2), $Fr$ can be written

(3.3) $$\begin{eqnarray}Fr=2^{1/2}Cy^{2}/({\rm\pi}{\it\beta}x^{2}),\end{eqnarray}$$

where

(3.4) $$\begin{eqnarray}y=(aM)^{1/4}(a{\it\omega}/N)^{1/2}\end{eqnarray}$$

is a length scale determined by the grid size and motion, and by the initial uniform buoyancy frequency, $N$ .

Figure 4.

Notional sketch of the production and maintenance of intruding layers in type (a) experiments. In this vertical section, the oscillating grid is at the left and the quiescent uniformly stratified fluid to the right. Three layers emerging from the turbulent region and intruding into the quiescent fluid are drawn, A, B and C, with return flows, D and E, between A and B and between B and C, respectively. Vertical profiles of mean density are shown in the quiescent region at (a), near the edge of the turbulent region at (b), within the turbulent region at (c) and close to the oscillating grid at (d), and labelled ${\it\rho}_{a}$ – ${\it\rho}_{d}$ , respectively. The averaged scale, $h$ , is that commonly used to represent layer height. It may, as sketched, exceed that thickness of the regions of constant density in the intrusions emerging at the edge of the turbulent region. The length of vertical arrows at (c) indicate how the magnitude of the mean vertical flux of buoyancy varies with the density gradient, increasing the density in regions of convergence (‘con.’) and decreasing density in regions of divergence (‘div.’), and so tending to maintain the layers of small (negative) vertical density gradient in the regions feeding the intrusions and enhancing the gradients above and below these regions. The sketch differs notably from that of Browand et al. (Reference Browand, Guyomar and Yoon1987, figure 11) and reproduced by Maffioli et al. (Reference Maffioli, Davidson, Dalziel and Swaminathan2014, figure 1) in that the intruding layers are shown to begin within the turbulent region, being initially formed by the collapse of large-scale overturning eddies or ‘blobs’ that are relatively uniform in density. The vertical collapse and horizontal spreading of the blobs is asymmetrical, favouring motion away from the grid, because of the horizontal variation in Reynolds stress across their width. Internal shear waves (not drawn and of undetermined phase in relation to the intrusions; see § 6) are found to the right of the intrusions, propagating ahead of them.

All the experiments of type (a) listed in table 3 find that ${\it\eta}$ , the distance of the edge of the turbulent region from the mid-position of the oscillating grid, and the thickness, $h$ , of intrusions measured at the edge of the turbulent region, are linearly proportional to $y$ , i.e.

(3.5) $$\begin{eqnarray}{\it\eta}={\it\alpha}y\end{eqnarray}$$

and

(3.6) $$\begin{eqnarray}h={\it\gamma}y,\end{eqnarray}$$

where ${\it\alpha}$ and ${\it\gamma}$ are non-dimensional coefficients that vary from one set of experiments to another. (The thickness of intrusive layers is approximately equal to the distance between neighbouring intrusions and, where several adjoining intrusions can be identified in experiments, is conveniently, easily and commonly estimated as the average distance between them. As sketched later in figure 4, however, this scale may exceed that of intrusions at the edge of the turbulent region.) The consistency found between the three experiments suggests that the Hopfinger & Toly scaling, (3.1) and (3.2), is valid for the three different grid types described in table 1, type (a) experiments. Browand et al. (Reference Browand, Guyomar and Yoon1987) find values of ${\it\alpha}$ and ${\it\gamma}$ equal to $1.34\pm 0.27$ and 0.34, respectively. The respective values are 0.94 and 0.34 in Thorpe’s (Reference Thorpe1982) experiments, and 1.34 and 0.32 in Ivey & Corcos’s (Reference Ivey and Corcos1982) (see Thorpe Reference Thorpe1982). Browand et al. use a vertical horizontally oscillating grid with mesh and bar size identical to one used by Hopfinger & Toly. Extrapolating from Hopfinger & Toly’s results, they adopt the values $C=0.29$ and ${\it\beta}=0.26$ . From (3.3) and (3.5), the Froude number at the edge of the turbulent region, $x={\it\eta}$ , can be estimated: $Fr=0.28$ . This is consistent with the observation by Dickey & Mellor (Reference Dickey and Mellor1980) that the character of motion in the wake of a horizontal grid moving vertically upwards in a uniformly stratified fluid changes from turbulence to internal waves where $Fr=0.27$ . Using (3.2), (3.5) and (3.6), at the boundary of the turbulent region, $h/l={\it\gamma}/{\it\beta}{\it\alpha}=1.01\pm 0.20$ , or $h\approx l$ at $x={\it\eta}$ as speculated by Thorpe (Reference Thorpe1982). Values of $C$ and ${\it\beta}$ are unknown for the grids used by Thorpe and Ivey & Corcos but can be inferred as $C=0.204$ , 0.267 and ${\it\beta}=0.37$ , 0.247, respectively by making the similarity assumption that $Fr=0.28$ and $h=l$ at $x={\it\eta}$ .

Table 4.

Values obtained from Browand et al.’s (Reference Browand, Guyomar and Yoon1987) type (a) experiments using $C=0.29$ , ${\it\beta}=0.26$ , ${\it\alpha}=1.34$ , ${\it\gamma}=0.34$ and listed in order of increasing $x/{\it\eta}$ through the turbulent region. ( $x/{\it\eta}=1$ is the location of its mean boundary. $h/L_{O}$ increases with $x/{\it\eta}$ .)

The local Ozmidov length scale, $L_{O}={\it\varepsilon}^{1/2}/N^{3/2}$ , equals

(3.7) $$\begin{eqnarray}L_{O}=2^{3/4}C^{3/2}y^{3}/({\rm\pi}^{3/2}x^{2}{\it\beta}^{1/2}),\end{eqnarray}$$

taking the rate of dissipation of turbulent kinetic energy per unit mass, ${\it\varepsilon}$ , as $w^{3}/l$ . Using (3.3)–(3.6), corresponding values of $x/{\it\eta}$ , $h/L_{O}$ and $Fr$ from Browand et al.’s experiments are found and specified in table 4. The ratio $h/L_{O}$ exceeds 1 at $Fr=1$ and, as shown by (3.3), $Fr$ decreases as $x/{\it\eta}$ increases through the turbulent region. At the edge of the turbulent region, $h$ (and therefore $l$ ), is much greater than the scale of the largest overturning eddies, $L_{O}$ , so that the turbulent motion is largely horizontal.

As conjectured by Maffioli et al. (Reference Maffioli, Davidson, Dalziel and Swaminathan2014) the local Froude number, Fr, provides values that determine the character of the turbulent flow. Further evidence that a Froude number characterises the state of the stratified turbulent motion is provided by the stability curves in figure 3: layers form only when $Fr_{0}$ is less than a value of approximately unity. As outlined in appendix B, the local buoyancy Reynolds number, $I$ , does not provide such a similarity parameter. Moreover, the local Reynolds number, $Re=wl/{\it\nu}$ , is independent of $x$ and therefore also cannot be an effective parameter to define the local state of turbulence.

In summary, the size, $h$ , of layers in the type (a) experiments is given by $h={\it\gamma}y$ , with ${\it\gamma}=0.33\pm 0.01$ . The bar scales of the three experiments listed in table 1, $b=1.0~\text{cm}$ , 0.64 cm and 1.27 cm, respectively, do not appear to provide the size or scaling of the layer thickness, $h$ , that might be expected if the layers result from zigzag instability of a vortex street (e.g. $h\approx 4b$ ; § 2.2). Indeed how the instability of a vortex pair or vortex street might apply to the disorganized mêlée of vortices shed by a rapidly oscillating grid is not obvious.

3.2 The origin of layers

3.3 Sustaining layers within the turbulent region

3.4 The speed of intrusions and internal waves

3.4.4 The speed of the intrusions in the grid mixing experiments

Thorpe (Reference Thorpe1982) and Browand et al. (Reference Browand, Guyomar and Yoon1987) find that the mean non-dimensional speed of intrusions, d $l_{i}$ /d $t$ , is approximately given by

(3.8) $$\begin{eqnarray}(1/Nh)\text{d}l_{i}/\text{d}t={\it\chi}(Nt)^{-q},\end{eqnarray}$$

where $l_{i}$ is the length of an intrusion and $h$ is the intrusion height. In Browand et al.’s experiments the non-dimensional coefficient ${\it\chi}$ varies from approximately 0.49 to 1.0 as the exponent, $q$ , varies from 0.3 to 0.5 when $Nt$ exceeds a value of approximately 4 and before internal waves reflect from the end of the tank. In Thorpe’s experiments ${\it\chi}\sim 0.69$ and $q\sim 0.56$ when $2\leqslant Nt\leqslant 20$ . These values give intrusion speeds, $\text{d}l_{i}/\text{d}t$ , ranging from $0.25Nh$ to $0.32Nh$ in Browand et al.’s experiments and equal to $0.19Nh$ in Thorpe’s, at the non-dimensional time, $(Nt)_{1}\approx 10$ (table 1), when intrusions are observed to emerge from the turbulent region. (Greater speeds are observed at smaller $Nt$ .) These speeds exceed Maurier & Linden’s estimate, $0.125Nh$ , of the speed of intermediate intrusions confined between levels separated in the vertical by $h$ (§ 3.4.2). They are indicative of different forcing processes. Unlike the constant speed intrusions formed by lock-breaking and continually supplied by fluid from behind the ‘lock’, or sustained by mixing from the mean level of a vertically oscillating horizontal grid (De Silva & Fernando Reference De Silva and Fernando1998), the type (a) intrusions continually decelerate (as in (3.8)). This is consistent with their formation being transient and not sustained as in the ‘lock-break’ scenario.

The initial speeds of advance of intrusions from the turbulent region, ranging from $0.19Nh$ to $0.32Nh$ , exceed the horizontal group velocity, $c_{gx}=N/m=Nh/2{\rm\pi}\approx 0.16Nh$ , of internal shear waves with vertical wavelength, $h$ , which therefore cannot propagate ahead of the intrusions. The advance of a vertical array of intrusions separated by a vertical length, $h$ , will however generate a spectrum of internal waves. It is shown in appendix D that only waves with vertical scale equal to or exceeding $2h$ can propagate ahead of the intrusions. This is in accordance with the observed wavelengths, ${\it\lambda}_{z}\sim 2h$ (§ 3.4.1). Because intrusion speeds decrease with $Nt$ , however, waves of vertical scale $h$ are able to advance faster than the intrusions when $Nt$ is sufficiently large. Like the intrusions themselves, the scale of the waves dominant at large times must be related to the presently unknown spatial and temporal distribution of collapsing mixed regions within the turbulent region.

3.5 Summary of §§ 3.1–3.4

It is yet to be shown whether zigzag instability can generate the layers observed in the type (a) experiments (§ 3.1), but it appears unlikely except perhaps as a grid of vertical bars is first set into motion (§ 2.1). Holford & Linden (Reference Holford and Linden1999a ) find that the PPI mechanism may help sustain, but not cause, layers in the turbulent region. The local Froude number, $Fr$ , characterises the variation of the state of turbulence with distance from its generation by the moving grid. The height of overturns, $L_{max}$ , well within the turbulent layer (e.g. at $x/{\it\eta}\sim 0.4$ ) substantially exceeds the layer height, $h$ (§ 3.2.2). Overturns are consequently sufficiently large to produce intruding layers of the height, $h$ , observed at the edge of the turbulent region. In § 3.2.1 it is shown that the variation in time of the intrusions’ horizontal extent is consistent with their starting at about the time when the grid is first set in motion. There is a similarity of layers on the two sides of a grid oscillating in the centre of a tank (§ 3.2.2). These conclusions imply that layers in continuously forced, type (a) (and possibly transiently forced, type (c)) experiments are formed soon after the grid stirring begins and may have connection through an oscillating grid.

In § 3.4 it is found that the leading internal waves propagating ahead of the intrusions have a vertical wavelength of approximately $2h$ , twice the thickness of the layers; waves of smaller vertical scale do not travel fast enough to advance more rapidly than the intrusions when they are first observed, emerging from the turbulent region. The sometimes remarkable vertical periodicity and regularity of intrusions remains unexplained.

4.1 Stability curves

Three experiments of type (b) in which the whole tank is stirred are listed in table 1. Layers are not observed until after stirring has continued for times exceeding some thousands of buoyancy periods. The stability curve determined empirically by Park et al. (Reference Park, Whitehead and Gnanadeskian1994) is shown in figure 3, together with a neighbouring (hatched) region in which they report that it is uncertain whether layers are formed or not. The stability curve separates two regimes recognised in their experiments: a regime of sufficiently low $Re_{0}$ and low $Fr_{0}$ in which layers are observed and a regime at higher $Re_{0}$ and $Fr_{0}$ in which no layers appear. Parameter values of two of Park et al.’s experiments, one in each regime, are given in table 1 and shown as circles in figure 3. The experiments in which layers are observed by Ruddick et al. (Reference Ruddick, Mcdougall and Turner1989) lie within Park et al.’s layer regime even though the apparatus and experimental details differ (e.g. Park et al. have one stirring rod, Ruddick et al. have many). Ruddick et al.’s experiment in which no layers form (point marked as ‘A’ in figure 3) lies very close to the (imprecise) position of the stability curve, consistent with the expectation from Park et al.’s study that the turbulent flows are at best weakly unstable.

As mentioned in § 2.3, Holford & Linden (Reference Holford and Linden1999a ) find that PPI is not a cause of layers. Rather they conclude that vertical mixing resulting from bending of vertical vortices shed by the moving grid bars causes the layers. Their explanation does not, however, account for the height of observed layers. They establish an empirical stability curve dividing the regions in which layers are formed from that in which they are absent. The curve depends on the grid scale parameter $S$ . Two of the curves are shown in figure 3. These correspond to the value $S=0.39$ in Holford & Linden’s experiments referred to in table 1 and shown as points in figure 3, and to $S=0.226$ corresponding to Park et al.’s experiments (although there, with only one stirring bar, the mesh size, $M$ , is uncertain. It is somewhat arbitrarily taken as the tank width. This may account for the difference between the Park et al.’s stability curve and Holford & Linden’s curve $S=0.226$ ).

4.2 The effect of turbulence ahead of the grid

Ambient turbulence in the tank, encountered and supplemented by the stirring grid after its first cycle, may affect the formation of vortices in the grid wake which are found by Holford & Linden to be necessary for the generation of layers. Turbulence produced by the moving grid decays in time. The repetition period of stirring by the cycling grid depends on position along the length of the tank. At the centre this period is $L_{s}/U$ , where $L_{s}$ is the horizontal length of grid motion, and at the ends of the grid stroke it is $2L_{s}/U$ . The r.m.s. time period is $T_{\ast }=\surd (8/3)(L_{s}/U)$ . (This is found by integrating and summing the squares of $X$ and $(1-X)$ , where $XL_{s}$ is the distance along the tank from the end of a grid stroke, then taking the square root.) $NT_{\ast }$ characterises the non-dimensional mean time between grid stirring events. Suppose that the processes producing layers as the grid passes by (e.g. those to be described in § 4.4) are only effective if the turbulence left by previous cycles of the grid has decayed sufficiently. Marginal conditions for layer generation exist when the time, $T_{d}$ , for turbulence to collapse is equal to $T_{\ast }$ , i.e. when

(4.1) $$\begin{eqnarray}NT_{d}=\surd (8/3)(NL_{s}/U),=\surd (8/3)(L_{s}/b)Fr_{0}^{-1}.\end{eqnarray}$$

If the non-dimensional time for the decay of turbulence in a stratified fluid is known as a function of $Re_{0}$ this equation defines the stability boundary separating conditions in which layers can or cannot be formed. Values of $NT_{d}$ smaller than $\surd (8/3)(L_{s}/b)Fr_{0}^{-1}$ imply conditions in which turbulence has decayed substantially between grid strokes so that the processes leading to layers can operate, unaffected by the residual turbulence. Alternatively, given the relation between $Re_{0}$ and $Fr_{0}$ defined by the stability curves of figure 3 and values of $L_{s}/b$ in the related experiments, (4.1) can be used to determine the value of $NT_{d}$ as a function of $Re_{0}$ for which marginal conditions are found.

Table 5.

The variation of non-dimensional turbulent decay times, $NT_{d}$ , with $Re_{0}$ , inferred from (4.1) and the stability curves of figure 3, in type (b) experiments.

Points at various values of $Re_{0}$ (with corresponding values of $Fr_{0}$ ) on the stability curves of Park et al. (Reference Park, Whitehead and Gnanadeskian1994) and Holford & Linden (Reference Holford and Linden1999a ) shown in figure 3 are selected. Table 5 gives values of the non-dimensional turbulence collapse time, $NT_{d}$ , determined using (4.1) for which conditions are marginal and turbulence does not affect the process of layer formation. These values increase with $Re_{0}$ . They span the times $(Nt)_{1}\sim 10$ for the emergence of layers from the turbulent region in type (a) experiments, and those of numerical experiments mentioned in § 1.3 and listed in table 3. In the latter $Nt\sim 2{\rm\pi}$ and 10 are found by Venayagamoorthy & Stretch (Reference Venayagamoorthy and Stretch2006) and by Maffioli et al. (Reference Maffioli, Davidson, Dalziel and Swaminathan2014, and private communication with Dr A. Maffioli, June, 2015) to characterise the cessation of overturning and the suppression of turbulent mixing when $Re_{0}\sim 4000$ and 8000, respectively. Diamessis & Nomura (Reference Diamessis and Nomura2004) find $Nt=4.5$ when the volume fraction of overturns tends to zero, when $Re_{0}\sim 600$ .

These times imply that in the type (b) experiments the production of overturning eddies by the turbulence virtually ceases between successive mixing by grids in those type (b) experiments listed in table 1 in which layers are formed. For example, selecting two type (b) experiments listed in table 1 where stirring produces layers, the time $\surd (8/3)(L_{s}/b)Fr_{0}^{-1}$ is 14.6 in Park et al.’s figure 8 at $Re_{0}=548$ , and 16.8 in Holford & Linden’s figure 6 at $Re_{0}=1100$ . Both exceed values of the non-dimensional turbulent decay times, $NT_{d}$ , in table 5. In these experiments, on average the grids advance into relatively weak residual turbulence left from their last cycle. Coherent vortices (§ 2.2) shed by the grid may then survive in the turbulent motion left from earlier grid stirring. In contrast, layers are not observed in conditions in which a vortex wake may be suppressed by turbulence still active from earlier grid stirring.

Figure 5.

The variation of $h/b$ with $U/Nb\,(=Fr_{0})$ for type (b) experiments. Points (●) are from Ruddick et al. (Reference Ruddick, Mcdougall and Turner1989), crosses ( $\times$ ) are from Park et al. (Reference Park, Whitehead and Gnanadeskian1994) and open circles (○) are from Holford & Linden (Reference Holford and Linden1999a ). The fitted long dashed line is $h/b=3.03U/Nb$ , and the short dashed lines, $(3.03\pm 0.3)U/Nb$ , (4.4), indicate its possible uncertainty. The full line is the relation (4.6) derived in § 4.4 in which vortices shed by the grid bars are supposed to be bent by internal waves moving with the grid at speed $U$ . The dotted curve is the best fit, (5.1), determined by Maffioli et al. (Reference Maffioli, Davidson, Dalziel and Swaminathan2014) from their laboratory experiments of type (c), although these are in the range $16<U/Nb<74$ , beyond that of the type (b) experiments (see figure 6). The stippled region indicates the range of $h/b$ observed by Bosco & Meunier (Reference Bosco and Meunier2014).

4.3 The layer height

Layer heights in type (b) experiments are found to be related to $U/N$ . Park et al. (Reference Park, Whitehead and Gnanadeskian1994), stirring with single vertical cylindrical bars of diameter, $b=1.29$ , 2.26 and 3.37 cm, find

(4.2) $$\begin{eqnarray}h~(\text{cm})=2.6U/N+1~\text{cm},\end{eqnarray}$$

independent of $b$ , with $h$ , $U$ and $N$ in cgs units, over a range of $U/N$ from approximately 0.5 to 3 cm. Holford & Linden (Reference Holford and Linden1999a ) use a variety of grids composed of 1–8 square vertical bars, $b$ ranging from 0.65 to 3.15 cm. They conclude

(4.3) $$\begin{eqnarray}h=3.07U/N,\end{eqnarray}$$

again independent of $b$ , in the range of $U/N$ from approximately 0.7 to 4.2 cm. (As mentioned in § 1.2, Ruddick et al. (Reference Ruddick, Mcdougall and Turner1989) have a complex stirring device with cylindrical bars of diameter 1.2 cm. Given values of $h~(\text{cm})$ are $2.72U/N$ for $U/N=0.73~\text{cm}$ and $3.27U/N$ for $U/N=0.64~\text{cm}$ , fairly close to and bracketing values in (4.3). Here $U$ is taken as $a{\it\omega}/\surd 2$ where ${\it\omega}$ is the grid frequency and $a$ is the amplitude of the grid motion.) Figure 5 is a graph showing the variation of $h/b$ with $U/Nb$ for the type (b) experiments. There is general alignment of points between the lines

(4.4) $$\begin{eqnarray}h/b=(3.03\pm 0.3)U/Nb\end{eqnarray}$$

(or $h/b=(3.03\pm 0.3)Fr_{0}$ ).

The vertical scale of the instability in the wake of a horizontally moving vertical cylinder found by Bosco & Meunier (Reference Bosco and Meunier2014) to be $(4\pm 1)b$ , independent of Froude number, is shown by the stippled region of figure 5. It is inconsistent with the type (b) experiments. As shown in figure 3, however, Bosco & Meunier’s study is at lower Reynolds number than that of the type (b) experiments. Further study is merited.

4.4 Internal waves and vortices

Although internal waves are not reported in the long-running experiments of type (b), it is probable that they are generated, particularly in the relatively low forcing conditions (larger $NT_{d}$ ) when layers are observed and the turbulence in the fluid into which the grid moves is relatively weak. Suppose that the motion of the grid through the tank at speed $U$ generates horizontally travelling internal waves. The optimally generated waves are those with group speed, $c_{gx}=N/m$ (and, as shown in appendix C, an equal phase speed), that match (and remain in phase with) the grid: $c_{gx}=U$ , where $m=2{\rm\pi}/{\it\lambda}_{z}$ is the vertical wavenumber and ${\it\lambda}_{z}$ is the vertical wavelength of the waves. The latter is therefore given by

(4.5) $$\begin{eqnarray}{\it\lambda}_{z}=2{\rm\pi}U/N,\end{eqnarray}$$

provided that vertical wavelengths of appropriate scale are excited. Using the relation between $U/Nb$ and $h/b$ (4.4) of figure 5 it follows that ${\it\lambda}_{z}=(2.07\pm 0.21)h$ ; internal shear waves that propagate with the speed of the grid have a vertical wavelength of approximately twice the observed layer height.

Figure 6.

The variation of $h/b$ with $U/Nb\,(=Fr_{0})$ for (i): the type (a) experiments by Thorpe (Reference Thorpe1982) indicated by the hatched range, including the two referred to in table 3 shown by ●, and Browand et al. (Reference Browand, Guyomar and Yoon1987), marked by bold range bars (with an off-scale point at $h/b=8.7,U/Nb=180$ omitted); (ii) the type (b) experiments (indicated by the dashed line at relatively small $U/Nb$ and shown in greater detail in figure 5); and (iii) type (c) experiments, where the circles (○) are points from Maffioli et al.’s (Reference Maffioli, Davidson, Dalziel and Swaminathan2014) experiments fitted by the long-dashed line, (5.1), and where the vertical bars indicate uncertainty, and the crosses ( $\times$ ) are from the experiments by Liu et al. (Reference Liu, Maxworthy and Spedding1987) (see appendix E).

Holford & Linden (Reference Holford and Linden1999b ) describe the development of vertical vortices in the lee of eight vertical bars with $b=0.65~\text{cm}$ in type (b) experiments. They observe that in experiments with $Re_{0}<140$ and, approximately $1.0<Fr_{0}<1.9$ , the vortices are bent in the $x$ -direction of the grid motion, becoming sinusoidal in the vertical. The effect of bending the vertical vortices is to produce a horizontal component of vorticity in the $x$ -direction, the maximum (positive) and minimum (negative) horizontal components of $x$ -vorticity in each vertical wavelength both tending to overturn isopycnals. This leads to layers with a vertical height half that of the bent vortices.

The possible existence and effect of shear waves is not recognised by Holford & Linden. The agreement of scales of layers and waves supports the conclusion that the mechanism causing the layers is an interaction of the vertical vortices and the internal shear waves generated by the moving vertical grid bars. As noted in § 3.4.1, the internal waves generate two shear layers in each vertical wavelength, one of positive horizontal vorticity and one negative. Bending the vertical vortices, these shear layers result in mixed layers of height $h={\it\lambda}_{z}/2$ or, from (4.5),

(4.6) $$\begin{eqnarray}h/b={\rm\pi}Fr_{0},\end{eqnarray}$$

implying a value of $h/b$ well within the scatter of observed points in figure 5.

What happens when the grid bars reverse their horizontal motion at the two ends of the tank? The conditions at the upper and lower ends of the bars determine the phase of waves and the location of associated shear layers along the vertical bars. The phase can adjust by half a vertical wavelength when the direction of bar motion reverses; the particle motions within the shear layers composing the waves continue to be forced even though the direction of grid motion and wave propagation is reversed at the end of each grid stroke. Since layers are formed by mixing at the levels where the internal wave shear is a positive or negative maximum, the mean horizontal flow induced there by the internal waves is zero; the vertical phase of the horizontal flow induced by the waves is ${\rm\pi}/2$ out of phase with that of the shear. It is therefore not expected that the layers will be in relative motion. (No examination of the relative mean flow in the layers appears to have yet been made to confirm or refute this conclusion.)

The long times, $(Nt)_{1}$ , required for the formation of layers in the type (b) experiments (table 1) are presumably a consequence of the relatively weak forcing of internal waves by the grids. The effect of molecular viscosity, ${\it\nu}$ , and diffusivity, ${\it\kappa}$ , on the internal shear waves is very small, tending slightly reduce their horizontal phase speed and to damp them by a factor $\exp (-{\it\delta}Nt)$ , where ${\it\delta}=\,({\it\nu}+{\it\kappa})m^{2}/2N$ (Ostrovsky & Zaborskikh Reference Ostrovsky and Zaborskikh1996); ${\it\delta}$ is of order $10^{-4}$ for typical values of ${\it\nu}$ , ${\it\kappa}$ , $m$ and $N$ in the experiments. Residual turbulent motions are relatively weak ahead of the grid when layers are formed, but their dissipative, diffusive and dispersive effects on the internal waves are presently unknown.

4.5 Boundary-driven layering

Dr Holford and Professor Linden have provided a copy of a draft referred to in Holford & Linden (Reference Holford and Linden1999a ) as Holford & Linden (Reference Holford and Linden1999b ) that was submitted to ‘The Physics of Fluids’ but not published. In it they describe how, when $90<Re_{0}<300$ and $2.7<Fr_{0}<7.1$ , layers develop sequentially over a time period of typically an hour $(Nt\sim 6\times 10^{3})$ from the upper and lower boundaries of the laboratory tank in their type (b) experiment with $b=0.65~\text{cm}$ . This is a parameter regime in which figure 3 indicates that layers are unlikely to be formed in the other type (b) experiments. The layer height is $(1{-}2)U/N$ , less than that in (4.2) and (4.3), and therefore indicative of a different form of layer formation. (Williamson Reference Williamson1996, described the local distortion of vortex streets near the ends of a long moving cylinder in an unstratified fluid. Because of buoyancy effects, such distortion is unlikely to be a significant effect in the stratified fluids considered here.)

4.6 Summary of §§ 4.1–4.5

Layers in type (b) experiments are formed only in conditions in which overturning within the decaying turbulent flow left by one stroke of the moving grid has virtually ceased before the next arrives. The height, $h$ , of layers is consistent with their generation through the bending of vertical wake vortices by internal shear waves moving at the speed of the grid. Once formed, layers may be supported by the PPI mechanism (Holford & Linden Reference Holford and Linden1999a ).

Figure 7.

The variation in the mean height, $h$ , of the intrusions with the length scale, $y$ . Open circles (○) are points determined by Thorpe (Reference Thorpe1982, taken from his figure 6b) in type (a) experiments, and closed circles (●) from Maffioli et al. (Reference Maffioli, Davidson, Dalziel and Swaminathan2014, their figure 15) in type (c) experiments using $y=(aM)^{1/4}(U/N)^{1/2}$ where $U$ is the steady grid speed and $a$ the amplitude of grid motion. Points ( $\times$ ) are from Liu et al.’s (Reference Liu, Maxworthy and Spedding1987) type (c) experiments (see appendix E). The line is $h=0.34y$ .

5.1 The layer height

In these experiments a grid is moved for a short time through a uniformly stratified layer, producing a turbulent cloud that then collapses. In the experiments by Liu et al. (Reference Liu, Maxworthy and Spedding1987) and Maffioli et al. (Reference Maffioli, Davidson, Dalziel and Swaminathan2014) the grid moves through only half of the tank length. As turbulence decays, layers appear as intrusions into the quiescent fluid from the boundary of the turbulent layer. Maffioli et al. use a grid of vertical square bars with $b=0.64~\text{cm}$ and find

(5.1) $$\begin{eqnarray}h/b=2.4Fr_{0}^{0.313}\end{eqnarray}$$

is a best fit to data, with $16<Fr_{0}<74$ (or approximately $5.7~\text{cm}<h<9~\text{cm}$ ). As shown in figure 5, (5.1) differs from the linear relationships of the type (b) experiments both in shape and in their ranges; the range of $Fr_{0}=U/Nb$ is approximately 0.4–1.8 in the type (b) experiments, much less than the range of $Fr_{0}$ in Maffioli et al.’s experiments. Considerable scatter of points is found in figure 6 where $h/b$ is plotted versus $U/Nb\,(=Fr_{0})$ using data from the type (a) experiments of Thorpe (Reference Thorpe1982) and Browand et al. (Reference Browand, Guyomar and Yoon1987), the type (c) experiments of Liu et al. and Maffioli et al., and the type (b) experiments (shown as the dashed line $h/b=3.03U/Nb$ taken from figure 5). No universal relation between $h/b$ and $U/Nb$ (or $y/b$ ) appears to exist; the scaling of the layer heights depends on the different processes through which layers are formed or sustained in each type of experiment.

Maffioli et al. argue that turbulent mixing causes the formation of mixed packets of fluid within the turbulent cloud that collapse producing the laterally spreading intrusions, much as described in § 3.3 in relation to the type (a) experiments. There is a further similarity: if the factor $a{\it\omega}$ in the expression (3.4) for $y$ is replaced by the steady grid speed $U$ in Maffioli et al.’s type (c) laboratory experiments, there is a fair agreement between Maffioli et al.’s observed intrusion heights with those of the type (a) experiments, as shown in figure 7. Appendix E is a note explaining how points in in figures 6 and 7 from Liu et al. (Reference Liu, Maxworthy and Spedding1987) are determined.

The similarity of the scale of $h$ in type (c) transient turbulence experiments with that found in the type (a) maintained mixing experiments lends support to the conclusions of § 3.2.1 that layers are formed, and their vertical scale is determined, at a very early stage of grid motion.