2.1 The breaking wave model

A simple but precise representation of finite amplitude progressive internal waves is adopted to describe the flow field just as it becomes unstable and mixing commences: in the Boussinesq approximation, an exact solution of the governing equations is available that describes internal waves of frequency, $\unicode[STIX]{x1D70E}$ , propagating through a uniformly stratified fluid of constant buoyancy frequency, $N$ , and density $\unicode[STIX]{x1D70C}_{0}(1-N^{2}z/g)$ and zero mean flow (Thorpe Reference Thorpe2005; § 2.3.2). The horizontal, $x$ , $y$ , and vertical, $z$ , components of velocity are

respectively, where the wavenumber is ( $k,m$ ), $\unicode[STIX]{x1D701}=(kx+mz-\unicode[STIX]{x1D70E}t)$ is the wave phase, $a$ is the wave amplitude, $s=am$ is defined as the wave steepness, $\unicode[STIX]{x1D70C}_{0}$ is the mean fluid density and $f$ is the Coriolis frequency. The maximum slope of isopycnals is $s\tan \unicode[STIX]{x1D703}/(1-s)$ , where $\tan \unicode[STIX]{x1D703}=k/m$ . In the wave-perturbed flow the density is

and

An example of the waves is shown in figure 1. The fluid density and velocity fields are similar on surfaces of constant phase, $\unicode[STIX]{x1D701}$ , lying at angle $\unicode[STIX]{x1D703}$ to the horizontal.

Figure 1.

Isopycnal surfaces perturbed by an internal wave with $s=1.5$ and $\unicode[STIX]{x1D703}=30^{\circ }$ , those surrounding $x=z=0$ representing the local density field at the onset of convective instability and turbulence. Stippled regions show sloping regions of wave overturn and static instability occurring over a height, $h_{\ast }$ , that are produced in the density field by wave motions when $s>1$ . The directions of the phase, $\boldsymbol{c}$ , and group velocity, $\boldsymbol{c}_{g}$ , are indicated. The horizontal extent of overturns is $l$ .

For simplicity (and until the constraint is relaxed in § 5) it is now assumed that the waves representing the local flow in which turbulence is about to occur are all of the same steepness, $s$ , and frequency, $\unicode[STIX]{x1D70E}$ ; in this model only such waves contribute to mixing. To cover their full range of variation, it is generally sufficient to examine the variation of critical features than control the stability of the flow (such as the slope of isopycnals or Richardson number) with the phase, $mz$ , over one vertical wavelength at $x=0$ and $t=0$ . This is appropriate provided that the wave-induced flow represents the local flow field at the onset of instability. If $s>4.712$ , the maximum density in the wave field near $x=0$ may be exceeded in an overlying location (above a height $2\unicode[STIX]{x03C0}/m$ , Thorpe (Reference Thorpe1994b ); see § 3.1 later) and unless the actual flow has similar properties, the local fit is appropriate only within one wavelength surrounding $x=t=0$ .

The extreme local values of isopycnal slope or small Richardson number that result in the breaking of individual waves advance through the fluid at the velocity, $\boldsymbol{c}_{b}=\boldsymbol{c}+\boldsymbol{c}_{g}$ (Thorpe Reference Thorpe1999a ), and $\boldsymbol{c}_{b}$ is generally different from either the phase velocity, $\boldsymbol{c}$ , or the group velocity, $\boldsymbol{c}_{g}$ , which are at right angles to one another (Mowbray & Rarity Reference Mowbray and Rarity1967). The magnitude of the speed of advance of the internal wave breaking region is found to be $c_{b}=N/K$ , where $K=(k^{2}+m^{2})^{1/2}$ is the wavenumber of the breaking wave, or

The angle, $\unicode[STIX]{x1D703}$ , is the inclination of the group velocity to the horizontal as shown in figure 1. The wave frequency is given by the dispersion relation,

where $f$ is the Coriolis parameter. The vector $\boldsymbol{c}_{b}$ is inclined at an angle $\unicode[STIX]{x1D703}_{b}=-\tan ^{-1}(F^{2}\cot \unicode[STIX]{x1D703})$ to the horizontal, where $F=f/N$ . The breaking region is consequently nearly horizontal if $F^{2}\ll \tan \unicode[STIX]{x1D703}$ or when $\unicode[STIX]{x1D70E}\gg f$ . The wave-induced motion is then predominantly in the $x$ – $z$ plane, the case considered in § 3. For the near-inertial waves with $\unicode[STIX]{x1D70E}\approx f$ considered in § 4, the angle, $\unicode[STIX]{x1D703}$ is small and $\unicode[STIX]{x1D703}_{b}$ may differ substantially from zero.

2.2 The energy of the internal waves

The energy in the wave field is of importance because, in the process of breaking, it provides the energy of the turbulent flow. The potential energy of the wave-disturbed density field described by (2.2) relative to $z=0$ integrated over one vertical wavelength and per unit horizontal area at the horizontal location and time of breaking, $x=t=0$ , is

The terms in the last expression contain two contributions,

from the ambient density, $\unicode[STIX]{x1D70C}_{0}[1-(N^{2}z/g)]$ , within which a wave is propagating, and

derived from the wave-induced perturbation of density. In the Boussinesq approximation, the corresponding local kinetic energy of the wave is

there being none from the (zero) mean flow.

The kinetic energy of the internal waves per unit mass (or the kinetic energy density) is found by integration over one vertical ( $-\unicode[STIX]{x03C0}/m$ to $\unicode[STIX]{x03C0}/m$ ) and one horizontal (0 to $2\unicode[STIX]{x03C0}/k$ ) wavelength at $t=0$ and division by $\unicode[STIX]{x1D70C}_{0}$ :

The potential energy density per unit mass is

where $\unicode[STIX]{x1D702}(x,z_{0})$ is the displacement of isopycnals from their level $z_{0}$ in the ambient density profile, $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{0}(1-N^{2}z/g)$ , and is given by

(Thorpe Reference Thorpe1968a ). The mean wave energy per unit mass is

The ratio of the potential energy density to the kinetic energy density, $R$ , is shown in figure 2 as a function of $s$ when $F=0$ . For small $s$ there is equipartition of energy ( $R\approx 1$ ; Yih Reference Yih1960) but when $s>1$ and isopycnals are folded and dense fluid is carried over less dense as illustrated in figure 1, the potential energy begins to significantly exceed the kinetic. The potential energy is independent of $F$ but the kinetic energy varies with $F$ as in (2.10), and $R$ is consequently a function of $F$ (Gill Reference Gill1982; Thorpe & Umlauf Reference Thorpe and Umlauf2002).

Figure 2.

The ratio of the wave potential energy density to the kinetic energy density, $R$ , as a function of $s$ when $F=0$ .

These contributions to the available energy will later be used to assess the relative magnitude of energy dissipated in wave breaking following overturn of isopycnals or at small Richardson number (resulting in KHI) in the exact wave solution.

The potential energy of the wave disturbance (2.8) is linearly proportional to the wave steepness, $s$ , or wave amplitude, $a$ . The gain in potential energy per unit wavelength and width in the $y$ -direction caused by the presence of the waves, described by (2.11), is however proportional to the vertical integral of $\langle \unicode[STIX]{x1D702}^{2}\rangle$ , the horizontal average of the square of the vertical displacement of density, $\unicode[STIX]{x1D702}$ , from its level in the mean density field, $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{0}(1-N^{2}z/g)$ (Yih Reference Yih1960; Thorpe Reference Thorpe1968a ), and is therefore proportional to $a^{2}$ . This difference in dependency on $a$ is because the local flow field prior to the onset of turbulence is here represented solely by the extreme flow, either of the overturning isopycnals in the case of convective instability or of Richardson number in the case of KHI, within the wave represented by the exact wave solution, and not by some average of its spatial and temporal variation. Furthermore, the representative waves will generally be of finite amplitude, and measures derived from linear small amplitude estimates of the waves, such as the wave energy density, $\unicode[STIX]{x1D6FA}=\unicode[STIX]{x1D70C}_{0}a^{2}\unicode[STIX]{x1D70E}^{2}(N^{2}-f^{2})/[2(\unicode[STIX]{x1D70E}^{2}-f^{2})]$ (Thorpe Reference Thorpe2005; equation (2.20)), and the ratio of potential to kinetic energy, $R=[1+2(F\cot \unicode[STIX]{x1D703})^{2}]^{-1}$ (Thorpe & Umlauf Reference Thorpe and Umlauf2002) are no longer valid.

2.3 The adopted methodology

Our approach is to calculate the change in the energy per unit horizontal area during mixing at the location where the flow is represented by the finite amplitude internal wave solution, and hence to estimate the loss of energy to turbulence. For each type of breaking wave, it is supposed that when some extreme value of $s$ is reached mixing, with conservation of mass and momentum, sets in instantaneously over a vertical range $z_{1}\leqslant z\leqslant z_{2}$ , a range that depends on the type of wave breaking, convective or shear, as specified later. This leads to a mean transfer of energy into turbulence per unit horizontal area, $E$ (i.e. a mean transfer of energy, $E/(z_{2}-z_{1})$ , per unit volume in $z_{1}\leqslant z\leqslant z_{2}$ ). It is further supposed that negligible energy is radiated from the breaking region by upward or downward propagating internal waves and that breaking events are discrete. The mean rate of dissipation of turbulent kinetic energy per unit mass in the turbulent region produced by wave breaking is therefore

where it is supposed that the turbulence generated by the wave breaking decays in a time $rN^{-1}$ , where $r$ is a non-dimensional constant. The turbulent region increases in volume with a speed $c_{b}$ , and hence the rate of production of turbulent kinetic energy per unit mass by a single breaker increases at a rate $E\boldsymbol{c}_{b}/\unicode[STIX]{x1D70C}_{0}(z_{2}-z_{1})$ .

This approach may be compared to that adopted in studies of breaking deep-water surface waves by Pizzo & Melville (Reference Pizzo and Melville2016) and Pizzo, Deike & Melville (Reference Pizzo, Deike and Melville2016), where analysis and numerical modelling benefit from being supplemented by laboratory experiments and field observations. It is found that 95 % of the energy lost from the breaking surface waves is dissipated in the first 4 wave periods after the breaking event, little energy is radiated (rather than transmitted) from the breaking of focusing wave groups, but a robust coherent vortex is formed that lasts for more than 50 wave periods after breaking (Rapp & Melville Reference Rapp and Melville1990; Melville, Veron & White Reference Melville, Veron and White2002; Pizzo & Melville Reference Pizzo and Melville2013). The energy flux in the vortex however amounts to only 2–4 % of the dissipated energy flux in the breaking wave (Pizzo et al. Reference Pizzo, Deike and Melville2016). A note about the vortices produced by the breaking internal waves is provided in appendix G, but henceforth any quasi-steady vortices generated by breaking internal waves are neglected (or regarded as part of the turbulent motion).

2.4 The breaking parameter $\unicode[STIX]{x1D6EC}_{I}$

Following Phillips’ (Reference Phillips1985) use of a parameter usually denoted by $\unicode[STIX]{x1D6EC}$ to describe the area of breaking waves on the ocean surface we adopt an internal wave breaking parameter, $\unicode[STIX]{x1D6EC}_{I}$ , defined as the mean area of the fronts of breaking regions per unit volume. (This differs from Thorpe (Reference Thorpe2010b ), who describes the parameter as the area projected onto the vertical.) The mean production of turbulent kinetic energy per unit mass of the fluid by the representative breaking waves is therefore

with $c_{b}$ given by (2.4). In a steady state in which the representative wave breaking provides all the turbulence this is equal to the mean rate of dissipation of turbulent kinetic energy per unit mass, denoted by $\unicode[STIX]{x1D700}$ .

The parameter, $\unicode[STIX]{x1D6EC}_{I}$ , is yet to be established by direct observation in the ocean. The dissipation rate, $\unicode[STIX]{x1D700}$ , is, however, related to $K_{\unicode[STIX]{x1D70C}}$ , the diapycnal diffusivity of density:

(Osborn Reference Osborn1980), where $\unicode[STIX]{x1D6E4}$ is an efficiency factor. Although subject to further investigation and debate (e.g. see Arneborg (Reference Arneborg2002), Thorpe (Reference Thorpe2005), § 6.7; Chalamalla & Sarkar (Reference Chalamalla and Sarkar2015), Salehipour et al. (Reference Salehipour, Peltier and Mashayek2015)), for simplicity the commonly used value, $\unicode[STIX]{x1D6E4}=0.2$ , is adopted here. If $K_{\unicode[STIX]{x1D70C}}$ is known and $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}_{I}$ , equations (2.15), (2.16) and (2.4) lead to

Since turbulence generated by wave breaking decays in a time $rN^{-1}$ , the length of the turbulent regions generated by the wave breaking is

and the fraction of the water column in which turbulence occurs is therefore

It follows from (2.14), (2.15) and (2.19) that

as expected, since $\unicode[STIX]{x1D700}_{m}$ is the dissipation rate in the active turbulent regions and $\unicode[STIX]{x1D700}_{I}$ is the mean dissipation over the whole fluid.

If the vertical scale, $H$ , of density overturns observed in the ocean is equated to the scale ( $z_{2}-z_{1}$ ) of the region mixed by convection or KHI in the breaking internal waves, it follows that a non-dimensional form of the mean transfer of energy, $E$ , into turbulence per unit horizontal area can be written as

2.5 Data used to quantify the model

Measured values of quantities appearing on the right-hand side of (2.21) are available to illustrate and to evaluate the validity of the methodology developed in following sections. At $28\,^{\circ }$ N, $155\,^{\circ }$ W, in the Pacific Ocean Gregg (Reference Gregg1980, Reference Gregg and Imberger1998) reports $K_{\unicode[STIX]{x1D70C}}\approx 2\times 10^{-5}~\text{m}^{2}~\text{s}^{-1}$ in mid-water depths of 800–1200 m where $N\approx 3.1\times 10^{-3}~\text{s}^{-1}$ . Turbulent patches are typically $H\approx 5$  m high. (Gregg finds some that are even larger but 5 m is chosen as a representative value.) The Coriolis parameter, $f$ , at $28\,^{\circ }$ N gives $F=f/N=0.022$ . Whilst the region is not recognised as a ‘hot spot’ for ocean mixing, the value of $K_{\unicode[STIX]{x1D70C}}$ and the corresponding value of $\unicode[STIX]{x1D700}$ inferred from (2.16) are typical and representative of oceanic values in the observed depth range (Waterhouse et al. Reference Waterhouse, MacKinnon, Nash, Alford, Kunze, Simmons, Polzin, St Laurent, Sun and Pinkel2014). Gregg identifies and quantifies turbulence from zero crossings of the vertical temperature gradient, finding $Q$ in the range 0.07 to 0.36. An intermediate value of $Q$ , approximately 0.16, is derived by Gregg & Sanford (Reference Gregg and Sanford1988), at $34\,^{\circ }$ N $127\,^{\circ }$ W. They use a more precise method, recognising turbulence as being active when the rate of dissipation of turbulent kinetic energy per unit mass obtained from microstructure profiling exceeds $16\unicode[STIX]{x1D708}N^{2}$ , where $\unicode[STIX]{x1D708}$ is the molecular kinematic viscosity, i.e. when the buoyancy Reynolds number, $Re_{b}=\unicode[STIX]{x1D700}/\unicode[STIX]{x1D708}N^{2}>16$ . Further discussion of $Re_{b}$ is provided in § B.2.

Gregg’s values of $K_{\unicode[STIX]{x1D70C}}$ , $N$ , $F$ , $H$ and $Q$ are adopted later in comparing observations with theoretical estimates and, in particular, in finding the characteristics of internal waves that, in breaking, can support the observed mixing. Taking $\unicode[STIX]{x1D6E4}=0.2$ and using Gregg’s values, the relation (2.16) between $\unicode[STIX]{x1D700}$ and $K_{\unicode[STIX]{x1D70C}}$ leads to an estimate of the mean rate of dissipation of turbulent kinetic energy, $\unicode[STIX]{x1D700}=9.6\times 10^{-10}~\text{W}~\text{kg}^{-1}$ , $=\unicode[STIX]{x1D700}_{\ast }$ , say, and (2.21) becomes

The broad and, because of the method by which it is obtained, uncertain range of 0.07 to 0.36 for $Q$ found by Gregg will be shown to provide only a weak constraint on estimated quantities related to the energy transfer by breaking waves, and Gregg and Sanford’s value, $Q=0.16$ (close to the mean, 0.215, of Gregg’s broad range) is therefore generally used later as a control value.

A distinction is drawn between the scale, $H=5$  m, characterising the size of turbulent patches, those that here are represented by internal waves breaking locally by convective instability (see figure 5, later) or by KHI (figure 8), and the r.m.s. scale of overturns within the patches, characterised by the Thorpe scale, $L_{T}$ , or equivalently (Dillon Reference Dillon1982) by the Ozmidov length scale, $L_{O}=\unicode[STIX]{x1D700}^{1/2}N^{-3/2}$ . The latter can be estimated as 0.18 m taking $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}_{\ast }=9.6\times 10^{-10}~\text{W}~\text{kg}^{-1}$ and $N=3.1\times 10^{-3}~\text{s}^{-1}$ or as $0.3<L_{O}~(\text{m})<0.68$ taking $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}_{I}$ using (2.20) with $0.07<Q<0.36$ . (For convective instability direct numerical simulation (DNS) experiments by Scotti (Reference Scotti2015), find that the Thorpe scale, $L_{T}$ , derived from the displacements, $D$ – $d$ , greatly exceeds $L_{O}$ . For KHI, the DNS model of Smyth, Moum & Caldwell (Reference Smyth, Moum and Caldwell2001), finds that $L_{O}$ is substantially greater that $L_{T}$ in the overturning billows.) Observations of the mean ratio of $L_{T}$ to patch size range from about 0.07 in the main thermocline of the North Atlantic (Moum Reference Moum1996) to 0.17 in Lake Banyoles, Catalonia, Spain (Planella Morato, Roget & Lozovataky Reference Planella Morato, Roget and Lozovataky2011). With $L_{O}=0.8L_{T}$ (Dillon Reference Dillon1982) and a patch size, $H$ , of 5 m, the range $0.3<L_{O}~(\text{m})<0.68$ gives $0.075<L_{T}/H<0.17$ , approximately spanning the range observed and pointing to a general consistency of the chosen values.

For brevity Gregg’s and Gregg & Sanford’s data described above are henceforth referred to by ‘GS’.

3.1 Convective overturn

It is evident from (2.3) that in the presence of internal waves with $s<1$ the fluid is always statically stable, i.e. $\text{d}\unicode[STIX]{x1D70C}/\text{d}z<0$ . If, however, $s>1$ the upward vertical density gradient, $\text{d}\unicode[STIX]{x1D70C}/\text{d}z$ , is positive for some wave phase $\unicode[STIX]{x1D701}$ , implying static or, when motions set in, convective instability at some locations in the waves. As shown by the stippled regions of figure 1, when $s>1$ sloping statically unstable regions in which the density increases upwards are formed. Their height is

the vertical distance derived from (2.3) between the two levels at which $\text{d}\unicode[STIX]{x1D70C}/\text{d}z=0$ . Isopycnals are folded over horizontal lengths, $l$ , given by

that increase as $s$ increases and $\unicode[STIX]{x1D703}$ decreases. Convective instability and wave breaking follows if the Rayleigh number, Ra, of the overturning regions is sufficiently large (Thorpe Reference Thorpe1994a ). Based on GS, the Rayleigh number corresponding to the height, $H$ , $Ra=N^{2}H^{4}/\unicode[STIX]{x1D708}\unicode[STIX]{x1D705}_{T}$ , is approximately $1.7\times 10^{9}$ , where $\unicode[STIX]{x1D708}$ where is the kinematic viscosity ${\approx}1\times 10^{-6}~\text{m}^{2}~\text{s}^{-1}$ and $\unicode[STIX]{x1D705}_{T}$ is the thermal conductivity ${\approx}1.4\times 10^{-7}~\text{m}^{2}~\text{s}^{-1}$ . This value of Ra is well beyond that at which convective motions become turbulent at a Prandtl number appropriate to the ocean (Turner Reference Turner1973). A cautionary note about the form of wave breaking is provided in § C.2.

Figure 3.

The vertical density profile at $x=0$ , $t=0$ in convectively breaking waves; $s>1$ . The points A and B at levels $z_{A}$ and $z_{B}$ , respectively, where the vertical density gradient is zero mark the ends of the statically unstable gradient, $\text{d}\unicode[STIX]{x1D70C}/\text{d}z>0$ . The density at D (or C), levels $z_{D}$ and $z_{C}$ , respectively, is equal to that at A (or B). Although the vertical density gradient is negative in the region D–B implying static stability, there is overlying fluid, B–A–C, with parts denser than fluid elements in D–B so that convection may extend downwards to D. (The layer A–C is similarly unstable having less dense fluid below it.) The whole layer, D–C, is thus potentially unstable and prone to mixing. As shown by Davies Wykes & Dalziel (Reference Davies Wykes and Dalziel2014) however, convection may result in the generation of a mixing layer spanning only the depth range F–E.

When $s>1$ , the vertical density profile of figure 1 at $x=0$ , $t=0$ , is as sketched in figure 3. The points A and B at $z=z_{A}$ and $z_{B}$ , respectively, where the vertical density gradient is zero, mark the ends of the statically unstable region with $\text{d}\unicode[STIX]{x1D70C}/\text{d}z>0$ ; $z_{A}-z_{B}=h_{\ast }$ . The density at C (or D), $z=z_{C}$ (or $z_{D}$ ) respectively, is equal to that at B (or A). The points $z_{C}$ and $z_{D}$ lie within one vertical wavelength (i.e. $-\unicode[STIX]{x03C0}/m<z_{D}<z_{C}<\unicode[STIX]{x03C0}/m$ ) provided $s<4.712$ . Points E and F ( $z_{F}<0<z_{E}$ ) are where the density equals $\unicode[STIX]{x1D70C}_{0}$ , that at $z=0$ . Using (2.2) and (2.3):

(recalling $1/s<1$ , consistent with a wave slope exceeding unity for possible convective wave breaking), $mz_{C}$ is given by

and

The non-dimensional heights, $mz_{A}$ , $mz_{C}$ and $mz_{E}$ depend only on $s$ and are as shown in figure 4, and $z_{B}=-z_{A}$ , $z_{D}=-z_{C}$ and $z_{F}=-z_{E}$ .

Figure 4.

The variation of $mz_{A}$ , $mz_{C}$ and $mz_{E}$ with $s-1$ .

The unstable density profiles or overturns are analysed in appendix A using the method devised by van Haren & Gostiaux (Reference van Haren and Gostiaux2014).

3.2 Energy lost to turbulence

Thorpe (Reference Thorpe2010b ) adopts a model in which convection leads to mixing, homogenising the density over the height $h_{\ast }$ , between levels $z_{A}$ and $z_{B}$ of figure 3. The effect of mixing is to transfer potential and kinetic energy to turbulent kinetic energy. Although $\text{d}\unicode[STIX]{x1D70C}/\text{d}z<0$ is negative in the region D–B, implying static stability, at each level there is overlying denser fluid in B–A–C, from which convection may extend. (Similarly the layer A–C appears to be convectively unstable, having less dense fluid below it.) The whole layer, D–C, is thus potentially convectively unstable and may be involved in mixing. Choices limited by the conservation of mass and momentum are made for the (‘final’) density, $\unicode[STIX]{x1D70C}_{F}$ , and the velocity, $u_{F}$ , following mixing. The final density is supposed to be stably stratified, $\text{d}\unicode[STIX]{x1D70C}_{F}/\text{d}z\leqslant 0$ , and its range is within that of the original density; mixing cannot increase (decrease) the density to above (below) its maximum (minimum) value within the mixed region.

Three alternative models denoted Con1–Con3 are devised to characterise the possible density and velocity profiles after mixing and to allow some assessment of the sensitivity of the model results. The density of Con1 is shown in figure 5(a). Following convective mixing that occurs when the density gradient, $\text{d}\unicode[STIX]{x1D70C}/\text{d}z$ , reaches its positive maximum, a statically stable continuous linear profile of density is taken to be formed between $z_{D}$ and $z_{C}$ :

For small $s$ , this corresponds closely to the rearranged dashed profiles of figure 11(a) in appendix A to which (without diffusion) the elements of fluid would collapse as a consequence of convection. In Con2 the density becomes uniform with $\unicode[STIX]{x1D70C}_{F}=\unicode[STIX]{x1D70C}_{0}$ between levels $z_{C}$ and $z_{D}$ (figure 5 b). In both models the assumption is made that, as a result of the convective mixing, the velocity becomes uniform between levels D and C of figure 3 and equal to the mean velocity $\langle u\rangle =[\int u\,\text{d}z]/(z_{C}-z_{D})$ with integration from $z_{D}$ to $z_{C}$ , thus conserving momentum (figure 5 c).

Figure 5.

A sketch showing the profiles of density, $\unicode[STIX]{x1D70C}$ , and $x$ -velocity component, $u$ , in convective breaking. The finer lines are the profiles of $\unicode[STIX]{x1D70C}$ and $u$ before mixing is induced by wave breaking and the thicker lines, $\unicode[STIX]{x1D70C}_{F}$ and $u_{F}$ , those after mixing. (a) Density, $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D70C}_{F}$ (model Con1); (b) density, $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D70C}_{F}$ (Con2); (c) $x$ -velocity component, $u$ and $u_{F}$ (Con1 and 2); (d) density, $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D70C}_{F}$ (Con3); and (e): $x$ -velocity component $u$ and $u_{F}$ (Con3). Dashed lines in (a), (b) and (d) are the mean density, $\unicode[STIX]{x1D70C}_{0}(1-N^{2}z/g)$ .

The density, $\unicode[STIX]{x1D70C}_{F}$ , of the third model, Con3, is prompted by the results and analysis of laboratory experiments by Davies Wykes & Dalziel (Reference Davies Wykes and Dalziel2014) in which convection occurs across a horizontal statically unstable interface between two stably stratified fluids. The experiments reduce the statically unstable region between A and B in figure 3 to a narrow interface and do not include the effects of horizontal motion, $u$ . Convective mixing results in a nearly uniform layer extending between the levels equivalent to E and F of figure 3, corresponding closely, in Davies Wykes & Dalziel’s terminology, to ‘perfect mixing efficiency’. The final density and velocity profiles in Con3 are chosen similarly, as in figures 5(d) and 5(e), respectively. The energy of internal waves radiated upwards or downwards from the convective mixing region in the laboratory experiments is less than 0.2 % of the initial available potential energy and, as assumed above, is very small. In all three models rotation is disregarded by taking $F=0$ so that $v$ and $\unicode[STIX]{x1D703}_{b}$ are zero, the latter implying that breaking advances horizontally.

The energy lost to turbulence per unit horizontal area during the breaking process in $z_{D}\leqslant z\leqslant z_{C}$ in Con1 and Con2, or in $z_{F}\leqslant z\leqslant z_{E}$ in Con3, the net initial energy minus the final energy, is

where

is the kinetic energy loss, with final kinetic energy, $KE_{cF}=(\unicode[STIX]{x1D70C}_{0}/2)\int u_{F}^{2}\,\text{d}z$ , and

with $PE_{cF}=g\int z\unicode[STIX]{x1D70C}_{F}\,\text{d}z$ , is the potential energy loss relative to $z=0$ per unit horizontal area in mixing the layer, where integration is over the range, $z_{D}\leqslant z\leqslant z_{C}$ for Con1 and Con2 or $z_{F}\leqslant z\leqslant z_{E}$ for Con3, $u$ and $\unicode[STIX]{x1D70C}$ being evaluated at $x=t=0$ . Possible limiting (extreme) values of $E_{c}$ are discussed later in § B.2.

For Con1 and Con2 the vertical distance, $2z_{C}$ , between points D and C in figure 3 is taken to be equal to $H$ , the size of turbulent patches; GS. This gives

with $mz_{C}$ given by (3.4).

The kinetic and potential energies, $KE_{c}$ and $PE_{c}$ are expressed in non-dimensional forms in terms of $\unicode[STIX]{x1D70C}_{0}N^{2}H^{3}$ in (B 1) and (B 2) of § B.1 and, together with $E_{c}/(\unicode[STIX]{x1D70C}_{0}N^{2}H^{3})$ derived using (3.7), depend only on $s$ and $\unicode[STIX]{x1D703}$ .

Using Davies Wykes & Dalziel’s (Reference Davies Wykes and Dalziel2014) results, it is shown in § C.1 that provided ( $s-1$ ) is sufficiently large or the angle, $\unicode[STIX]{x1D703}$ , is small, the growth time of convective instability is small in comparison to the period of time in which the density gradient in an internal wave with $s>1$ is negative at $x=z=0$ . Breaking by convective instability is then relatively instantaneous or, with respect to the growth of instability, the wave may be regarded as ‘frozen’.

The fraction of the water column that is turbulent, $Q$ , is given by (2.19) in terms of $\unicode[STIX]{x1D703}$ , $m$ (3.10), $\unicode[STIX]{x1D6EC}_{I}$ and the non-dimensional decay time constant $r$ ( $=\,r_{c}$ , say). Smyth, Zavialov & Moum’s (Reference Smyth, Zavialov and Moum1997) observations of the decay time of an isolated near-surface turbulent layer gives $r_{c}\approx 6$ . The laboratory experiments by Davies Wykes & Dalziel (Reference Davies Wykes and Dalziel2014) find that growth ceases at a value of $r_{c}$ (their value $\unicode[STIX]{x1D70F}\approx 4$ ) that decreases with $s$ from approximately 10.4 at $s=1.4$ to 4.8 at $s=3$ , with an approximate variation

appropriate for Con3. (Decay of turbulence generated by convective wave breaking will generally occur at times less than half the wave period, i.e. $r_{c}N^{-1}<\unicode[STIX]{x03C0}\unicode[STIX]{x1D70E}^{-1}$ if $\unicode[STIX]{x1D703}\leqslant 32^{\circ }$ when $r_{c}=6$ , or if $\unicode[STIX]{x1D703}\leqslant 15^{\circ }$ when $r_{c}=12$ .)

Equation (2.17) provides a means to estimate the non-dimensional breaking parameter, $\unicode[STIX]{x1D6EC}_{I}H$ , with $z_{2}-z_{1}=2z_{C}$ , $=H$ , for Con1 and Con2, or $z_{2}-z_{1}=2z_{E}$ , $=H$ , for Con3. For selected values of $s$ and $\unicode[STIX]{x1D703}$ , $mz_{C}$ (3.4), $mz_{E}$ (3.5) and $E_{c}/(\unicode[STIX]{x1D70C}_{0}N^{2}H^{3})$ are known and the size of $\unicode[STIX]{x1D6EC}_{I}H$ required to support mixing can be estimated using the observed values of $K_{\unicode[STIX]{x1D70C}}$ , $N$ and $H$ .

3.3 Results; convective instability

For Con1 and 3 and with $Q$ within the range, 0.07–0.36 (GS; § 2.5), most of the energy lost to turbulence is derived from potential energy, i.e. $PE_{c}>KE_{c}$ , but for Con2 $PE_{c}<KE_{c}$ , in the same range of $Q$ . For a given model, the non-dimensional energy transfer to turbulence, $E_{c}/(\unicode[STIX]{x1D70C}_{0}N^{2}H^{3})$ , is a function only of the wave characteristics, $s$ and $\unicode[STIX]{x1D703}$ , as shown in figure 6. For Con1:

for Con2:

and for Con3:

when $Q$ is in Gregg’s range, $0.07<Q<0.36$ .

Figure 6.

The non-dimensional energy transferred to turbulence by convective breaking of waves as a function of $s-1$ at $\unicode[STIX]{x1D703}=10^{\circ }$ , $30^{\circ }$ and $50^{\circ }$ , together with (bold lines) values at $Q=0.07$ , 0.16 and 0.36; GS. (a) The full lines show Con1 and the dashed lines are Con2. (b) The same for Con3.

For Con1 and Con2 to match the observations of GS, the model estimates of the non-dimensional transfer of energy to turbulence must be equal to $G=0.0116/Q$ , given by (2.22) using $r=r_{c}\approx 9$ , but with an uncertainty in $G$ of approximately 30 % to allow for that in $r_{c}$ . The values of $G$ at $Q=0.07$ , 0.16 and 0.36 are indicated by the horizontal lines in figure 6(a). Breaking waves within a range of characteristics, $s$ and $\unicode[STIX]{x1D703}$ , covered by these lines can, alone, produce the observed mixing. For model Con1, values of $Q$ within the range, 0.07 to 0.36, are found only between $s=1.49$ at $\unicode[STIX]{x1D703}=10^{\circ }$ (at the limit $Q=0.36$ ) and $s=4.91$ at $\unicode[STIX]{x1D703}=50^{\circ }$ (when $Q=0.07$ ). (The latter $s$ exceeds 4.712 and is strictly beyond the range of model validity; § 3.1.) Higher values of $s$ are found for Con2; the broad range of $Q$ estimated by Gregg (Reference Gregg1980, Reference Gregg and Imberger1998) provides only a weak constraint on the possible range of wave characteristics, $s$ and $\unicode[STIX]{x1D703}$ , involved in wave breaking. Values of $s$ at Gregg & Sanford’s (Gregg & Sanford (Reference Gregg and Sanford1988)) value, $Q=0.16$ , are 2.15 and 2.50 for Con1 at $\unicode[STIX]{x1D703}=10^{\circ }$ and $50^{\circ }$ , respectively, and 2.77 and 4.00 for model Con2 at the same values of $\unicode[STIX]{x1D703}$ . Within these narrower ranges of $s$ , equation (3.12) for Con1, and (3.13) for Con2 with $G\equiv E/(\unicode[STIX]{x1D70C}_{0}N^{2}H^{3})=0.0725$ at $Q=0.16$ provide the values of wave steepness required for representative waves with given $\unicode[STIX]{x1D703}$ (or, using $\unicode[STIX]{x1D70E}=N\sin \unicode[STIX]{x1D703}$ , of a given wave frequency) to break and produce sufficient mixing to support the observed mixing. For example, for Con1, the required steepness is

Using (3.11) and (2.22), $G\equiv E_{c}/(\unicode[STIX]{x1D70C}_{0}N^{2}H^{3})=0.0185/(Qs^{1.06})$ for Con3. The values of $G(s)$ at $Q=0.07$ , 0.16 and 0.36 are indicated in figure 6(b). The curve $Q=0.16$ intersects the calculated non-dimensional values of $E_{c}/(\unicode[STIX]{x1D70C}_{0}N^{2}H^{3})$ at $s=2.43$ and 2.75 when $\unicode[STIX]{x1D703}=10^{\circ }$ and $50^{\circ }$ , respectively, between the respective values of Con1 and Con2.

The size of waves required to support the observed mixing can be determined: (3.10) and $s=am$ are used to obtain $a/H=s/2mz_{C}$ , and a fit,

to the curve in figure 4 relating $mz_{C}$ to $s$ when $0.2<(s-1)<2$ , leads to

providing an expression for the required amplitude of the representative waves. Taking the above estimates of $s$ and $\unicode[STIX]{x1D703}$ in model Con1 at $Q=0.16$ to be typical, $a/H=0.47$ at $s=2.15$ , the lower limit at $\unicode[STIX]{x1D703}=10^{\circ }$ ( $\unicode[STIX]{x1D70E}/N=0.174$ ), and $a/H=0.49$ at $s=2.50$ , the upper limit at $\unicode[STIX]{x1D703}=50^{\circ }$ ( $\unicode[STIX]{x1D70E}/N=0.766$ ). The corresponding vertical and horizontal wavelengths, $\unicode[STIX]{x1D706}_{v}$ and $\unicode[STIX]{x1D706}_{h}$ (or $2\unicode[STIX]{x03C0}m^{-1}$ and $\unicode[STIX]{x1D706}_{v}/\tan \unicode[STIX]{x1D703}$ , respectively) are given in non-dimensional forms $\unicode[STIX]{x1D706}_{v}/H$ and $\unicode[STIX]{x1D706}_{h}/H$ , as 1.36, 7.74 and 1.23, 1.03, respectively. Using $H=5$  m, the vertical wavelengths are typically 6 m, and the horizontal wavelengths lie between approximately 5 and 45 m. The speeds of the breaking region, $c_{b}$ , and the phase speeds, $c=c_{b}\sin \unicode[STIX]{x1D703}$ are typically $5\times 10^{-3}~\text{m}~\text{s}^{-1}$ , small relative to typical oceanic flow speeds of $0.1~\text{m}~\text{s}^{\text{-1}}$ (and are also relatively small for waves breaking by KHI).

The non-dimensional breaking parameter, $\unicode[STIX]{x1D6EC}_{I}H$ , and contours of $Q$ are shown in figures 7(a) (Con1), 7(b) (Con2) and 7(c) (Con3). $\unicode[STIX]{x1D6EC}_{I}H$ is typically in the range of 0.05 to 0.3 (or $0.01<\unicode[STIX]{x1D6EC}_{I}~(\text{m}^{-1})<0.06$ if $H=5$  m), $10^{\circ }<\unicode[STIX]{x1D703}<50^{\circ }$ and when $Q$ is in the observed range, $0.07<Q<0.36$ .

Figure 7.

Contours of constant $10^{2}\unicode[STIX]{x1D6EC}_{I}H$ (labelled full lines) and $Q$ (dashed lines) in the ( $s-1$ , $\unicode[STIX]{x1D703}$ ) plane for models (a) Con1; (b) Con2; (c) Con3.

The energy lost, $E_{c}$ , in $z_{D}\leqslant z\leqslant z_{C}$ for Con1 and Con2 (or $z_{F}\leqslant z\leqslant z_{E}$ for Con3) may be compared to that available in the same layer in the waves immediately before breaking, $E_{cw}$ , given by (B 3) in § B.1. The ratio $E_{c}/E_{cw}$ increases with $s$ at given $\unicode[STIX]{x1D703}$ but generally lies between 0.36 and 0.94 (Con1), 0.29 and 0.62 (Con2) or 0.35 and 0.58 (Con3) when $Q$ is in the observed GS range: to support the required mixing a substantial fraction of the localised wave energy must necessarily be dissipated and transferred to turbulence by convective overturn. The mean energy per unit mass, $E_{w}^{\ast }$ , at the location where breaking of individual waves is imminent, lies in the range $1.9\times 10^{-5}~\text{J}~\text{kg}^{-1}$ to $7.7\times 10^{-5}~\text{J}~\text{kg}^{-1}$ . Significantly, this is much less than the known mean energy of the internal wave field of approximately $1\times 10^{-3}~\text{J}~\text{kg}^{-1}$ (§ 1.1; an approximation to that in the region of GS data). It follows that even at the extreme conditions where breaking is imminent the local energy of the waves, of whatever steepness, required to produce the required mixing can be accommodated by the energy available in the oceanic internal wave field and is a relatively small energetic component of the whole field. The mean rate of dissipation of turbulent kinetic energy per unit mass in the turbulent region produced by wave breaking is, by (2.20), $\unicode[STIX]{x1D700}_{m}=\unicode[STIX]{x1D700}_{I}/Q$ , exceeding the overall volume mean dissipation rate, $\unicode[STIX]{x1D700}_{\ast }$ , as they should, for values of $Q$ ( ${<}1$ ) within Gregg’s range.

3.4 Summary; convective wave breaking

Three models of the mixing produced by convective wave breaking, Con1–3 shown in figure 5, are considered. They are distinguished by the degree of mixing produced in the density field and the range over which mixing occurs. Relations may be found, e.g. equation (3.15), for the steepness of waves of given frequency which represents the flow prior to mixing and which, in breaking, provide mixing sufficient to support the values of GS. Waves must be of rather small scale, both in amplitude (typically 2–3 m), wavelength (order at most a few tens of metres) and propagation speed (typically $5\times 10^{-3}~\text{m}~\text{s}^{-1}$ ), and have a steepness of at least 1.49 (Con1 when $Q=0.36$ and $\unicode[STIX]{x1D703}=10^{\circ }$ ), with values of critical steepness increasing with the wave propagation direction, $\unicode[STIX]{x1D703}$ . The wave breaking parameter, $\unicode[STIX]{x1D6EC}_{I}$ , must be of order 0.3 $H^{-1}$ , but the limiting value also depends on $\unicode[STIX]{x1D703}$ (figure 7). The figure shows that, for each model, the higher the wave frequency, the greater is the wave steepness required to provide mixing at the rates required. The energy lost from waves in the breaking process, $E_{c}$ , is generally a substantial fraction of that locally available, $E_{cw}$ , in the breaking internal wave. The energy of the waves that in breaking support the mixing necessary to sustain the stratification is however considerably less than that available in the wave field. This implies that, although the required steepness of a wave breaking locally by convective instability is rather high, the wave field contains adequate energy to support the required rates of mixing and to sustain the oceanic stratification. Regions of locally high energy density, and high $s$ , may not, however, be located where the representative waves have the required frequencies.

But what about breaking through KHI?

4.1 Conditions for breaking

Thorpe (Reference Thorpe1999b ) examines the conditions for the onset of KHI in an internal wave propagating in a uniformly stratified fluid, with $s<1$ to ensure static stability. Including rotation the disturbances to the flow given by (2.1) and (2.3) that first become unstable are directed in the $x$ -direction. Again limiting consideration to $x=t=0$ , these equations are used to determine the gradient Richardson number related to flow in the $x$ -direction:

Varying $z$ at given $s$ and $\unicode[STIX]{x1D703}$ , the minimum value of $Ri_{x}$ is

If $F=f/N=0$ it follows that $Ri_{xc}\geqslant \{\!2\cos ^{2}\unicode[STIX]{x1D703}\! )^{-1}\geqslant 1/2$ , so that in the absence of rotation the Richardson number in the wave-generated flow always exceeds 0.25 and (at least in a local quasi-steady flow approximation) the flow will be stable to small disturbances by the Miles–Howard theorem. Only if $F>0$ is $Ri_{xc}<0.25$ , making KHI possible. The height, $z_{0}$ , at which the Richardson number is minimum is given by

If $s$ and $F$ are given, equation (4.2) may be solved to find $\unicode[STIX]{x1D703}$ , the direction of group propagation of a wave with steepness, $s=am$ , necessary to produce a chosen minimum Richardson number, $Ri_{xc}<0.25$ . If $F^{2}\ll 1$ as at the location of the observations by GS, but non-zero, the direction is given approximately by $\unicode[STIX]{x1D703}=F\sqrt{[q/(1-q)]}$ , where $q=2Ri_{xc}[1-\sqrt{(1-s^{2})}]~({<}1/2)$ or, when both $s$ and $F$ are small, $\unicode[STIX]{x1D703}\approx FsRi_{xc}^{1/2}$ . The angle, $\unicode[STIX]{x1D703}$ , is consequently small when $F=0.022$ (as in GS), implying that the breaking waves have a near-horizontal group velocity, and that (from (2.5)), the breaking internal waves must be near inertial (i.e. with $\unicode[STIX]{x1D70E}$ close to $f$ ) for KHI to occur.

Measurements of turbulence in near-inertial waves by Alford & Gregg (Reference Alford and Gregg2001) in the Banda Sea suggest that their breaking is indeed initiated by KHI.

4.2 Energy lost to turbulence; the models

Assumptions are now made about the flow transition produced by KHI. To devise models, it is supposed that KH billows occur and generate turbulence at level $z_{0}$ where the minimum Richardson number, $Ri_{xc}$ , is reached in a wave with characteristics $\unicode[STIX]{x1D703}$ (or $\unicode[STIX]{x1D70E}$ ) and $s$ . The turbulent motion results in homogenisation of fluid in a region surrounding the level $z_{0}$ where both the density and the horizontal components of velocity become constant and equal to their mean values before breaking. The density, velocity and Richardson number profiles are sketched in figure 8. (A similar model of wave breaking is adopted by Boguski & Garrett (Reference Boguski and Garrett1993). The more natural assumption – that the final density and velocity profiles are continuous and linear rather than uniform in this region – does not provide for conservation of mass and momentum.)

Figure 8.

A sketch showing the profiles of (a) density; (b) horizontal velocity components; and (c) gradient Richardson number, $Ri_{x}$ , in wave breaking through KHI. The finer lines are the profiles of $\unicode[STIX]{x1D70C}$ , $u$ and $v$ before mixing is induced by wave breaking, and the thicker lines, $\unicode[STIX]{x1D70C}_{F}$ , $u_{F}$ and $v_{F}$ , those after mixing. The minimum Richardson number, $Ri_{xc}$ , is at level $z_{0}$ . The values of $z$ ( $z_{1}$ and $z_{2}$ ) at $Ri_{x}=Ri_{F}$ define the vertical limits of mixing (or the size, $H$ , of a turbulent patch) and are within one vertical wavelength, $-\unicode[STIX]{x03C0}/m<z<\unicode[STIX]{x03C0}/m$ of the wave.

For how long does the wave-induced local flow at $x=z=0$ remain unstable to KHI? The times, $t_{1}$ and $t_{2}$ , at which the gradient Richardson number is 0.25, implying that the flow becomes potentially unstable, are given in § D.1. It is shown there that when $F\ll 1$ , unless $Ri_{xc}$ is close to 0.25, the growth times of KHI are generally much less than the period of time, $t_{1}$ to $t_{2}$ , in which instability may occur; the wave field may therefore be regarded as frozen during the period of growth of KHI. It is supposed that the flow in which turbulence is about to set in is represented by the wave model at the time at which $Ri_{xc}$ is reached and that mixing occurs instantaneously at this time.

The time for turbulence to collapse following KHI, $rN^{-1}$ , is generally much greater than the times for the growth of disturbances, and corresponds to a non-dimensional decay constant $r=r_{s}\approx 25$ (Thorpe Reference Thorpe1973). Provided $F\ll 1$ , decay will generally occur at times much less than half the wave period, i.e. $r_{s}N^{-1}\ll \unicode[STIX]{x03C0}\unicode[STIX]{x1D70E}^{-1}$ , for the small values of $\unicode[STIX]{x1D703}$ associated with near-inertial waves. Generally a local change in the direction of shear in wave propagation will not affect the decay of turbulence following KHI. (The non-dimensional times, $N(t_{2}-t_{1})$ , during which $Ri_{x}<0.25$ , are 100.6, 80.5, 62.2 and 42.1, for $Ri_{xc}=0.05$ , 0.1, 0.15 and 0.2, respectively, all greater than the non-dimensional time, $r_{s}\approx 25$ .) But, to repeat: in the models any growth of billows or turbulence developing during the time period $t_{1}$ to $t_{0}$ between the Richardson number reaching 0.25 and $Ri_{xc}$ are not accounted for in the estimations of mixing.

For simplicity the bounds of the mixed region are set to levels $z_{1}$ and $z_{2}$ where $Ri_{x}=Ri_{F}$ , with either $Ri_{F}=0.32$ (a value corresponding to that to which turbulence is observed to collapse following KHI in laboratory and numerical experiments as discussed in § D.2), or $Ri_{F}=0.25$ , the Miles–Howard stability limit. The flow after transition has a minimum gradient Richardson number ${\geqslant}$ 0.25 and is therefore (linearly) stable by the Miles–Howard theorem. Using (4.1) and (4.2) an equation is found for the non-dimensional heights, $\unicode[STIX]{x1D701}=mz$ , at which $Ri_{x}=Ri_{F}$ in a wave in which the minimum Richardson number is $Ri_{xc}$ :

where $\unicode[STIX]{x1D712}=2[1-(1-s^{2})^{1/2}]Ri_{xc}/(s^{2}Ri_{F})$ . Equation (4.4) is a quadratic giving two roots, $\unicode[STIX]{x1D701}_{1}$ and $\unicode[STIX]{x1D701}_{2}$ , equal to $mz_{1}$ and $mz_{2}$ , respectively. It may be shown that

In KHI there is generally a loss of kinetic energy and an increase in potential energy. The energy lost and locally transferred to turbulence per unit horizontal area at $x=t=0$ by wave breaking is

Here

is the kinetic energy lost, with final kinetic energy, $KE_{cF}$ , equal to $(\unicode[STIX]{x1D70C}_{0}/2)\int (u_{F}^{2}+v_{F}^{2})\,\text{d}z$ or ( $\unicode[STIX]{x1D70C}_{0}/2$ ) $(\langle u\rangle ^{2}+\langle v\rangle ^{2})(z_{2}-z_{1})$ if, as in the models, $u_{F}=\langle u\rangle$ and $v_{F}=\langle v\rangle$ where $\langle \cdots \rangle$ denotes mean values over the range, $z_{1}\leqslant z\leqslant z_{2}$ (or $\unicode[STIX]{x1D701}_{1}<\unicode[STIX]{x1D701}<\unicode[STIX]{x1D701}_{2}$ ) of integration determined by the choice of $Ri_{F}$ . The gain in potential energy is

where $PE_{sF}=g\int z\unicode[STIX]{x1D70C}_{F}\,\text{d}z$ is the final potential energy. As in § 3.2, choosing $H=z_{2}-z_{1}$ to represent the size of the mixing region gives an equation for the vertical wavenumber,

( $\unicode[STIX]{x1D701}_{2}-\unicode[STIX]{x1D701}_{1}$ ) given by (4.5) is independent of $s$ , and consequently $m$ is also independent of $s$ at fixed $Ri_{xc}$ and $Ri_{F}$ . The vertical wavelength, $2\unicode[STIX]{x03C0}/m$ , tends to infinity as $Ri_{xc}$ tends to $Ri_{F}$ when $\unicode[STIX]{x1D701}_{2}-\unicode[STIX]{x1D701}_{1}$ tends to zero.

Equations for the energies non-dimensionalised by $\unicode[STIX]{x1D70C}_{0}N^{2}H^{3}$ are given in § E.1 in terms of $\unicode[STIX]{x1D701}_{1}$ and $\unicode[STIX]{x1D701}_{2}$ . As in § 3.2, equation (2.17) provides an equation to estimate the breaking parameter, $\unicode[STIX]{x1D6EC}_{I}$ , non-dimensionalised with $H$ :

The fraction of the water column that is turbulent, $Q$ , is given by (2.19) in terms of $m$ (4.9), $\unicode[STIX]{x1D6EC}_{I}$ and the non-dimensional decay time constant $r_{s}$ . Contours of $\unicode[STIX]{x1D6EC}_{I}H$ and $Q$ are shown in figures 10(a) and 10(b), respectively, as functions of the wave characteristics, $s$ and $\unicode[STIX]{x1D703}$ , when $Ri_{F}=0.32$ . (Contours of $\unicode[STIX]{x1D6EC}_{I}H$ and $Q$ when $Ri_{F}=1/4$ are similar to those of $Ri_{F}=0.32$ except close to the curve $Ri_{xc}=0.25$ where, from (4.5), ( $\unicode[STIX]{x1D701}_{2}-\unicode[STIX]{x1D701}_{1}$ ) tends to zero, leading to singularities in $KE_{s}$ given by (E 1) in § E.1.)

4.3 Results; KHI

The breaking internal waves are specified by wave steepness, $s$ , and a frequency determined from $\unicode[STIX]{x1D703}$ through the dispersion relation, equation (2.5). For values of $Ri_{xc}$ less than approximately 0.19, equivalent values of energy-related quantities estimated with $Ri_{F}=0.25$ and 0.32 are of comparable magnitude and are not very sensitive to its choice. Attention is focussed on the model with $Ri_{F}=0.32$ . The lines $Q=\text{constant}$ correspond to constant values of $E_{s}/(\unicode[STIX]{x1D70C}_{0}N^{2}H^{3})$ as shown in figure 9 when $Ri_{F}=0.32$ . The curves of constant $\unicode[STIX]{x1D703}$ drawn in figure 9 are terminated at the Miles–Howard stability limit, $Ri_{xc}=0.25$ above which the flow is likely to be stable. Points between the lines $Q=0.07$ and 0.36 in this figure determine the range of characteristics of the breaking waves that can support the mixing observed by Gregg’s (Reference Gregg1980, Reference Gregg and Imberger1998) observations. For example, the observed mixing can be sustained by waves with $\unicode[STIX]{x1D703}=0.2^{\circ }$ (or a frequency, $\unicode[STIX]{x1D70E}$ , of 1.013 $f$ using $F=f/N=0.022$ ; GS) provided their steepness, $s$ , is between 0.32 (at $Q=0.36$ ) and 0.61 ( $Q=0.07$ ) or, using (4.2), for $Ri_{xc}$ between 0.233 and 0.0591, respectively.

Figure 9.

The variation of the non-dimensional energy transfer to turbulence in waves breaking through KHI with $Ri_{F}=0.32$ as a function of the wave steepness, $s$ , at set values of $\unicode[STIX]{x1D703}$ (deg). Values of $Q$ (bold lines) determined by GS and the curve $Ri_{xc}=0.25$ are also shown.

More generally the required wave steepness can be estimated by expressing the non-dimensionalised energy loss to turbulence, $E_{s}/\unicode[STIX]{x1D70C}_{0}N^{2}H^{3}$ , as a function of the wave characteristics, $s$ and $\unicode[STIX]{x1D703}$ , and of $F$ , given approximately by

where $\unicode[STIX]{x1D703}$ is measured in radians. Relations between the wave frequency and its steepness, those required for dissipation through breaking to be consistent with GS, are now obtained using (4.11) and (2.22) with $E=E_{s}$ and $r=r_{s}$ . For example, when $Q=0.16$ , equation (2.22) gives $E_{s}/\unicode[STIX]{x1D70C}_{0}N^{2}H^{3}=0.202$ , and with this value (4.11) leads $s=2.64(1\pm 0.041)(\unicode[STIX]{x1D703}/F)$ . Using (2.5) at $\unicode[STIX]{x1D70E}=f+$ the required wave steepness is

increasing as $\unicode[STIX]{x1D70E}$ increases. When $\unicode[STIX]{x1D703}\ll F$ this corresponds approximately to $Ri_{xc}=0.14$ ; a minimum Richardson number of this size is sufficient to provide mixing at the rate required to sustain the observed stratification provided $Q=0.16$ .

Most of the kinetic energy lost in breaking, $KE_{s}$ , is transferred to turbulence; only a relatively small proportion ranging from approximately 10 %–37 % contributes to an increase in potential energy. For $Ri_{F}=0.32$ the efficiency of wave breaking, defined here as $PE_{s}/KE_{s}$ , is approximately given by

increasing as $Ri_{xc}$ increases from approximately 0.12 at $Ri_{xc}\approx 0.05$ to 0.4 at $Ri_{xc}\approx 0.2$ . $KE_{s}/KE_{sw}$ is the ratio of wave kinetic energy lost to turbulence, $KE_{s}$ , to the kinetic energy available in the breaking wave, $KE_{sw}$ , for which an equation is given by (E 3) in § E.1. Generally less than 40 % of the kinetic energy initially available in the internal waves representing the flow is transferred to turbulence or to the increase of potential energy.

The magnitude of the non-dimensional wave breaking parameter, $\unicode[STIX]{x1D6EC}_{I}H$ , is shown in figure 10(a). The values of $\unicode[STIX]{x1D6EC}_{I}$ required to support turbulent dissipation at the rate $\unicode[STIX]{x1D700}_{\ast }=9.6\times 10^{-10}~\text{W}~\text{kg}^{-1}$ inferred by GS are typically of order $(5{-}15)\times 10^{-3}H^{-1}$ . $Q$ , shown in figure 10(b), exceeds the lower bound, 0.07, and spans most of the observed range, 0.07 to 0.36, over some range of $\unicode[STIX]{x1D703}$ and with $0.1<s<0.8$ when, approximately, $0.05\leqslant Ri_{xc}<0.19$ . When $Ri_{xc}<0.05$ the estimated values of $Q$ are lower than the observed; the predicted mixing is then more intense than observed. The curve $Q=0.16$ corresponds to (4.12) when related to $\unicode[STIX]{x1D703}$ by (2.5).

Figure 10.

Contours of (a) $10^{3}\unicode[STIX]{x1D6EC}_{I}H$ , and (b) $Q$ in the ( $s$ , $\unicode[STIX]{x1D703}$ ) plane for waves breaking through KHI with $Ri_{F}=0.32$ . The curves $Ri_{xc}=0.25$ , 0.19 and 0.05 are shown by the dashed lines.

As in the case of convective instability described in § 3.3, the rates of dissipation of turbulent kinetic energy per unit mass in the breaking region, $\unicode[STIX]{x1D700}_{m}$ , exceed the mean rate, $\unicode[STIX]{x1D700}_{\ast }$ . The mean energy of the waves per unit mass at the point of their breaking is $E_{w}^{\ast }$ given by (2.13). Excluding the values affected by the limit at $Ri_{F}=0.25$ and those when $Q$ is outside the range 0.07–0.36, the values of $E_{w}^{\ast }$ range from $2.48\times 10^{-4}~\text{J}~\text{kg}^{-1}$ to $6.16\times 10^{-4}~\text{J}~\text{kg}^{-1}$ . This is a substantial fraction of the estimated total mean energy, approximately $1\times 10^{-3}~\text{J}~\text{kg}^{-1}$ , of the oceanic internal wave field.

A selection of the wave characteristics contributing to the generation of turbulence spanning a range of values of $Q$ and determined from figure 10(b) are listed in table 1. The non-dimensional wave amplitude, $a/H$ , is equal to $s/(\unicode[STIX]{x1D701}_{2}-\unicode[STIX]{x1D701}_{1})$ , with $\unicode[STIX]{x1D701}_{2}-\unicode[STIX]{x1D701}_{1}$ given by (4.5). With $H=5$  m, wave amplitudes are generally 0.4–3 m, and vertical and horizontal wavelengths 13–30 m and 2–18 km, respectively.

Table 1.

Characteristic measures of representative waves breaking by KHI with $Ri_{F}=0.32$ and leading to the mixing required to support the observed ocean stratification: the wave steepness, $s$ , and direction, $\unicode[STIX]{x1D703}$ , for values of the fractional volume of turbulence, $Q$ , spanning Gregg’s range, together with the gradient Richardson number, $Ri_{xc}$ , the non-dimensional vertical range of mixing after breaking, $\unicode[STIX]{x1D701}_{2}-\unicode[STIX]{x1D701}_{1}$ , the vertical wavenumber, $m$ , and wavelength, $\unicode[STIX]{x1D706}_{v}=2\unicode[STIX]{x03C0}/m$ , the horizontal wavelength, $\unicode[STIX]{x1D706}_{h}=\unicode[STIX]{x1D706}_{v}\cot \unicode[STIX]{x1D703}$ , and the wave amplitude, $a=s/m$ .

The (unlikely) case in which KHI produces no change in the density profile and $E_{s}$ is consequently maximised is considered in § E.2. Smaller values of $s$ are then required for the resulting turbulence to sustain the mixing observed by GS.

4.4 Summary; wave breaking by KHI

In local regions where they represent breaking by KHI the internal waves must be near inertial. Two model representations of breaking are examined, those in which Richardson numbers, $Ri_{F}$ , after mixing tends to either 0.32 or 0.25. These prove to produce similar estimates of the non-dimensional energy transfer to turbulence provided mixing occurs when the gradient Richardson number, $Ri_{xc}$ , is less than approximately 0.19. For simplicity attention is focused on $Ri_{F}=0.32$ . Most of the kinetic energy lost in breaking, $KE_{s}$ , is transferred to turbulence, a relatively small proportion contributing to an increase in potential energy, $PE_{s}$ . When turbulence sets in with $Ri_{x}<0.05$ mixing results in relatively large transports of energy to turbulence, $E_{s}$ , and the fraction of fluid that must be turbulent, $Q$ , to lead to the observed diffusivity, $K_{\unicode[STIX]{x1D70C}}$ , is less than the smallest observed, i.e. 0.07. To produce mixing consistent with GS, the amplitudes of the breaking waves must be of order 3 m, and their vertical and horizontal wavelengths of order 30 m to 18 km, respectively, although the horizontal wavelength of the wave preceding turbulence is unlikely to be representative of the size of a local turbulent patch. The values of $\unicode[STIX]{x1D6EC}_{I}$ needed to sustain mixing at the rates required are typically of order $(5{-}15)\times 10^{-3}H^{-1}$ . The internal wave energy per unit mass, $E_{w}^{\ast }$ , required to sustain mixing at the rate observed is generally a substantial fraction of the $1\times 10^{-3}~\text{J}~\text{kg}^{-1}$ estimated to be available in the total internal wave field.