2.1. Length, width and breadth of reflecting groups

The wavelength of waves with wavenumber (K, M) is 2π/κ, where κ = (K ² + M ²)^1/2, and since K_I/M_I = K_R/M_R = t_β, the ratio of the wavelengths of the waves in incident and reflected groups is λ_I/λ_R = M_R/M_I. The time over which a wave group with length L_I and speed c_gI encounters a slope is the same as that over which it departs but with length L_R and speed c_gR. It follows that ratio of the lengths of reflecting groups measured in the directions of their propagation is

(2.1)\begin{equation}{L_I}/{L_R} = {\boldsymbol{c}_{gI}}/{\boldsymbol{c}_{gR}},\end{equation}

which is found using (1.8) and (1.5) and shown in figure 4. The ratios at zero incident azimuth, θ_I = 0°, are equal to R (defined in § 1.3.6) when β > α and, with the adopted selection of incident waves, (−R) when β < α. The lengths of wave groups are generally reduced on reflection (i.e. L_I/L_R > 1) at small or moderate θ_I, but the lengths of reflected groups generally exceed those of the incident groups when θ_I > 130°.

Figure 4.

The ratio of the lengths of incident and reflected groups, L_I/L_R, measured in the directions of their propagation, as a function of the incident azimuthal angle, θ_I: (a) α < β and (b) α > β.

The ratio of the widths, W_I/W_R, of the incident and reflected wave groups can be found as described in appendix C.1

(2.2)\begin{equation}{W_I}/{W_R} = {s_{({\alpha ^/} + \beta )}}/\sin \{ {\cos ^{ - 1}}[{s_{{\alpha ^/}}}{s_\beta } + {c_{{\alpha ^/}}}{c_\beta }{c_{({\theta _I}\text{--}{\theta _R})}}]\} .\end{equation}

This ratio is shown in figure 5 for various α and β. It generally decreases with increasing θ_I. Widths of the reflected groups are substantially less than those of the incident at small θ_I but increase to values that are generally greater when θ_I > 60°, i.e. when W_I/W_R = 1. The relative breadths, B_R, of the reflected wave groups are found in appendix C.2. The ratio B_I/B_R is given by (A3) and (A4) and is shown in figure 6 as a function of θ_I for various α and β. The breadths of the incident and reflected groups are approximately equal for θ_I < 40° and when α < β, although the breadth of the incident groups exceeds that of the reflected by approximately 20 % near θ_I = 160°, compared to the changes in length and width, the breadth of groups is only moderately changed on reflection.

Figure 5.

The ratio of the widths of the incident and reflected groups, W_I/W_R, as a function of the incident azimuthal angle, θ_I. Panel (a) α < β and (b) α > β. The parameter θ_Ic is defined in § 1.3.4.

Figure 6.

The ratio of the breadths of the incident and reflected groups, B_I/B_R, as a function of the incident azimuthal angle, θ_I, at specified values of (a) α < β and (b) α > β.

The ratio L_IW_IB_I/L_RW_RB_R provides a measure of the relative volumes of the incident and reflected wave groups. Its variation with θ_I may be inferred from figures 4–6. When α < β, the volume of an incident wave group is generally greater than that of the reflected group at small θ_I; the ratio L_IW_IB_I/L_RW_RB_R equals R ² and exceeds unity at θ_I = 0°. The ratio, however, decreases as θ_I increases. For example, if (α, β) = (3°, 6°) the ratio reaches unity when θ_I = 37° and if (α, β) = (20°, 30°) it reaches unity when θ_I = 96°. Examples of the variation of the volume ratio near critical slopes are shown later in table 1. The changes in volume may be due in part to variations in the shape of groups on reflection, considered in the next section.

The ratios of group dimensions on reflection take no account of the changes to the amplitude or slope of waves within a reflecting group that may, according to the criteria used to determine the boundaries of a group, result in a variation of its extent.

2.2. Group shape

Insight into the properties of the reflected groups may be provided by choosing particular forms for the incident groups. For example, the reflection of a subcritical group with α < β, relatively large breadth, and azimuth θ_I = 0°, is described in appendix D and shown in figure 7(a). The incident group is selected to have a plane front normal to the direction of propagation. The plane front of the incident group is OE, normal to c_gI. Unless α + β = 45°, the front of the reflected group is skewed, no longer normal to the propagation direction. The angle CAB between the reflected front, along AB, and one normal to the propagation direction, the line AC, is equal to ${\tan ^{ - 1}}[(1 - {t_{(\alpha + \beta )}})/{t_{(\alpha + \beta )}}{t_{(\beta - \alpha )}}]$ and this may be large. E.g. if α = 6° and β = 15°, angle CAB = 84.4°. A case when α > β is illustrated in figure 7(b). The front of the incident group is AD normal to I ₁ and I ₂. The front of the reflected group is OB, twisted from the direction normal to R ₁ and R ₂.

Figure 7.

(a,b) The reflection of wave groups with α + β < 90° and with fronts, OE in (a; α < β) and AD in (b; α > β), that are planar and normal to the group boundaries, I ₁ and I ₂, respectively The wave azimuth is θ_I = 0°. The reflected groups have fronts along (a) AB and (b) OB. All fronts are shown by bold lines. In (c) the incident wave group (stippled) has an elliptical shape with maximum width CB. It propagates between I ₁ and I ₂ at direction β > α, to the horizontal with θ_I = 0°. The reflected group propagates between R ₁ and R ₂. E and F in the reflected group correspond to C and B, respectively.

The reflection of an elliptically shaped subcritical group is sketched in figure 7(c). One axis, CB, extends across the group width from I ₁ to I ₂. Its distortion on reflection can be described as for the incident group in figure 7(a): if the incident and reflected groups are each assumed to remain unchanged as they approach or recede from the boundary, the line across the width of the reflected group, EF, corresponding to CB in the incident group, is skewed as is the front AB in figure 7(a), leading to a twisting of the incident elliptical group shape.

The propagation direction of a group is given by β and azimuthal angle, θ. The latter changes on reflection (§ 1.3.3). As shown in appendix E the waves in the lower part of an incident group reach the sloping boundary at a time different from the upper, an earlier time if α^/ + β < 90°. Generally therefore the group is twisted on reflection from the finite length of the slope. The lower waves of the incident group shown in figure 19(a) are displaced, after reflection, by a horizontal distance, d, given by (A5a) and shown in figure 19(b), from the waves in the upper part of the reflected group. The plane of the reflected waves is no longer vertical as is that of the incident waves in figure 1, but is tilted through an angle proportional to d/W_I. Although moderate at small azimuthal angle, θ_I, and as θ_I approaches 180°, it is shown in figure 8 that d/W_I may exceed 3, with relatively severe twisting of the reflected group, when θ_I is in a range greater than about 80°.

Figure 8.

Twisting of a reflected wave group: the variation of d/W_I with incident azimuthal angle θ_I at three specified values of α and β, where d is the displacement shown in figure 19(b) and given by (A5).

3.1. The height, h, and area, A, of wave group interaction regions

The extent and effectiveness of the interactions between incident and reflected waves described in § 1.3.7, or the energy transferred from the waves to their interaction products, will depend on how far from the slope the incident and reflected wave groups co-exist. Figure 9(a) shows the vertical plane at azimuth angle, θ_I, in which an incident wave group of width, W_I, is travelling between the lines I ₁ and I ₂ with an inclination, β, to the horizontal that exceeds α. The upper edge of the group is labelled I ₂ and the lower edge, I ₁, as in figure 1. The sloping boundary lies along the line ODA at angle α^/ to the horizontal X axis. The group encounters and reflects from the slope between points O and A. The reflected group propagates at azimuthal angle, θ_R, in the vertical plane shown in figure 9(b). The waves in the incident wave group that are reflected from the point O travel along line OE, labelled R ₁. The projection of OE from the azimuthal direction θ_R onto direction θ_I in figure 9(a), is OB. This gives the location of the reflected waves travelling along R ₁. The position of the point B indicates the maximum extent of the region above the slope within which the incident and reflected waves may co-exist and interact; the length of the vertical line, BD = h, say, is the greatest vertical distance from the slope of the region where both incident and reflected wave groups may be present and may interact. At normal incidence when θ_I = θ_R = 0°, the line OB is that of the reflected waves in the group which originate along I ₁ in the incident group. Figure 9(b) takes account of the difference between θ_R and θ_I (e.g. as shown in figure 2). Using simple geometry it can be shown that

(3.1a)\begin{equation}h/{W_I} = 2[{s_\beta }{c_{({\theta _I}\text{--}{\theta _R})}} - {c_\beta }{c_{{\theta _I}}}{t_\alpha }]/\{ {s_{2\beta }}[1 + {c_{({\theta _I}\text{--}{\theta _R})}}]\} ,\end{equation}

when α < β or

(3.1b)\begin{equation}h/{W_I} = 2[{s_\beta }{c_{({\theta _I}\text{--}{\theta _R})}} + {c_\beta }{c_{{\theta _I}}}{t_\alpha }]/\{ {s_{2\beta }}[1 - \textrm{ }{c_{({\theta _I}\text{--}{\theta _R})}}]\} ,\end{equation}

if α > β.

Figure 9.

(a) The vertical (X, Z) plane at azimuthal angle, θ_I, in which an incident wave group of width, W_I, is travelling downwards from the left with inclination, β, to the horizontal. The upper edge of the waves in the group is labelled I ₂ and the lower edge, I ₁, as in earlier figures. (b) The vertical plane of the reflected group with azimuthal angle, θ_R.

The area in which the incident and reflected waves of the group coincide in the vertical plane at angle θ_I (the area of the interaction triangle OBA in figure 9a) is given by A where

(3.2)\begin{equation}A/W_I^2 = (h/{W_I}){c_{{\alpha ^/}}}/[2{s_{(\beta + {\alpha ^/})}}].\end{equation}

Figures 10 and 11 show values of h/W_I and $A/W_I^2$, respectively, as functions of the azimuthal angle of the incident wave, θ_I, at various α and β. Values increase with θ_I but are relatively insensitive to an increase of θ_I when θ_I is less than 30°.

Figure 10.

The non-dimensionalized height, h/W_I, of the interaction region of incident and reflected groups, as a function of θ_I at specified slope and wave propagation angles, α and β: (a) α < β and (b) α > β.

Figure 11.

The non-dimensionalized area, $A/W_I^2$, of the interaction region of incident and reflected groups as a function of θ_I at specified slope and wave propagation angles, α and β: (a) α < β and (b) α > β.

Wave interactions within the area A are examined by Leclair et al. (Reference Leclair, Raja and Staquet2020) for an incident beam of two-dimensional plane waves of finite width, but only with θ_I = 0° and α < β. They calculate the transfer of energy to a second harmonic wave of frequency 2σ within the area A and take account of its radiation, showing that the effects of viscosity will eventually dominate over those of nonlinear transfer as the slope angle, α, increases.

3.2. Spatial and temporal limits on h

Whether interaction can actually occur over the full depth, h, given by (3.1), or over the area A of (3.2), depends on further conditions. The first is spatial: because the wave azimuth changes on reflection, it is necessary that the reflected wave group is broad enough at location E in figure 9 to extend from section θ_R (figure 9b) to θ_I (figure 9a); the breadth of the wave group, B_R, must be sufficiently large. This is so when

(3.3)\begin{equation}{B_I}/{W_I} \gt {({B_I}/{W_I})_0},\end{equation}

where

(3.4a)\begin{equation}{({B_I}/{W_I})_0} = 1/[{s_\beta }(1 + {c_{({\theta _I}\text{--}{\theta _R})}})],\end{equation}

if α < β, or

(3.4b)\begin{equation}{({B_I}/{W_I})_0} = 1/[{s_\beta }(1 - {c_{({\theta _I}\text{--}{\theta _R})}})],\end{equation}

if α > β, where ${c_{({\theta _I}\text{--}{\theta _R})}} = \cos ({\theta _I} - {\theta _R})$. (B_I/W_I)₀ is shown in figure 12 as a function of the direction θ_I, for several values of β and α. When B_I/W_I < (B_I/W_I)₀, the height of the region of possible interaction between the incident and reflected waves is reduced to h(B_I/W_I)/(B_I/W_I)₀; the height, h, given by (3.1) is therefore an upper bound for the extent of the possible interaction region adjoining the slope.

Figure 12.

The lower bound of the wave group dimensional ratio, (B_I/W_I)₀, satisfying the spatial condition that the incident and reflected groups co-exist to height h from the slope, as a function of the incident wave direction θ_I, (a) α < β and (b) α > β.

A second condition for interaction to be possible to height h is temporal: the incident and reflected wave groups are both present at the same time at height, h, only if the front of a reflected wave group has propagated upwards to height, h, in a time before the rear of the incident wave group reaches the height, h. Only then can interaction be possible over the full height, h. This condition depends on the length of the incident wave group, L_I, and can be written

(3.5)\begin{equation}{L_I}/{W_I} \gt {({L_I}/{W_I})_0},\end{equation}

where the minimum value of L_I/W_I is

(3.6a)\begin{equation}{({L_I}/{W_I})_0} = 2{c_{({\theta _I}\text{--}{\theta _R})}}[({\boldsymbol{c}_{gI}}/{\boldsymbol{c}_{gR}}) + 1]/\{ {s_{2\beta }}[1 + {c_{({\theta _I}\text{--}{\theta _R})}}]\} \text{--}{({t_\beta })^{ - 1}},\end{equation}

when α < β, or

(3.6b)\begin{equation}{({L_I}/{W_I})_0} = 2{c_{({\theta _I}\text{--}{\theta _R})}}[1 - \textrm{(}{\boldsymbol{c}_{gI}}/{\boldsymbol{c}_{gR}}\textrm{)}]/\{ {s_{2\beta }}[1 - {c_{({\theta _I}\text{--}{\theta _R})}}]\} + {({t_\beta })^{ - 1}},\end{equation}

when α > β. (A version of (3.6a) is given by Thorpe (Reference Thorpe2001), but only when θ_I = 0° and even then incorrectly.) Examples of the variation of (L_I/W_I)₀ with θ_I are shown in figure 13. As in the spatial condition (3.3), the height of the region of possible interaction between the incident and reflected waves is reduced to h(L_I/W_I)/(L_I/W_I)₀ when L_I/W_I < (L_I/W_I)₀.

Figure 13.

The lower bound of the wave group dimensional ratio, (L_I/W_I)₀, satisfying the temporal condition that the incident and reflected groups are present at the same time at a position at a height h from the slope as a function of the incident wave direction θ_I, for specified slope angles, α, and group velocity directions, β: (a) α < β and (b) α > β.

A further temporal condition is necessary for interactions to lead to a significant effect on mixing: the duration of interaction between the incident and reflected waves must exceed the time required for growth of the instabilities produced by the wave interactions to reach finite amplitude. This time is proportional to the inverse growth rate of instabilities described in § 1.3.7, a time commonly proportional to N ⁻¹ but which depends on their nature. This condition implies that even greater values of ${({L_I}/{W_I})_0}$ than given by (3.5) will be required for interactions to lead to substantial mixing to height h.

As can be seen from figures 12 and 13, values of the limits, (B_I/W_I)₀ and (L_I/W_I)₀, are relatively large at some θ_I. These are lower bounds of the group dimension ratios B_I/W_I and L_I/W_I, and extreme values larger than those given by (3.3) and (3.5) are required for turbulent mixing caused by wave--wave interaction to occur. Although large values of B_I/W_I and L_I/W_I coincide with those at which wave groups have their greatest duration (§ 1.2.3), the latter (large L_I/W_I) being characteristic of wave rays or beams, it is likely that the requirement for extreme values of the ratios will not be satisfied in general. Mechanisms that generate groups with specific, large or small, dimension ratios are yet to be classified. Consequently, rather than being driven by wave interactions, in some regions close to a sloping boundary mixing is more commonly driven through an alternative mechanism. This may be through turbulence induced by mean flow Ekman layers (§ 1.1) or by the breaking of waves in incident or reflected wave groups through convective instability or K–H instability. Mixing through wave breaking is most likely after waves have steepened on reflection (see (1.10)) or when the minimum Richardson number is substantially reduced after wave reflection. Such near-boundary breaking in wave groups is considered in the following section.

4.1. Continuous breaking and mixing

A patch of continuous mixing within a boundary layer parallel to a sloping boundary is promoted by an internal wave group if the direction of a breaking wave in the group, γ, is equal to that of the effective slope, α^/, i.e. from (1.4) and (1.7), when

(4.1)\begin{equation}{t_\beta } = {F^2}/{t_\alpha }{c_\theta }.\end{equation}

This relationship is shown in figure 14 for a typical value F = 0.1 at various values of θ_I specified on the curves. There are no solutions of (4.1) when α or β is small, or when ${t_\alpha }{t_\beta } \lt {F^2}$, below the curve θ = 0° in figure 14. The frequency, σ_b, of waves that satisfy (4.1) is given by (1.1)

(4.2)\begin{equation}{({\sigma _b}/f)^2} = [{F^2} + {({t_\alpha }{c_\theta })^2}]/[{F^4} + {({t_\alpha }{c_\theta })^2}].\end{equation}

Figure 14.

Labelled contours of constant azimuth angle (θ measured in degrees) where continuous mixing occurs in the (α, β) plane from (4.1) when F ( = f/N) = 0.1.

Condition (4.1), whilst necessary, is not sufficient. It must be geometrically possible that c_b lies along the effective slope. Taking for example the wave reflection of an incident wave at zero azimuth, θ_I = 0°, as shown in figure 3, (a(i)) and (b(i)), the sloping boundary plane lies in quadrants 1 and 3 (where X and Z are both positive or both negative). Although being of different magnitude, the phase vector, c (1.2a), and the breaking wave vector, c_b (1.3b), lie in the same quadrants because their components have the same signs. The incident phase velocity, c_I, has both K and M > 0 (figure 3, parts (i)) lying in the first quadrant and therefore so too does c_bI, which may consequently be parallel to the slope. The reflected wave phase velocities, c_R, have K > 0 and M < 0 when α < β, or K < 0 and M > 0 when α > β, as in figure 3(a(i),b(i)), respectively, and so lie in the fourth and second quadrants, respectively. Neither are in the same quadrants as the sloping boundary and c_bR cannot be parallel to the boundary, even when (4.1) is satisfied.

4.2. Successive breaking

If the velocity, c_g, of a group in which a wave is breaking is parallel to the effective slope, successive waves will break in a direction along the slope at time intervals approximately equal to the wave period, T, and separated by distances of Tc_g. A line of mixed patches will then be formed (e.g. as illustrated by Thorpe Reference Thorpe1999b; his figure 4). The condition that c_g is parallel to the effective slope is

(4.3)\begin{equation}{t_\beta } = {t_\alpha }{c_\theta }.\end{equation}

At θ = 0° this gives α = β, and the wave frequency is equal to the critical frequency, σ_c. This is not so if θ ≠ 0°. After reflection, can the group velocity, c_gR, for waves travelling in direction θ_R be parallel to the effective slope in this direction? Substituting ${\theta _R} = {\cos ^{ - 1}}({t_\beta }/{t_\alpha })$ from (4.3) into (1.6) gives ${\theta _I} = {\cos ^{ - 1}}( - {t_\beta }/{t_\alpha }) = {\theta _{Ic}}$; thus waves with incident azimuth angle ${\cos ^{ - 1}}( - {t_\beta }/{t_\alpha })$ will reflect at angle ${\theta _R} = {\cos ^{ - 1}}({t_\beta }/{t_\alpha })$ parallel to the effective slope defined in § 1.3.5. This implies that the corresponding incident waves must travel tangentially to the effective slope. Unless the slope happens to be convex (when effects not considered here may occur, as they are by Gilbert & Garrett Reference Gilbert and Garrett1989; Legg & Adcroft Reference Legg and Adcroft2003) it is difficult to see how such waves travelling tangentially to a sloping boundary may be generated and occur naturally.

4.3. When does wave breaking adjacent to a slope have its greatest effect?

The breaking of incident or reflected wave groups will have a more substantial and extensive effect in mixing the internal surf zone when the waves that are breaking remain within the surf zone, i.e. when the group velocity vectors, c_gI or c_gR, are in directions that remain within the boundary layer or nearly parallel to the effective slope at which waves meet the boundary. Mixing initiated by waves breaking within the internal surf zone will continue to be generated within the zone if the direction, γ, of c_b lies between the lines OA (representing the effective slope) and OB (the upper limit of the reflecting wave group) in figure 9(a). This leads to the conditions ${t_{{\alpha ^/}}} \le {t_\gamma } \le \tan (\textrm{angle}\;\textrm{BOC})$ or

(4.4)\begin{equation}{t_\alpha }{c_{{\theta _I}}} \le {F^2}/{t_\beta } \le {t_\beta }{c_{({\theta _I}\text{--}{\theta _R})}},\end{equation}

when the breaking of incident or reflected waves remains in and repeatedly produces mixing in the internal surf zone. Figure 15 shows the lower (i) and upper (ii) bounds on F ²/t_β in (4.4) at various α for a typical value, F = 0.1. At zero azimuth the upper limit of (4.4) implies that β ≥ tan⁻¹F = 5.71°. Points in the stippled area of the figure satisfy both the conditions of (4.4). For the selected value of F, there appear to be no values of β that satisfy (4.4) when α > 6°. Using the relation (1.6) between θ_I and θ_R it may be shown that, in general, there are no solutions of (4.4) when β < tan⁻¹F. When α < β (subcritical waves) solutions are found in a limited range of θ_I < 90° and in a narrow range of β (consistent with figure 15). When α > β (supercritical waves) solutions are found only in a limited range of θ_I below and adjoining 90°.

Figure 15.

The upper (labelled (i)) and lower (labelled (ii)) bounds of F ²/t_β given by (4.4) when F = 0.1. Shown is the variation of θ_I with β for various slope angles, α, when (a) α = 2°; (b) α = 5° and (c) α = 5.71°. The curves are at points in the stippled areas that satisfy condition (4.4) for the breaking of waves in wave groups to be sustained near the slope.

5.1. Analytical results

The most stable and therefore persistent wave groups in mid-water are those with large L_I/W_I and B_I/W_I (§ 1.2.3). Some of these will eventually reach the sloping boundaries of the ocean and be reflected. There they will modify the internal surf-zone.

Several consequences of the interaction of wave groups with a sloping boundary have been considered. On reflection, changes occur to the dimensions and volume of groups. In § 2.1 these changes are shown to depend on the azimuthal directions, θ_I, of the incident wave groups, as well as on α and β: the relative lengths and widths of the incident and reflected groups, L_I/L_R and W_I/W_R, both decrease with increasing θ_I, at least when θ_I < 90°. The ratio of volumes of the incident and reflected groups is given by L_IW_IB_I/L_RW_RB_R. At θ_I = 0°, the ratio is greater than unity; the volume of an incident wave group exceeds that of its reflection. The relative volume of reflected groups decreases as θ_I increases. At a sufficiently large θ_I, depending on α and β, the volume of an incident group becomes equal to that of its reflection and, at even larger θ_I, its volume will increase on reflection. There are also structural changes. Examples in appendix D with θ_I = 0° show that the shape of the front (or rear) of groups can be changed on reflection. When θ_I ≠ 0°, changes in the orientation of the groups’ boundaries cause groups to become twisted on reflection.

Interactions, including resonant interactions, may occur between the waves in groups that are incident on a slope and those in their reflected groups. The types of possible interactions are summarized in § 1.3.7. For such interactions and subsequent turbulent mixing to be possible there must be a region within which the incident and reflected wave groups co-exist. In § 3 it is shown that there is a region of maximum height, h, and area, A, above a plane slope within which such co-existence is possible. The size of h is subject to conditions on the dimensions of the incident group. A spatial condition demands that the incident and reflected groups must co-exist in space to the distance, h, above the slope. This is satisfied only when B_I/W_I is sufficiently large, i.e. only if B_I/W_I exceeds the value (B_I/W_I)₀, given by (3.4). Otherwise the thickness of the interaction region, h, is fractionally reduced to h(B_I/W_I)/(B_I/W_I)₀. A temporal condition requires that the incident and reflected wave groups must also co-exist in time over the full height, h. This is satisfied only if the ratio L_I/W_I of an incident group's dimensions exceeds a value, (L_I/W_I)₀, given by (3.6). The height of the interaction region is otherwise reduced to h(L_I/W_I)/(L_I/W_I)₀. The height, h, given by (3.1) is therefore an upper bound for the extent of the possible wave interaction region adjoining the slope. As shown in figure 13 (L_I/W_I)₀ may be relatively large, especially when α > β. A further limit is, however, required if there is to be a significant contribution of wave interaction to mixing: the duration of interaction between the incident and reflected waves must exceed the time required for growth to finite amplitude of the instabilities produced by wave interactions; larger values of L_I/W_I than (L_I/W_I)₀ are required if interactions to height h are to lead to turbulence. The time required for an instability to lead to mixing may, however, be reduced if earlier breaking leaves a substantial disturbance on which instability can grow; the intermittency of wave groups may be an important factor.

The spatial and temporal conditions on the size of the region in which interactions may occur coincide with those of the longest lived and therefore most stable wave groups (§ 1.2.3). Given their severity, however, the conditions appear to be unlikely to be satisfied on all oceanic slopes, and their turbulent boundary layers may be sustained by the breaking of waves in incident or reflected wave groups through convective instability or K–H instability rather than through resonant interactions.

Table 1 shows examples of how values of the parameters defining the changes in the dimensions of a group vary when its component waves are near critical. Examples are selected at α = 3° and 20°, typical of the continental slopes and of seamounts, for incident wave azimuths of 0° and 45°. The table shows how, in the chosen example, θ_I = 45°, the azimuthal direction of reflected waves, θ_R, is rotated towards the up- or down-slope direction in near-critical conditions. The group characteristics, L_I/L_R, when θ_I = 0° and 45° and W_I/W_R, when θ_I = 0°, approach maximum values as β tends to the critical direction of the component waves in the incident group. The ratio, L_IW_IB_I/L_RW_RB_R, of the volume of the incident and reflected groups also increases as β tends to α from above or below. Far greater changes in group volume occur near critical conditions at θ_I = 0° than at θ_I = 45°. Maxima near critical frequencies are not apparent in W_I/W_R when θ_I = 45°, or in B_I/B_R; evidently the group properties may differ from those of the component waves.

The breaking of incident or reflected wave groups will have a more extensive effect within the internal surf zone when the waves that are breaking there remain near the slopes, leading to repeated mixing within the internal surf zone. Conditions in which this occurs are summarized by (4.4) and marked by the stippled areas of figure 15. The wave groups must travel in directions β > α, with β in a moderate range of values that exceed tan⁻¹F, but with a broad azimuth angle, θ_I. These conditions are satisfied only at small α, typical of the continental slopes, and at moderate β.

5.2. Observations

Except for the definitive observations of K–H billows by van Haren & Gostiaux (Reference van Haren and Gostiaux2010) at 10–20 m above the sloping side of the Great Meteor Seamount in the Canary Basin, the available measurements in the boundary layer over a slope are generally insufficient to establish whether, when or where, the mixing is generated as a result of resonant interactions, or by convective or K–H instability.

The values, H, in table 2 are the heights above the sloping bottom to which Eriksen finds that the internal wave spectra are enhanced over a band of frequencies centred at the local internal wave critical frequency, σ_c. At the Muir Seamount (MS) spectral enhancement decays roughly exponentially from the sloping bottom with a vertical scale of about 100 m, the value taken to characterize H at the MS in table 2. The enhancement is studied in greater detail by Eriksen (Reference Eriksen1998) using data obtained on the side of the Fieberling Guyot (FG). High above the sloping bottom of the FG the internal wave spectra are in agreement with those predicted by Garrett and Munk (Munk Reference Munk1981). Variation of the observed wave spectra from the G-M model appears to be due to the presence of reflected waves and can be detected to a vertical distance, H, from the bottom of 750 m, the value given at the FG in table 2. Approximate measures of the frequency bands of spectral enhancement at about 100 m off the bottom at the MS and FG are given by Δσ in table 2. These are not symmetrical about σ_c; the upper values are almost twice those of the critical frequencies, σ_c, suggesting that a sub-harmonic resonance may be a contributor.

Table 2.

Parameter values characterizing observations reported by Eriksen (Reference Eriksen1982, Reference Eriksen1998) over the Muir Seamount (MS) and Fieberling Guyot (FG). ‘depth’ is the water depth at the location of observations. The height, H, of the boundary layer in which wave spectra are affected by wave reflections from the slope are those found by Eriksen (Reference Eriksen1982, Reference Eriksen1985, Reference Eriksen1998) and explained in § 5.1. The parameter Δσ gives the approximate range of wave frequencies around the critical frequency, σ_c, over which spectra are significantly enhanced above the Garrett--Munk spectra at a height of approximately 100 m above the slope.

The boundary layers of spectral enhancement of internal waves having thickness H in table 2 are distinct from those in which turbulent mixing occurs. At the FG Eriksen (Reference Eriksen1998) detects relatively frequent density overturns that occur between sensors separated vertically by 10 m. Their occurrence frequency decreases monotonically with height up to at least about 100 m off the bottom, a distance substantially less than H. At 20–30 m off the bottom Richardson numbers, Ri, < 0.25, indicative of impending or existing K–H instability occur for 23.1 % of the one year observation period. Approximately half of these low Richardson numbers are associated with negative density gradients (when Ri < 0), and may be a consequence of convective instability. Eriksen deduces that, rather than a superposition of tidal or lower frequency activity, the internal wave activity is responsible for the occurrence rates of unstable conditions; shear and vertical strain fields are dominated by near-critical internal waves. It is not possible, however, from the moored instruments to distinguish between K–H instability and convective instability as sources of overturns; Eriksen is unable, unambiguously, to establish the origin of the overturns. There appears to be a lack of suitable observations of the time and spatial variation of the density structure over a relatively uniform and gentle continental slope to which the present theoretical study can be related. There are, moreover, no observations of the dimensions of wave groups that can provide guidance.

We may, nevertheless, examine Eriksen's data further to see whether (or not) they conform to our analytical results. Table 3 provides values of parameters at the MS and FG that are derived from the analysis of §§ 2–4, taking β as the inclination, β_{M 2}, of internal waves of tidal semi-diurnal M ₂ frequency found using (1.1). This is less than the slope angles, α, and the M ₂ internal tidal waves are therefore supercritical. Tabulated values of L_I/L_R and (L_I/W_I)₀ are found at θ_I = 0° from (2.1) and (3.6), respectively, but are representative of values in the range of azimuths, 0° ≤ θ_I ≤ 30°. The lengths of reflected groups are less than the incident and, for resonant interactions to be possible, L_I/W_I must be at least greater than 2.6.

The lengths h/W_I in table 3 are the non-dimensional distances from the slope to which the incident and reflected waves may both be present provided that values of B_I/W_I exceed (B_I/W_I)₀ and L_I/W_I exceed (L_I/W_I)₀. If h is set equal to Eriksen's observed distance of about 100 m for the extent of the turbulent boundary layer at the FG, then the tabulated value of h/W_I at the FG, 2.92, implies that W_I is approximately 34 m. Using the tabulated values of (B_I/W_I)₀ and (L_I/W_I)₀ at the FG, B_I and L_I, however, need to exceed 260 m and 91.4 m, respectively, if the regions of wave interactions are to extend to the height of 100 m to which overturns provide evidence of turbulent mixing. The corresponding vertical area at the FG, A, is 3.29 × 10³ m². The up-slope length to which a wave group must extend is l = 72.7 m, equal to ${W_I}/{s_{(\alpha + \beta )}}$ or 2A/(hc_α), and B_I and l define the smallest dimensions of regions on the slope required for incident and reflecting wave groups to support mixing through interactions to a height of 100 m. Values of β and length scales are summarized in table 4. Values B_I, L_I and l at the FG defining the smallest sizes of incident and reflected wave groups that can co-exist and therefore possibly interact, are less than the 2.5 km along- and up-slope length of uniformity of the 24.4° slope of the FG around Eriksen's moorings, implying that the assumption that the slope is relatively extensive and uniform may be valid. Provided that incident M ₂ internal tidal wave groups with dimensions B_I > 258 m and W_I > 90.4 m exist, incident and reflected wave groups can co-exist and may interact within the 100 m thick region of observed overturn. No call is required for mixing by waves breaking through K–H instability or convective overturn.

Comparable values for subcritical waves of frequency 1.1σ_c and 1.2σ_c at the FG are also given in table 4. The minimum length scales at frequency 1.1σ_c are relatively large; the vertical size of a wave group is W_I/c_β = 1.8 km, comparable to the water depth of 1.415 km at FG where Eriksen's moorings were located, while l exceeds the 2.5 km dimensions of the smooth slope. The analysis is therefore unlikely to be valid for the reflection of wave groups containing waves of frequency 1.1σ_c. The values at a frequency of 1.2σ_c appear more possible although, from (1.10) with θ_I = 0°, it is found that the slopes of waves in the reflected groups exceed those in the incident wave groups by a factor of 75.6 implying that even the reflection of incident waves of moderate slope may be unstable to convective instability. The length scales are proportional to h and would be doubled if h = 200 m.

The necessary (but not sufficient) condition for continual wave breaking near the slope, (4.4), cannot be satisfied at either the MS or the FG; the slopes, α, are too great. Values of σ_b/f given in table 3 are slightly greater than unity implying that the frequency of breaking waves that will result in a breaking layer that travels along the slope slightly exceeds the inertial. Disturbances of such frequency and period are evident in Eriksen's (Reference Eriksen1998) spectra at FG.

Table 3.

The value of β_{M 2} is the inclination of the M ₂ internal tide to the horizontal found using (1.1) when the wave frequency, σ_{M 2}, is that of the internal M ₂ (semidiurnal) tide, 1.40 × 10⁻⁴ rad s⁻¹. Values of parameters, L_I/L_R to (L_I/W_I)₀ at the Muir Seamount (MS) and Fieberling Guyot (FG) are derived from the analysis when θ_I = 0° and β = β_{M 2} with other values taken from table 1. Here, γ_{M 2} is the inclination of the breaking wave direction (1.4) to the horizontal for waves of M ₂ frequency. The calculated ratios T_d/T are those of the duration of turbulence caused by convective instability or by K--H instability, T_d, divided by the wave period, T, for waves of M ₂ frequency.

Table 4.

Values of β and length scales from the FG for incident waves of frequencies σ_{M 2}, 1.1σ_c and 1.2σ_c. The group lengths W_I, B_I, L_I and l must be exceeded if wave interactions are possible to h = 100 m.

The observations by van Haren & Gostiaux (Reference van Haren and Gostiaux2010) at the Great Meteor Seamount where K–H billows are observed are in a water depth of 549 m over a slope ≈ 4°. Temperature variations are dominated by supercritical waves of M ₂ frequency (β_{M 2} ≈ 1.5°). Values comparative to those at the FG are h ≈ 50 m, W_I = 59.8 m, B_I = 1,140 m, L_I = 1,370 m and l = 623.9 m. These are relatively moderate values. On reflection an incident M ₂ internal wave is increased in steepness by a factor 4.8. It appears possible that, in addition to the K–H instability made evident by billows, resonant wave interactions will also be possible above the Seamount.

Following Kaminski & Smyth (Reference Kaminski and Smyth2019) and as noted in § 5.1 in relation to the intermittency of wave groups, some account should, in general, be taken of the effects of pre-existing turbulence when assessing the new contribution of breaking incident internal wave groups to boundary layer turbulence. The ratios, T_d/T, given by (A1) and (A2) in appendix B and displayed in table 3 for the M ₂ internal tides at the MS and FG are less than unity, implying that turbulence will not persist over the time, T, between the onset of breaking events in a group. Only at wave frequencies exceeding those of the M ₂ tide is turbulence produced by one breaking wave on the steep slopes of the MS and FG likely to persist until its successor is about to break, when it may consequently affect the successor's instability and subsequent transition to turbulence.