Abstract

A series of two-dimensional numerical simulations examine the breaking of first-mode internal waves at isolated ridges, independently varying the relative height of the topography compared to the depth of the ocean h0/H0; the relative steepness of the topographic slope compared to the slope of the internal wave group velocity γ; and the Froude number of the incoming internal wave Fr0. The fraction of the incoming wave energy, which is reflected back toward deep water, transmitted beyond the ridge, and lost to dissipation and mixing, is diagnosed from the simulations. For critical slopes, with γ = 1, the fraction of incoming energy lost at the slope scales approximately like h0/H0, independent of the incoming wave Froude number. For subcritical slopes, with γ < 1, waves break and lose a substantial proportion of their energy if the maximum Froude number, estimated as Frmax = Fr0/(1 − h0/H0)2, exceeds a critical value, found empirically to be about 0.3. The dissipation at subcritical slopes therefore increases as both incoming wave Froude number and topographic height increase. At critical slopes, the dissipation is enhanced along the slope facing the incoming wave. In contrast, at subcritical slopes, dissipation is small until the wave amplitude is sufficiently enhanced by the shoaling topography to exceed the critical Froude number; then large dissipation extends all the way to the surface. The results are shown to generalize to variable stratification and different topographies, including axisymmetric seamounts. The regimes for low-mode internal wave breaking at isolated critical and subcritical topography identified by these simulations provide guidance for the parameterization of the mixing due to radiated internal tides.

1. Introduction

The breaking of low-mode internal waves at sloping topography is the focus of this study. These waves, generated principally by winds and tidal or subinertial flow over topography, transport mechanical energy around the ocean. When the internal waves break, that energy is dissipated, and some fraction may be used for mixing. Questions to be examined in this study are therefore where do the internal waves break, relative to the topography, and where is the mixing distributed?

The role of sources of mechanical energy such as the tides and winds in mixing the ocean has been described in Munk and Wunsch (1998), Wunsch and Ferrari (2004), St. Laurent and Garrett (2002), and Garrett and Kunze (2007). A complex chain of processes ultimately converts barotropic tidal energy to mixing. Considerable progress has been made in understanding the generation of baroclinic tides (Althaus et al. 2003; Nash et al. 2006; Garrett and Kunze 2007; Rudnick et al. 2003; Egbert and Ray 2000; Carter et al. 2008; Merrifield and Holloway 2002; Zilberman et al. 2011) and developing theoretical models of the conversion of barotropic tidal energy into baroclinic energy (Bell 1975; Balmforth et al. 2002; Llewellyn Smith and Young 2002; St. Laurent et al. 2003; Khatiwala 2003; Llewellyn Smith and Young 2003; Nycander 2005; Balmforth and Peacock 2009). A variety of processes gives rise to local breaking of the baroclinic tides near the generation site including nonlinear lee waves (Klymak et al. 2008; Legg and Klymak 2008; Levine and Boyd 2006), boundary processes (Aucan et al. 2006; Lee et al. 2006; Gayen and Sarkar 2011), internal tide beams (Johnston et al. 2011), and nonlinear wave–wave interactions (Nikurashin and Legg 2011; Carter and Gregg 2006). Current estimates for the local fraction of dissipation are in the range of 10%–40%, dependent on the topography, tidal forcing, and latitude (St. Laurent and Nash 2004; Klymak et al. 2006; Alford et al. 2011). The majority of the internal tide energy therefore propagates far away from the generation site, usually in the form of low vertical modes (which are least susceptible to immediate breaking) (Ray and Mitchum 1996; Alford and Zhao 2007; Zhao et al. 2010). How and where these low modes eventually break is the focus of this study.

One possibility is that the low modes undergo parametric subharmonic instability at the critical latitude where the tidal frequency is exactly twice the Coriolis frequency (Alford 2008; MacKinnon and Winters 2005; MacKinnon et al. 2013a,b). Another possibility is that scattering from small-scale random topography causes a cascade of energy from low to high modes and ultimately dissipation (Buehler and Holmes-Cerfon 2011; St. Laurent and Garrett 2002). A third possibility, the focus of this study, is that the low-mode internal waves break at large-amplitude topography far from the generation site (Muller and Liu 2000; Johnston and Merrifield 2003; Johnston et al. 2003; Nash et al. 2004, 2007; Martini et al. 2011).

When internal waves propagating from deeper to shallower water reflect from sloping topography, an increase in energy density results. If the topography is at the critical angle (i.e., where the topographic slope is equal to the slope of the wave group velocity vector) the resultant very high energy density adjacent to the slope leads to overturning and the transfer of energy to turbulence (Wunsch 1969; Eriksen 1982, 1985; Ivey and Nokes 1989; Slinn and Riley 1996; Ivey et al. 2000; Nash et al. 2004). Previous studies have shown that mixing occurs in the form of an upslope propagating bore (Cacchione and Wunsch 1974; Venayagamoorthy and Fringer 2007). Mixing is not confined to the critical angle, but rather occurs for a range of near-critical angles dependent on the incoming wave Froude number and can occur for monotonic slopes with a variety of shapes (Legg and Adcroft 2003). One focus of this study will be a full quantitative examination of the variation of total dissipation on the near-critical slope as a function of topographic height, complementing the phenomenological study with a small number of simulations in this regime performed by Johnston and Merrifield (2003).

In the present study, the focus is primarily on the mixing induced by low-mode scattering at critical and subcritical slopes. Apart from one example of a supercritical slope for comparison, and two examples with variable slopes, supercritical slopes are not examined here because they have already been considered extensively by other authors, particularly Klymak et al. (2013), Kelly and Nash (2010), Muller and Liu (2000), Johnston and Merrifield (2003), Cacchione and Wunsch (1974), Legg and Adcroft (2003), and Hall et al. (2013). A subcritical slope will transmit most of the incoming wave energy, reflecting little, and from Legg and Klymak (2008) does not allow for lee-wave generation, and so the subcritical slope problem might seem straightforward and uninteresting. However, for the purposes of climate model parameterizations, it is essential to distinguish between scenarios in which all the incoming wave energy is transmitted to the shelf, perhaps as a solibore (Mackinnon and Gregg 2003; Shroyer et al. 2011) where dissipation will have little influence on the large-scale open ocean, and scenarios where some of the wave breaking occurs over the continental slope, in deeper water. One focus of this study will therefore be the location of wave breaking over subcritical slopes, and in particular the topographic depth at which breaking first occurs.

This study examines the interaction between low-mode waves and simple topographic slopes, varying the topographic height and slope and incoming wave Froude number. Related earlier studies include Cacchione and Wunsch (1974) (laboratory experiments with critical, subcritical, and supercritical slopes), Johnston and Merrifield (2003) (numerical simulations with three-dimensional topography and critical, subcritical, and supercritical slopes), Hall et al. (2013) (numerical simulations with subcritical, supercritical, and critical slopes, with a focus on variable stratification) and Venayagamoorthy and Fringer (2006, hereafter VF06) who examined low-mode internal wave scattering from slopes of varying steepness and for different incoming wave Froude numbers. The present study complements the earlier studies by independently varying all three of the slope steepness, the topographic height, and the incoming wave Froude number, and quantifying the dependence of energy loss due to wave breaking on these parameters. VF06 changed the topographic steepness by keeping the horizontal length scale of the topography fixed and varying the height, so that the two parameters were not independent. Cacchione and Wunsch (1974) examined the interaction of internal waves with slopes of varying steepness, but did not examine the role of Froude number, or of variations in the total elevation of the slope, and could not measure dissipation. Johnston and Merrifield (2003) do not discuss energy budgets or quantitative scaling, focusing instead on phenomenology of scattering for a small number of cases. Hall et al. (2013) concentrate on variations in steepness and stratification, keeping Froude number and slope height constant for each choice of stratification profile.

For simplicity most of the simulations in this study employ single ridge topographies, but a few simulations consider continental slope/shelf topography, and—in three dimensions—axisymmetric seamounts, to examine the sensitivity of results to the specifics of the topographic shape. The domain is forced by incoming first-mode waves, which have a single velocity zero crossing, separating the horizontal velocity anomaly maxima at top and bottom. Most of the simulations have uniform stratification: the results may be generalized to variable stratification by using a Wentzel–Kramers–Brillouin (WKB)-stretched vertical coordinate (Llewellyn Smith and Young 2003) as shown by Hall et al. (2013). Johnston and Merrifield (2003) also show qualitatively similar behavior for variable and constant stratification. A small number of simulations in this paper are repeated for variable stratification to quantify the dependence of wave energy loss on stratification profile. The goal of the study is to understand the fraction of incoming wave energy that is dissipated/reflected/transmitted at the slope, and the location of any dissipation relative to the topography, as a function of topographic height, steepness, and incoming wave amplitude.

Section 2 outlines the parameter space for the problem of low-mode internal waves encountering a single ridge, estimating the different regimes where wave breaking can occur. Section 3 describes the idealized numerical simulations designed to verify these wave-breaking regimes, and the energy budget diagnostics used in the analysis. In section 4, the results of the simulations for 2D ridge topographies and uniform stratification are described and compared with the theoretical regime predictions. Section 5 describes simulations with variable stratification, different topographic shapes, and in 3D, axisymmetric seamount topographies. Finally, the implications for large-scale model parameterizations of internal wave–driven mixing are discussed.

2. Theoretical context

Here, the behavior of internal waves normally incident on finite uniform slopes in two dimensions is summarized. Linear internal wave reflection theory from Phillips (1966) is used to make predictions of the change in Froude number on reflection and hence determine criteria for wave breaking.

Consider a domain (Fig. 1a) of maximum depth H0, and local depth H(x), where x is the horizontal location, with uniform buoyancy frequency N and Coriolis frequency f. Topography consists of a single ridge of topographic displacement h(x) and maximum height h0 and width 2L (where L is the horizontal distance from the bottom of the ridge to the top). (Although the simulations use a sinusoidal-shaped topography, for simplicity the slope sketched in Fig. 1 has a triangular shape, and the analysis is applicable to either shape.) The internal wave is characterized by a frequency ω, and has a first-mode form, such that the vertical wavelength of the wave λz is twice the depth of the fluid: λz = 2H. The wave horizontal velocity field has the form:

 
formula

where U is the peak horizontal velocity, which has an initial value in the deepest part of the domain (where H = H0) given by U0, and s is the wave characteristic slope:

 
formula

The parameter space is therefore defined by H0, h0, L, ω, N, f, λz and U0. From these eight physical parameters, a set of six nondimensional parameters can be constructed. One important nondimensional parameter, the relative steepness of the slope, is obtained from s:

 
formula

Critical slopes have γ = 1, sub- and supercritical slopes have γ < 1 and γ > 1, respectively. Note that for the sinusoidal slope shape used in the numerical simulations γ is not constant over the slope; the topography will be characterized by its maximum value of γ.

Fig. 1.

Schematic diagrams showing (a) the important topographic parameters for the single ridge scenario and (b) the width of the beam that intersects a topographic slope at the critical angle θ = atan(s), where s is the wave characteristic slope.

Fig. 1.

Schematic diagrams showing (a) the important topographic parameters for the single ridge scenario and (b) the width of the beam that intersects a topographic slope at the critical angle θ = atan(s), where s is the wave characteristic slope.

A second key nondimensional parameter associated with the wave is the Froude number

 
formula

which for a first-mode internal wave where λx = 2H/s becomes

 
formula

where Cp is the horizontal phase speed, and λx is the horizontal wavelength. When H = H0, Fr = Fr0 = U0πs/(H0ω).

In addition to γ and Fr0, the other nondimensional parameters are h0/H0, ω/N, ω/f, and λz/H0. In this study ω/N and ω/f are fixed, and forcing consists only of a mode-1 wave, that is, with λz/H0 = 2. Hence, this study examines the response of wave scattering to variations in γ, Fr0, and h0/H0.

Now consider the interaction of this low-mode internal wave with the sloping topography in several different limiting regimes, defined by the topographic slope and topography height (Fig. 2). In the first regime, the horizontal length scale over which the topography changes substantially is large relative to the horizontal wavelength: H0/(dh/dx) > λx. Substituting for λx for a first-mode wave: λx = λz/s = 2H0/s, where s is the wave slope, the slowly changing topography regime is given by γ < ½. In this slowly varying topography regime, the topographic slope is therefore subcritical. The change in wave properties as the wave travels up the slope can be estimated using WKB theory.1 Using WKB theory, assume that the wave propagates horizontally preserving its first-mode form; the change in the properties of the wave as the depth changes slowly can then be estimated. The vertical wavenumber m(x) = π/H(x), and the horizontal wavenumber k(x) = sm(x). Assuming the vertically integrated energy flux is conserved, then

 
formula

where E is the internal wave energy density, proportional to U2; U is the horizontal velocity amplitude, which varies as depth changes; and Cg is the horizontal group velocity, proportional to 1/m and hence to H for mode-1 waves. The Fr = U/Cp can therefore be estimated as a function of depth and initial wave Froude number, by substituting for U into Eq. (5):

 
formula

where Fr0 is the Froude number of the wave at depth H0. [Use of the exact Wunsch (1969) slope solutions also gives a 1/H2 dependence for the Froude number.] The first-mode wave will likely break, leading to mixing, when the Froude number is greater than order one, which occurs at a depth

 
formula

The minimum value of H(x) = H0h0, so wave breaking occurs when

 
formula

where Frmax is the maximum Froude number of the lowest mode, and Frcrit is the critical value of Froude number necessary for mixing. This criterion for breaking can be rewritten in terms of the relative height:

 
formula

For gently sloping topography, wave breaking likely occurs when Frmax is large, which from Eq. (10) occurs for topography of sufficient height, the critical height being dependent on the amplitude of the incoming wave. This region of parameter space where mixing on subcritical slopes is expected is identified by region A in Fig. 2 (although the WKB theory used here only formally applies for γ < ½).

Fig. 2.

Regime diagram, showing the parameter space defined by relative topographic height h0/H0 and relative topographic steepness γ. Regions A and C have a subcritical slope and a large and small h0/H0, respectively. Region B is the near-critical slope region, bounded by α1 < γ < α2, where and . Region D is the highly supercritical slope region examined in Klymak et al. (2013). Region E is intermediate between regions B and D, and region F is the small topography h0/H0 ≪ 1 regime.

Fig. 2.

Regime diagram, showing the parameter space defined by relative topographic height h0/H0 and relative topographic steepness γ. Regions A and C have a subcritical slope and a large and small h0/H0, respectively. Region B is the near-critical slope region, bounded by α1 < γ < α2, where and . Region D is the highly supercritical slope region examined in Klymak et al. (2013). Region E is intermediate between regions B and D, and region F is the small topography h0/H0 ≪ 1 regime.

A second regime is the near-critical slope regime considered in Ivey and Nokes (1989) and Legg and Adcroft (2003), where γ is close to unity. Wave breaking occurs when the reflected wave has a Froude number greater than unity, which from Legg and Adcroft (2003) occurs when

 
formula

Notice that in this regime there is a dependence on the topographic steepness, but no dependence on the topographic height relative to the water depth. This near-critical region is identified by region B in Fig. 2.

For a finite-length critical slope, only a fraction of the incoming wave energy will reflect from the slope, and all the beams that do not intersect the slope are assumed to pass beyond the ridge without loss of energy. On geometric grounds, for a triangular ridge (i.e., piecewise constant slope) all downward beams originating in a region of horizontal width Lx = 2L = 2h0/s will intersect the slope (Fig. 1b). The ratio of this length to the horizontal wavelength gives the fraction of the beams that reflect from the slope: Lx/λx = h0/H0. Most of the energy that is incident on the critical slope is expected to dissipate at or near to the slope, because the reflected wave will have very large Froude number, and therefore break at the slope. The fraction of the wave energy flux that is therefore unaffected by the slope, and transmitted in the forward direction, is predicted to be 1 − h0/H0.

A third regime is that of steep topography. Specifically when γ > 2.0, the topography can be approximated by a knife edge (St. Laurent et al. 2003), and analytical solutions for the reflected and transmitted waves calculated. Wave breaking occurs through the generation of lee waves near the ridge crest. Because this regime, identified as region D in Fig. 2, is covered in detail in Klymak et al. (2013), it is not the main focus here. Region E in Fig. 2 denotes the supercritical region where dynamics is intermediate between the near-critical and lee-wave regimes, a region that decreases in extent as the incoming wave Froude number increases (and region B therefore extends to higher γ values).

Finally, for topography that has a much smaller horizontal and vertical length scale than the incoming wave, the flow due to the low-mode wave at the topography is similar to a barotropic flow, and the response will therefore be similar to that due to a barotropic flow, that is, as predicted by the Bell (1975) theory for subcritical topography, or as predicted by Llewellyn Smith and Young (2003) for very steep topography. The limits of this regime are h0λz/2 → h0H0 and Lλx/2 → γh0/H0. This regime is shown by region F in Fig. 2.

The following regimes for low-mode internal wave scattering from isolated topography are therefore proposed, controlled by the parameters h0/H0, γ and Fr0: for low-amplitude topography, the response will resemble the internal tides generated by barotropic flow over the same topography (region F); for taller topography the subcritical regime will lead to wave breaking only when both the incoming wave Froude number and topographic height are large enough (region A); while, for critical topography, wave breaking along the slope will occur for all incoming Froude numbers and topographic heights, with the transmitted fraction of the energy equal to 1 − h0/H0 (region B).

The analysis has assumed a constant slope over the length of the topography, while sinusoidal topography has a variable slope. However, the criterion for breaking at near-critical slopes requires the slope to be within the range defined by Eq. (11), not for the slope to be exactly critical, so this criterion is satisfied over much of the sinusoidal slope when the maximum value of γ = 1.0. Similarly, the analysis for subcritical slopes assumes that γ ≤ 0.5, which will be the case for a sinusoidal slope with a maximum value of γ = 0.5. The sinusoidal shape of the topography does not therefore change the expected wave-breaking regimes.

3. Methods

a. Numerical model

A series of simulations are carried out with the Massachusetts Institute of Technology General Circulation Model (MITgcm) nonhydrostatic model, with the majority of simulations being in a 2D (x, z) plane, where flow is permitted in the y direction, but y derivatives are identically zero. A first-mode internal wave is forced at the left boundary and radiative boundary conditions are applied at the right boundary. All simulations have f = 8 × 10−5 s−1 (appropriate to subtropics) and an internal wave frequency of ω = 1.41 × 10−4 s−1 (the M2 tidal frequency). The Coriolis parameter has been deliberately chosen to give f > ω/2 so that parametric subharmonic instability is not expected to occur (MacKinnon and Winters 2005). The depth of the domain is 4700 m, and the length of the domain is 7λx for most simulations (larger for topography with a very large horizontal scale). Resolution is Δx = 320 m and Δz = 23.5 m in the horizontal and vertical directions, respectively, with a total of nx × nz = 1400 × 200 grid points. This rather low resolution enables a large number of simulations to be carried out until there is a balance between incoming and outgoing energy on the slope, allowing an extensive survey of the parameter space. Simulations in 3D, while feasible, require much more computing time, and so only two 3D simulations are performed to examine the role of finite topographic width. Moreover, a full understanding of parameter dependencies in 2D is a prerequisite to useful 3D calculation. Whereas each 2D simulation took approximately 1000 CPU hours, 3D simulations of the same length but slightly reduced horizontal resolution (nx × ny × nz = 800 × 400 × 200) took more than 200 times as long, that is, about 3 weeks per simulation on 400 processors. A parameter space survey of 30 simulations is therefore not feasible in 3D, and for this reason the main focus of this study is on 2D simulations.

Most simulations have a uniform buoyancy frequency of N = 8 × 10−4 s−1, with some additional simulations with variable stratification for comparison. This choice of stratification has been made to give a horizontal wavelength that is relatively small, to minimize the range of scales that need to be resolved (from the small scales of breaking to the large scales of the mode-1 internal wave). Although this value of stratification is relatively small compared to thermocline ocean values, parameters are nonetheless in the oceanographic regime of N > ω with ω/N = 0.18, so that multiple higher harmonics of the forcing frequency are possible. Prior to breaking, the linear waves are still in the hydrostatic regime (calculating the corrections to horizontal wavenumber and wave characteristic slope s using the full nonhydrostatic formulae gives a correction of about 2% compared to the hydrostatic estimates). For large topographies, Nh/U ≫ 1, another nondimensional measure of strong stratification.

In most simulations, the topography consists of a single ridge, with a piecewise sinusoidal form:

 
formula

for 3λx < x < 3λx + 2L, where L varies so that the maximum topographic slope is dh/dxmax = γs, and γ varies from 0.25 to 2. This form of the topography ensures there are no discontinuities in topographic height or slope, but also provides for a slope that is close to the maximum value over much of the change in elevation. By contrast a piecewise linear slope would have slope discontinuities at top and bottom, whereas a Gaussian slope would lack a region of close to uniform slope. The 3λx separation between the beginning of the topography and the left boundary attempts to minimize the influence of reflected waves on the boundary forcing. However, for scenarios where significant backward reflection occurs, the simulation has to be discontinued when those reflected waves reach the left boundary. Most simulations have from little to no reflection, so this is a problem for only a small number of the calculations to be described (in particular those with supercritical slope).

At the left boundary an internal wave is forced by prescribing velocities and temperature as a function of height and time, as in Legg and Adcroft (2003). An Orlanski radiative condition is applied at the right boundary to allow the internal wave energy to propagate out of the domain. Free-slip boundary conditions are applied at the bottom boundary, because resolution is not high enough to resolve the bottom boundary layer, and wave breaking rather than frictional processes are the focus of this study. At the top surface, a linearized free surface is applied. With this boundary condition, nonzero vertical motion is allowed at the top boundary and there is a nonzero contribution to pressure from the boundary displacement. However, the boundary displacement η is assumed to be small relative to the fluid depth, so that boundary conditions are applied at z = 0, rather than z = η (much as in the classical text book solution for linear surface waves).

The subgrid-scale scheme of Klymak and Legg (2010) is used, which estimates the vertical diffusivity based on diagnosed Thorpe scales whenever overturns appear, and uses a turbulent Prandtl number to then estimate vertical viscosity. Large values of vertical diffusivity and viscosity are therefore prescribed whenever overturns appear. This scheme is suitable for stratified flows where the largest scales of overturning are resolved, but the small scales of turbulence are not. Viscosity and diffusivity can be very small outside of overturning regions, and hence the linear waves will not be damped. This parameterization scheme has few tunable parameters, and the same standard values as in Klymak and Legg (2010) are used: the Thorpe scale and Ozmidov scale are assumed equal, the mixing efficiency has a value of 0.2, and the turbulent Prandtl number has a value of 1. Outside of overturning regions background vertical diffusivity and viscosity are set to 10−5 m2 s−1. Horizontal viscosity and diffusivity have constant values of 10−4 m2 s−1.

The parameters varied between simulations are h0 the maximum height of the ridge, γ the relative steepness of the topography, and U0 the velocity amplitude of the incoming wave. Table 1 lists the full range of simulations and relevant nondimensional parameters for the uniform stratification, sinusoidal ridge 2D simulations, and the parameter space covered is shown in graphical form in Fig. 3. For the default internal wave amplitude (U0 = 12 cm s−1) there are three series of simulations with progressively increasing h0/H0: γ = 1.0 (critical slope), γ = 0.5 and γ = 0.25 (subcritical slope). Other smaller series of simulations have a critical slope and U = 6 cm s−1, or a subcritical slope γ = 0.5 and U0 = 24 or 6 cm s−1.

Table 1.

Values of topographic and internal wave parameters for the standard configuration (constant stratification, 2D sinusoidal ridge) simulations.

Values of topographic and internal wave parameters for the standard configuration (constant stratification, 2D sinusoidal ridge) simulations.
Values of topographic and internal wave parameters for the standard configuration (constant stratification, 2D sinusoidal ridge) simulations.
Fig. 3.

The h0/H0, γ, and Fr0 parameter space covered by the standard configuration numerical simulations described in this paper.

Fig. 3.

The h0/H0, γ, and Fr0 parameter space covered by the standard configuration numerical simulations described in this paper.

The configuration in which the majority of calculations are carried out, with uniform stratification and 2D sinusoidal topography, will be referred to as the standard configuration. Additional simulations to be described in later sections examine the influence of variable stratification, variable topographic shape, and, using 3D simulations, finite topographic width.

b. Energy budgets: Reflectance, transmittance, and dissipation

The principal goal of this study is to determine the relative amounts of the incoming wave energy that are reflected, transmitted, and dissipated [as in Hall et al. (2013)] and their dependence on slope, topographic height, and Froude number. To this end the main diagnostic is the mechanical energy budget, comprising the kinetic energy and potential energy budgets, which for the Boussinesq model with free surface are as follows:

 
formula
 
formula

where u = (u, υ, w) represents the three-dimensional velocity field, b is the buoyancy, p is the pressure, ρ0 is the reference density, η is the sea surface height, g is the gravitational acceleration, ν is the model viscosity, κ is the model diffusivity, KE = uu/2 is the kinetic energy density, and PE = −bz is the potential energy density. Combining these gives

 
formula

This equation is integrated over an area A bounded by the topography and the top boundary at z = 0, and x1 < x < x2. (z = 0 is used at the top boundary, rather than z = η, consistent with the linearized free-surface implementation):

 
formula

where term 1 represents the mechanical energy tendency, term 2 is the horizontal divergence of the advective transport, term 3 is the horizontal divergence of the pressure transport, term 4 is the horizontal divergence of the diffusive transport, term 5 is the vertical transport at the top boundary, term 6 is the dissipation ϵ, always a sink of kinetic energy, and term 7 is the diffusive mixing term M, which tends to increase potential energy. Term 5 is nonzero because there is vertical motion at the top boundary, and to be consistent with the linearized free-surface formulation used in the simulation, the boundary of the integration domain is z = 0, not z = η. (In fact, this surface term is negligible as will be seen later.)

This energy equation is summarized as

 
formula

where angle brackets represent the area integral,

 
formula
 
formula
 
formula
 
formula
 
formula
 
formula

In a numerical simulation, the dissipation and diffusive mixing terms both include some component that can be explicitly calculated, dependent on the explicit model viscosity and diffusivity, as well as an implicit numerical component that cannot be directed evaluated. Klymak and Legg (2010) showed that for the subgrid-scale scheme used here, the dissipation diagnosed directly from the explicit viscosity compares very well with that estimated as a residual in the energy budget for hydrostatic calculations of internal wave generation and breaking at supercritical topography, but less well at critical slope topography. Buijsman et al. (2012) showed that this agreement is less satisfactory in nonhydrostatic calculations, where the explicitly calculated viscosity tends to overestimate the dissipation. Therefore, as in earlier studies by Hall et al. (2013) and VF06, the net loss of energy (ϵ − M in our notation) is estimated as the residual in Eq. (17), to obtain energetically consistent values for these terms. Our estimate for the net loss of energy is therefore

 
formula

Instantaneous values of these terms are dominated by the internal wave oscillations. A running time mean is therefore carried out, to average all terms over an internal wave period.

Note that M is not the term representing diabatic buoyancy flux from kinetic to potential energy, which as a conversion term does not appear in the combined mechanical energy equation, and which is usually associated with mixing (more properly vertical stirring, increasing available potential energy, followed by horizontal diffusion, mixing buoyancy and converting available potential energy to unavailable potential energy). Rather M is a conversion from potential to internal energy. The diabatic (i.e., irreversible) component of the buoyancy flux cannot be easily separated from the adiabatic component associated with the wave motion, both of which are combined in the term −wb. However, it is possible to estimate the irreversible component by assuming that once the incoming and outgoing wave fluxes have reached steady state, the advection of potential energy is largely balanced by the adiabatic (i.e., reversible) component of the buoyancy flux. The increase in wave period–averaged potential energy can then be equated with the sum of diabatic component of buoyancy flux and the diffusive mixing term:

 
formula

where B is the diabatic component of the conversion from kinetic to potential energy −wb. Hence, the combined dissipation and diabatic conversion is given approximately by

 
formula

Together, ϵ + B represent the irreversible loss of wave energy to dissipation and diabatic mixing.

In addition to the dissipation and mixing, other quantities of interest are the fraction of incoming wave energy flux that is transmitted beyond the topography, and the fraction that is reflected back toward the source. As in Hall et al. (2013), these quantities T = transmittance and R = reflectance are defined by

 
formula
 
formula

where (A + P + D)0 is the value of the incoming wave energy transport when there is no reflection, that is, the value imposed by the boundary forcing. Because in these simulations only the net wave energy transport can be diagnosed at any location (i.e., the difference between incoming and reflected), as in Hall et al. (2013) the incoming wave energy transport just before the encounter with topography is instead diagnosed from a flat-bottomed reference simulation (in which there is no backward reflection, and all of the incoming energy flux, except for a very small amount of dissipation, is transmitted). Alternatively, the incoming energy transport could be calculated analytically given the boundary conditions, but diagnosing the value from a numerical simulation ensures that any small dissipation taking place between boundary and topography is accounted for, as well as any deviation in wave behavior from linear theory. In comparing different simulations, it is most useful to similarly normalize the dissipation and mixing by the incoming energy flux. Then

 
formula

and

 
formula

4. Results

Two examples of the nonlinear wave phenomena seen in the simulations are shown in Fig. 4, for a ridge of h0/H0 = 0.64 and critical slope (Fig. 4a) and subcritical slope γ = 0.5 (Fig. 4b). Both snapshots are taken about midway through the simulation, when waves are highly nonlinear, but the behavior has not yet become obscured by mixing. (A later time is shown for the subcritical slope than for the critical slope because the wave energy takes longer to propagate to the top of the gentler slope.) In the critical slope case, overturning isopycnals are associated with strong upslope flow parallel to the slope. In the subcritical slope case, by contrast, the largest isopycnal displacements are high up in the water column, where the first-mode wave has increased in amplitude because of the reduced water column depth. For further information on the phenomenology of the wave breaking at critical and subcritical slopes the reader is referred to Ivey and Nokes (1989), Legg and Adcroft (2003), Venayagamoorthy and Fringer (2006, 2007), and Hall et al. (2013). This study complements those previous studies by focusing on the energy budget and the scaling of reflection, transmission, and dissipation with the imposed nondimensional parameters, and therefore, for conciseness and to avoid repetition of earlier work, additional snapshots of flow evolution are not shown. Because previous studies and these snapshots show that waves lose energy at topographic slopes through breaking, the energy budget and dissipation is a quantitative indicator of the energy lost from the waves to breaking processes. The subgrid-scale scheme, by providing large viscosities only when overturns are present, ensures that dissipation is associated with breaking.

Fig. 4.

Snapshots of the horizontal velocity (color, m s−1) and temperature field (contours) for (a) h0 = 3000 m and critical slope at 6.9 wave periods T after the start of the simulation; (b) h0 = 3000 m and subcritical slope at 9.2 wave periods after the start of the simulation.

Fig. 4.

Snapshots of the horizontal velocity (color, m s−1) and temperature field (contours) for (a) h0 = 3000 m and critical slope at 6.9 wave periods T after the start of the simulation; (b) h0 = 3000 m and subcritical slope at 9.2 wave periods after the start of the simulation.

Figure 5 shows an example of the energy budget terms calculated as in Eq. (24) for the simulation H3000crit, an internal wave encountering a critical slope of height 3000 m. Running averages have been made for all terms over two wave periods, to remove large oscillations associated with the incoming internal wave forcing. However, the potential energy tendency and potential energy advection terms, which vary the most over the wave period, still have large residual oscillations, probably because the integration volume does not cover an integer number of wavelengths. (Because the scattered waves have different wavelengths than the incoming waves, it is not possible to select a perfect integration volume for all calculations.) Typical features of the energy budget, seen in this and most other calculations, include an initial rise in incoming wave energy flux that is balanced by an increase in both kinetic and potential energy, with little initial dissipation. As the incoming energy flux reaches its equilibrium value, the kinetic energy tendency declines to zero, while the potential energy tendency stabilizes at a finite positive value. The outgoing wave energy flux rises at a later time because of the finite time for waves to propagate through the volume, and stabilizes at a value less than the incoming energy flux. The dissipation/mixing term, calculated as a residual, begins to rise around the time the incoming energy flux has reached its equilibrium. Ultimately the incoming energy flux is balanced by an increase in potential energy, a loss to dissipation, and some fraction of energy propagated out of the integration region. In the final two wave periods of the calculation, the incoming and outgoing energy fluxes are in approximately steady state, and so, for comparison between different simulations, the time mean of incoming and outgoing energy flux, potential and kinetic energy tendencies and dissipation are calculated over the final two wave periods. Some calculations with larger topography require longer integration times to reach this steady state in the energy fluxes. Some calculations become contaminated by reflected waves, or the approximate balance between incoming and outgoing fluxes and dissipation and potential energy increase is only transient as the background stratification changes due to mixing, and then time averages have to be made over a shorter period, with correspondingly larger error estimates. Because the dissipation is a residual between two terms with large fluctuations (the difference between the incoming and outgoing fluxes and the potential energy tendency), the standard deviation of the potential energy tendency is calculated as an estimate of the error in the time-mean dissipation.

Fig. 5.

The terms in the energy budget for a single example simulation: a critical slope ridge with height 3000 m. All terms are normalized by the incoming flux in the absence of topography, and a running mean over two wave periods is used. The terms are evaluated for the area bounded by the topography and top surface and x1 = 160 and x2 = 450 km.

Fig. 5.

The terms in the energy budget for a single example simulation: a critical slope ridge with height 3000 m. All terms are normalized by the incoming flux in the absence of topography, and a running mean over two wave periods is used. The terms are evaluated for the area bounded by the topography and top surface and x1 = 160 and x2 = 450 km.

Figure 6 shows the time-mean dissipation, averaged over the final two wave periods of the simulation, calculated as the residual of energy budget terms from (Fig. 6a) Eq. (24) and (Fig. 6b) Eq. (26), normalized by the incoming energy flux in a flat-topography calculation, for all simulations, as a function of normalized topographic height h0/H0. Error bars correspond to the standard deviation of the potential energy tendency. Points to notice are the following: for small ridge heights, the dissipation at subcritical slopes is negligible, while that at critical slopes is finite (as also seen in VF06). However, at larger ridge heights the dissipation at subcritical slopes increases, and matches or exceeds that at critical slopes. This latter point was not observed by VF06, because steepness and ridge height were not varied independently, so subcritical slopes were always of lower height than critical slopes. The ridge height at which dissipation rises for subcritical slopes depends on the amplitude of the incoming wave, with finite dissipation at smaller ridges for larger-amplitude waves, a new result that could not be determined from the experimental design of VF06. By contrast, the normalized dissipation is not sensitive to wave amplitude for critical slopes, a result also seen in VF06.

Fig. 6.

The normalized time- and area-averaged dissipation (a) (ϵM)* and (b) (ϵ + B)* for all standard configuration calculations.

Fig. 6.

The normalized time- and area-averaged dissipation (a) (ϵM)* and (b) (ϵ + B)* for all standard configuration calculations.

Figure 7 shows the proportion of incoming wave energy that is transmitted and reflected [calculated from Eqs. (28) and (27)] for all simulations. Reflectance is negligible for most subcritical slope calculations, except for h0/H0 = 0.21. This height corresponds to a regime where the topography is considerably smaller than half the vertical wavelength of the first mode, but still finite amplitude: the response is therefore similar to the wave generated by a barotropic flow at finite-amplitude topography. This is a completely linear effect, well predicted by Bell (1975), and corresponds to a scattering of incoming low-mode energy into backward (and forward)-propagating waves of smaller length scales. This increase in reflectance for moderate height subcritical slopes could not be determined from the simulations of VF06 where height and steepness did not vary independently. Critical slopes have a small finite reflectance, while, as expected, the supercritical slope reflects a significant proportion of energy. Transmittance more closely reflects the effect of dissipation, being approximately equal to 1 − ϵ* in the absence of reflection: transmittance declines steadily with increasing h0/H0 for critical slopes, and declines at large h0/H0 and large incoming wave amplitude for subcritical slopes. The roughly linear decrease in transmittance with h0/H0 and independence of wave amplitude for critical slopes is in accordance with the geometric argument given earlier. VF06 found a similar roughly linear decrease in transmittance in their single series of calculations where steepness and Froude number were fixed and topographic scales varied, at γ = 0.5.

Fig. 7.

The proportion of the incoming wave energy that is (a) transmitted and (b) reflected, for all standard configuration calculations.

Fig. 7.

The proportion of the incoming wave energy that is (a) transmitted and (b) reflected, for all standard configuration calculations.

In Fig. 8, the dissipation is replotted as a function of the estimated maximum Froude number for a mode-1 internal wave [Eq. (9)] for all calculations with steepness γ = 0.5. Compared to Fig. 6 where dissipation was plotted against h0/H0 only, the results collapse more closely now that the amplitude of the incoming wave is taken into account (although larger-amplitude waves still lead to greater dissipation, even at the same Frmax). Also shown in Fig. 8 are dissipation values for a series of simulations with much larger stratification (N = 5 × 10−3 s−1), more typical of the thermocline, and with a total ocean depth of only 2000 m instead of 4700 m as for the default simulations. A full listing of parameters for these additional simulations is given in Table 2. Despite the difference in parameter choices, the dissipation (as a fraction of incoming wave energy) scales similarly with the maximum Froude number, with a transition from almost no dissipation at low Frmax, to dissipation of most of the wave energy at Frmax ≈ 1.0. For subcritical waves therefore, the maximum Froude number estimate gives a reasonable guide to the onset of dissipation, which occurs for Frmax > 0.3.

Fig. 8.

The normalized time- and area-averaged dissipation (a) (ϵM)* and (b) (ϵ + B)* for all standard configuration calculations with γ = 0.5, as a function of the Frmax estimated from Eq. (9). Also shown in red, are dissipation values for simulations with uniform large stratification (N = 5 × 10−3 s−1), and a max depth of H = 2000 m, with other parameters given in Table 2.

Fig. 8.

The normalized time- and area-averaged dissipation (a) (ϵM)* and (b) (ϵ + B)* for all standard configuration calculations with γ = 0.5, as a function of the Frmax estimated from Eq. (9). Also shown in red, are dissipation values for simulations with uniform large stratification (N = 5 × 10−3 s−1), and a max depth of H = 2000 m, with other parameters given in Table 2.

Table 2.

Values of topographic and internal wave parameters for the four simulations with total depth 2000 m and large stratification, N = 5 × 10−3 s−1.

Values of topographic and internal wave parameters for the four simulations with total depth 2000 m and large stratification, N = 5 × 10−3 s−1.
Values of topographic and internal wave parameters for the four simulations with total depth 2000 m and large stratification, N = 5 × 10−3 s−1.

The area-integrated dissipation and net transmittance and reflectance provide overall guidelines of topographic scenarios in which low-mode internal waves deposit their energy at sloping topography: for critical topography the energy propagating beyond the slope decreases linearly as topographic height increases, while for subcritical topography a critical maximum Froude number (dependent on incoming wave amplitude and topographic height) must be surpassed before appreciable energy is lost. For global models of internal wave energy redistribution it is useful to know where in the water column this energy is lost and used for mixing. Direct calculations of mixing are difficult in relatively low-resolution simulations such as this, but the explicit component of dissipation can easily be calculated. Buijsman et al. (2012) shows that for nonhydrostatic simulations, the Klymak and Legg (2010) scheme tends to over report the dissipation. Nonetheless, this direct calculation can serve as a qualitative guide to the location of mechanical energy transfer from small-scale motion to mixing.

Figures 9 and 10 show the time-integrated dissipation for the critical (γ = 1.0) and subcritical (γ = 0.5) slope calculations with U0 = 12 cm s−1. For the lowest ridge heights (h = 200, 500 m), dissipation is largest near the topography in the beams aligned both forward and backward (relative to the incoming wave direction), in both critical and subcritical slope calculations. These calculations are in region F in Fig. 2, where the scattered wave resembles that generated by barotropic flow. However, the critical slope topography also shows enhanced dissipation in a region along the slope. As the topography becomes taller, the beam directed back toward the source of the low-mode wave becomes less prominent in both critical and subcritical topographies. For tall critical slope ridges, the dissipation is largest along the slope facing the incoming wave, and along the beam aligned in the forward direction, all the way to the surface, that is, extending far above the topography. For tall subcritical topography, the dissipation along the slope at depth remains small, but dissipation increases for the tallest topographies in an almost vertical column near the top of the ridge, all the way to the surface.

Fig. 9.

The time-averaged spatial distribution of dissipation for critical slope topography of different heights and wave amplitude U0 = 12 cm s−1, nondimensionalized by , plotted on a log10 scale.

Fig. 9.

The time-averaged spatial distribution of dissipation for critical slope topography of different heights and wave amplitude U0 = 12 cm s−1, nondimensionalized by , plotted on a log10 scale.

Fig. 10.

The time-averaged spatial distribution of dissipation for subcritical slope topography (γ = 0.5) of different heights and wave amplitude U0 = 12 cm s−1, nondimensionalized by , plotted on a log10 scale.

Fig. 10.

The time-averaged spatial distribution of dissipation for subcritical slope topography (γ = 0.5) of different heights and wave amplitude U0 = 12 cm s−1, nondimensionalized by , plotted on a log10 scale.

Figure 11 shows the time-integrated dissipation, normalized by the square of the wave amplitude, for subcritical slope γ = 0.5, with ridge height 3000 m, for different amplitudes of the incoming wave. As the incoming wave amplitude is increased, not only does the normalized dissipation increase, but the spatial distribution of that dissipation changes. For low-amplitude waves, dissipation is small except right at the topography, particularly at the peak. For intermediate-amplitude waves, the dissipation is low above deeper portions of the slope, but increases at the top of the slope, and extends all the way to the surface. For large-amplitude waves, the dissipation becomes finite amplitude lower down on the slope, and as a result dissipation is much greater on the side of the ridge facing the incoming wave, than on the leeward side. This dependence of the depth of onset of large-amplitude dissipation with incoming wave amplitude is consistent with wave breaking taking place when a critical Froude number is reached, with both incoming wave amplitude and fluid depth determining the Froude number. The critical Froude number is reached in deeper fluid for a larger-amplitude incoming wave.

Fig. 11.

The time-averaged spatial distribution of dissipation for subcritical slope topography (γ = 0.5) with height h = 3000 m, for different amplitudes of incoming waves, nondimensionalized by , plotted on a log10 scale.

Fig. 11.

The time-averaged spatial distribution of dissipation for subcritical slope topography (γ = 0.5) with height h = 3000 m, for different amplitudes of incoming waves, nondimensionalized by , plotted on a log10 scale.

Figure 12 shows that in contrast to the subcritical case, the distribution of dissipation at critical slopes is relatively independent of wave amplitude; however, the lower-amplitude case has a somewhat narrower band of enhanced dissipation, particularly above the top of the ridge.

Fig. 12.

The time-averaged spatial distribution of dissipation for critical slope topography (γ = 1.0) with height h = 3000 m, for different amplitudes of incoming waves, nondimensionalized by , plotted on a log10 scale.

Fig. 12.

The time-averaged spatial distribution of dissipation for critical slope topography (γ = 1.0) with height h = 3000 m, for different amplitudes of incoming waves, nondimensionalized by , plotted on a log10 scale.

5. Generalizing results to different stratification and topography

The simulations discussed thus far have been for uniform stratification and sinusoidal topography in two dimensions. Here each of these constraints are relaxed, to examine the sensitivity of the wave breaking to these assumptions.

a. Variable stratification

Two series of simulations have been carried out with a nonuniform stratification of the form

 
formula

where H is the depth of the fluid, δ is the vertical decay scale (chosen to be δ = H/3) and N1 = N0H/{δ[exp(H/δ) − 1]}, and N0 is the buoyancy frequency 8 × 10−4 s−1 used in the uniform stratification simulations. With this form of stratification, the vertically averaged stratification is unchanged compared to the N = 8 × 10−4 s−1 uniform stratification case, facilitating comparison:

 
formula

Hall et al. (2013) have previously shown that the interaction between internal wave and slope in uniform stratification and variable stratification is qualitatively similar if the vertical coordinate is transformed using a WKB scaling as proposed by Llewellyn Smith and Young (2002):

 
formula

To compare variable with uniform stratification, the topography in the variable stratification case is therefore chosen to give a sinusoidal topography on WKB transformation, that is,

 
formula

where h* has the same piecewise sinusoidal shape as for the uniform stratification.

Because the stratification is enhanced near the surface, the height of the ridge in the WKB-scaled coordinate is less than that in the variable stratification frame, that is, h* < h.

The two different series of simulations (listed in Table 3) correspond to topography with maximum slopes of γ = 1.0 and 0.5, which are sinusoidal when WKB scaled. The boundary forcing is now in the form of the first mode for the variable stratification

 
formula

where F(z) is the first-mode eigenvector found numerically from the internal wave equation with variable stratification and normalized so that

 
formula

to match the constant stratification case [where F(z) = cos(/H)]. Flow amplitude is therefore increased near the surface where stratification is stronger and decreased at depth where it is weaker.

Table 3.

Values of topographic and internal wave parameters for the simulations with variable (the v in the simulation name) stratification. For ease of comparison with the uniform stratification cases, the simulations are labeled by the WKB-scaled topographic height h*, not by the physical height h.

Values of topographic and internal wave parameters for the simulations with variable (the v in the simulation name) stratification. For ease of comparison with the uniform stratification cases, the simulations are labeled by the WKB-scaled topographic height h*, not by the physical height h.
Values of topographic and internal wave parameters for the simulations with variable (the v in the simulation name) stratification. For ease of comparison with the uniform stratification cases, the simulations are labeled by the WKB-scaled topographic height h*, not by the physical height h.

Snapshots of the flow (Fig. 13) show that the features of wave breaking are qualitatively similar, provided ridges of the same WKB-scaled height are compared. Strong surface stratification is associated with larger internal wave velocities, and more nonlinear displacements, as also noted by Johnston and Merrifield (2003). The greater nonlinearity in the stronger stratification cases can lead to thermocline-trapped high-frequency waves [i.e., solitary wave trains (Grisouard et al. 2011; Shroyer et al. 2011)], which propagate beyond the ridge.

Fig. 13.

Snapshots of the horizontal velocity (color, m s−1) and temperature field (contours) for simulations with a surface-enhanced stratification and a WKB-scaled height of h* = 3000 m (physical height of h0 = 4040 m): (a) critical slope at 6.9 wave periods after the start of the simulation and (b) subcritical slope at 9.2 wave periods after the start of the simulation.

Fig. 13.

Snapshots of the horizontal velocity (color, m s−1) and temperature field (contours) for simulations with a surface-enhanced stratification and a WKB-scaled height of h* = 3000 m (physical height of h0 = 4040 m): (a) critical slope at 6.9 wave periods after the start of the simulation and (b) subcritical slope at 9.2 wave periods after the start of the simulation.

If the dissipation and transmission of internal wave energy are compared for constant and variable stratification (Fig. 14), little difference is seen in the scaling behavior, provided that ridges of the same WKB-scaled height, not physical height, are compared, as shown by Hall et al. (2013). The enhanced thermocline stratification results in slightly higher dissipation, probably because of the smaller vertical length–scale waves excited in the thermocline.

Fig. 14.

The proportion of the incoming wave energy that is (a) dissipated and (b) transmitted, for the two series of simulations with surface-enhanced stratification, shown plotted against both physical height (red) and WKB-scaled height (green). For comparison, the two series of simulations with uniform stratification corresponding to the same steepness and incoming Froude number are shown in blue. Dotted lines indicate γ = 1.0 and solid lines indicate γ = 0.5.

Fig. 14.

The proportion of the incoming wave energy that is (a) dissipated and (b) transmitted, for the two series of simulations with surface-enhanced stratification, shown plotted against both physical height (red) and WKB-scaled height (green). For comparison, the two series of simulations with uniform stratification corresponding to the same steepness and incoming Froude number are shown in blue. Dotted lines indicate γ = 1.0 and solid lines indicate γ = 0.5.

b. Variable topographic shape

Whereas most of this study has focused on sinusoidal ridges, the most widespread large-amplitude topography is found in the form of continental slopes. Two simulations with the right slope replaced with a flat continental shelf at the maximum ridge height are compared with the ridge calculations corresponding to the same left slope and height (simulations listed in Table 4). The new topography is therefore of the form

 
formula

and

 
formula

The wave breaking over the slope is unchanged, compared to the ridge case. Dissipation continues over the shelf, because the energy density remains high here (Figs. 15a,b), so overall dissipation values are slightly higher than in the ridge case (Fig. 16). The main study focused on ridges primarily to isolate the behavior over the slope, where wave breaking can influence the large-scale circulation, in contrast to wave breaking on the shelf that is more relevant to the coastal circulation.

Table 4.

Values of topographic and internal wave parameters for the four simulations with continental slope/shelf topography.

Values of topographic and internal wave parameters for the four simulations with continental slope/shelf topography.
Values of topographic and internal wave parameters for the four simulations with continental slope/shelf topography.
Fig. 15.

The time-averaged spatial distribution of dissipation for slope-shelf topography with height h0 = 3000 m and forcing amplitude U0 = 12 cm s−1, with (a) critical slope (γ = 1.0), (b) subcritical slope (γ = 0.5), and (c),(d) variable slopes, consisting of 3 piecewise constant sections, each of height 1000 m. Dissipation is nondimensionalized by , plotted on a log10 scale.

Fig. 15.

The time-averaged spatial distribution of dissipation for slope-shelf topography with height h0 = 3000 m and forcing amplitude U0 = 12 cm s−1, with (a) critical slope (γ = 1.0), (b) subcritical slope (γ = 0.5), and (c),(d) variable slopes, consisting of 3 piecewise constant sections, each of height 1000 m. Dissipation is nondimensionalized by , plotted on a log10 scale.

Fig. 16.

The proportion of the incoming wave energy that is (a) dissipated and (b) reflected, for calculations with different topographic shape, all with uniform stratification and h0 = 3000 m.

Fig. 16.

The proportion of the incoming wave energy that is (a) dissipated and (b) reflected, for calculations with different topographic shape, all with uniform stratification and h0 = 3000 m.

Another aspect of real topography that is not captured by the sinusoidal topography is the possibility for slopes combining regions of different steepness. To examine the role of nonuniform steepness on the wave breaking, two simulations are carried out with slopes taking three different piecewise steepness values: subcritical, critical, and supercritical (CS3000variable1) and supercritical, critical, and subcritical (CS3000variable2), both with height of 3000 m, and slope/shelf shape. The topography is given by

 
formula
 
formula
 
formula

where x1 is the base of the slope; s1, s2, and s3 are the three different slopes; and h1 = h3/3, h2 = 2h3/3, and h3 = 3000 m. This change in topography has much more influence than the change from ridge to continental slope topography, leading to greater reflection (due to the supercritical slope section) and hence less dissipation (Fig. 16). The case with subcritical slope below and supercritical slope above has greater reflectance than vice versa: in the former, wave energy reflected in the forward direction from the subcritical slope then encounters the supercritical slope and is reflected back to deep water. The Froude number–based arguments developed to predict wave breaking on uniform slopes can be used to predict reflection and dissipation for these piecewise constant slopes by examining the incoming energy into each section separately, and considering how any energy reflected impacts the next section of slope. For CS3000variable1, about ⅔ of the energy incident on the slope is reflected and ⅓ is dissipated on the critical slope section; while for CS3000variable2, ⅓ is reflected, ⅓ is dissipated on the critical slope section, and ⅓ is propagated to shallower regions where—through shoaling—some further dissipation occurs.

c. Three-dimensional flow

Finally, results are generalized to three dimensions, whereas all simulations thus far have assumed 2D flow, equivalent to assuming an infinitely long ridge. Two simulations with axisymmetric sinusoidal topography, both of height 3000 m, one with γ = 1.0 and the other with γ = 0.5 (parameters given in Table 5) are compared with the 2D results. These simulations have the same dimensions in x and z as for the 2D simulations. In the y direction (normal to the wave propagation direction), the domain is 4λx across and periodic boundary conditions are used. The vertical resolution is unchanged, while horizontally stretched grids are used in both x and y directions, to allow resolution over the topography to be the same as in 2D, but using fewer total points: nx × ny = 800 × 400.

Table 5.

Values of topographic and internal wave parameters for the three-dimensional simulations with axisymmetric topography.

Values of topographic and internal wave parameters for the three-dimensional simulations with axisymmetric topography.
Values of topographic and internal wave parameters for the three-dimensional simulations with axisymmetric topography.

Unlike 3D barotropic flow over topography of finite width (Holloway and Merrifield 1999), where for Nh/U > 1 the flow is forced around rather than over the seamount, baroclinic wave interaction with topography depends only on the relative angles of the slope and wave group velocity vector. Waves incident normal to the slope therefore continue to propagate forward in the same plane over the topography, as shown by Johnston and Merrifield (2003), while waves impacting the edges of the bump, where the slope is at an oblique angle to the wave group velocity vector, may be deflected around the topography.

Snapshots of the velocity and flow field show that in the plane normal to the incident wave group velocity, coincident with the peak of the “seamount,” the behavior is qualitatively similar to the 2D case (Figs. 17a,b). Quantitative differences include an increase in the magnitude of the velocity anomalies. The snapshots of the horizontal velocity field near the top surface (Figs. 17c,d) show the effect of the topography on the initially plane wave: phase velocity is slowed in shallower water, while the wave amplitude is enhanced over the top of the ridge (effects also seen in the 2D calculations). The wave refraction by topography and resulting downstream interference were noted by Johnston and Merrifield (2003). Dissipation in the plane of maximum topographic height is also almost identical to that seen in 2D, and the vertically integrated dissipation seen in the horizontal plane shows that the dissipation is strongly correlated with the topography height (subcritical slope) or slope steepness (critical slope), as for 2D (Fig. 18). In conclusion, waves that are normally incident to a convex slope behave very similarly, in terms of their steepening and dissipation on the upstream slope, to the 2D infinite ridge case. Downstream effects, such as wave focusing, will of course be very different [as shown by Johnston and Merrifield (2003)], but these are not the focus of this paper. A slope that is concave in the direction normal to the incident wave group velocity (not considered here) will lead to focusing on the upstream side of the topography—a topic for future research.

Fig. 17.

Snapshots of (top) horizontal velocity (color, m s−1) and density (contours) and (bottom) horizontal velocity (color) with topography (contours) overlain, for 3D simulations with axisymmetric topography of height h0 = 3000 m. (left) Critical slope (γ = 1.0) and (right) subcritical slope (γ = 0.5) are shown.

Fig. 17.

Snapshots of (top) horizontal velocity (color, m s−1) and density (contours) and (bottom) horizontal velocity (color) with topography (contours) overlain, for 3D simulations with axisymmetric topography of height h0 = 3000 m. (left) Critical slope (γ = 1.0) and (right) subcritical slope (γ = 0.5) are shown.

Fig. 18.

The time-averaged spatial distribution of dissipation for axisymmetric 3D topography with height h0 = 3000 m and forcing amplitude U0 = 12 cm s−1, with (left) critical slope (γ = 1.0) and (right) subcritical slope (γ = 0.5). Vertical sections across the top of the topography (top) and vertically integrated dissipation with topographic contours overlain (bottom) are shown. Dissipation is nondimensionalized by , plotted on a log10 scale; vertically integrated dissipation is given in meters, also on a log10 scale.

Fig. 18.

The time-averaged spatial distribution of dissipation for axisymmetric 3D topography with height h0 = 3000 m and forcing amplitude U0 = 12 cm s−1, with (left) critical slope (γ = 1.0) and (right) subcritical slope (γ = 0.5). Vertical sections across the top of the topography (top) and vertically integrated dissipation with topographic contours overlain (bottom) are shown. Dissipation is nondimensionalized by , plotted on a log10 scale; vertically integrated dissipation is given in meters, also on a log10 scale.

6. Discussion and conclusions

This study has examined mode-1 internal waves incident on isolated topography, and the sensitivity of internal wave breaking and dissipation to the normalized topographic steepness γ, the nondimensional topographic height h0/H0, and the incident wave Froude number Fr0. Values of 0.25 < γ < 2, 0.0426 < h0/H0 < 0.8511, and 0.01 < Fr0 < 0.17 define the range of parameter space examined here. The principal result is the identification of the regimes in which breaking of low-mode internal waves at isolated topography will occur. As shown in earlier studies (Ivey and Nokes 1989; Slinn and Riley 1996), at critical slopes most of the wave energy that encounters the slope is transferred to small scales on scattering from the slope, and dissipation therefore takes place in a band adjacent to the slope and continues up into the water column in a band aligned with the wave characteristic (Johnston and Merrifield 2003). The present study shows that the fraction of wave energy that is unaffected by the topography can be calculated on geometrical grounds and is proportional to (1 − h0/H0), and approximately independent of wave amplitude. For subcritical slopes, wave breaking will occur if the shoaling depth causes a critical wave Froude number to be exceeded, where the maximum first-mode Froude number is Frmax = Fr0/(1 − h0/H0)2, a new result from this study. Empirically, the critical value of Frmax is found to be between 0.3 and 1.0. This result is shown to hold for very different values of ocean depth and stratification. This Froude number criterion for wave breaking helps to explain the results of the subcritical slope experiments of Cacchione and Wunsch (1974), where the measured increase in wave amplitude on the slope compared well with the exact theoretical solutions for high frequencies, but appeared to be capped at a maximum attainable value for lower frequencies. Because Froude number increases as frequency decreases, the critical Froude number would be reached at lower amplitude for lower frequencies.

In these simulations the spatial distribution of mixing over subcritical slopes is such that very little mixing occurs until the depth has shallowed so that the critical Froude number is reached, and then mixing is large and extends all the way to the surface. The two simulations with continental slope/shelf topography show that the same behavior holds for continental slopes, because most of the mixing occurs on the ridge slope facing the incoming wave [supported by the investigation of an internal wave propagating into deep water from a step by Chapman and Hendershott (1981)]. At critical continental slopes, mixing is expected to occur in a band over the critical slope, and extending up into the water column along the wave characteristic, while for subcritical continental slopes, mixing can occur at a depth where the critical Froude number is exceeded. Depending on the incoming wave Froude number and location of the shelf break, that depth might be shoreward of the shelf break, in which case the mixing would have less influence on the large-scale circulation, or wave breaking might concentrate mixing toward the ocean from the shelf break. From Alford and Zhao (2007) typical low-mode velocity amplitudes in deep water (5000 m) are about 2 cm s−1, and group velocities are about 3 m s−1. Using these values to calculate the deep-water Froude number, from Eq. (10), the low-mode waves are predicted to break when the water depth is between 750 and 400 m. However, it should be noted that, taking into account the nonuniform stratification, this refers to the WKB-scaled topographic depth, and the corresponding actual topographic depth would be shallower, as in the simulations with variable stratification in section 5a, assuming a stratification that is enhanced in the thermocline near the surface, relative to the abyss. For the wave breaking to occur on the continental slope, the critical Froude number needs to be reached at depths greater than that of the shelf break, typically about 140 m. This Froude number criterion for breaking of internal waves at a subcritical slope therefore suggests that smaller-amplitude internal waves may not break on the continental slope, but may propagate all the way onto the shelf. Indeed, remotely generated internal waves have been observed on the continental shelf (Nash et al. 2012). This implies that dissipation of a substantial fraction of the internal tide energy may occur on the shelf, just as much of the barotropic tide is dissipated in coastal seas.

An important result of these simulations is that, for both critical and subcritical slopes, large dissipation is not confined only to the region near the slope. For subcritical slopes, once wave breaking occurs, the dissipation extends all the way to the surface, while for critical slopes dissipation is enhanced along the characteristic in the forward direction, above the ridge crest. The critical slope result indicates that dissipation occurs along the direction of the propagating reflected wave, which is concentrated near the boundary. The subcritical slope result for dissipation is consistent with the direct breaking of the mode-1 wave (with maximum density perturbation at middepth and velocity maxima at top and bottom), following the shoaling-induced increase in amplitude.

How widespread these wave-breaking processes are in the ocean depends on the distribution of slope steepness. Figure 19 shows the regions of supercritical slope in red. The vast majority of the ocean topography has a subcritical slope. Supercritical slopes are found at submarine ridges and around most continental shelf boundaries. Near-critical slopes are not shown separately, because the range of slopes over which near-critical processes apply depends on the incoming wave Froude number. Every slope that passes from sub- to supercritical must nonetheless pass through a critical slope region, so critical slopes are found around most continental boundaries. There are, however, a few regions like the slopes near the northwest Australian, southern Brazilian, and southwest African coasts where the slope from abyss to shelf is almost entirely subcritical. Most large changes in depth are associated with changes in slope, from subcritical through critical and supercritical and back to subcritical on the shelf, and incoming waves will therefore experience a combination of reflection to deeper water, breaking near the boundary at the critical slope, and shoaling over the subcritical slope, with possible breaking if the incoming wave is of sufficiently high amplitude.

Fig. 19.

Locations of supercritical slope (red) superimposed on topographic height (gray shading, m). The topography is from the Smith and Sandwell database, and slope criticality is calculated using stratification from the global World Ocean Circulation Experiment dataset, interpolated to the Smith and Sandwell grid.

Fig. 19.

Locations of supercritical slope (red) superimposed on topographic height (gray shading, m). The topography is from the Smith and Sandwell database, and slope criticality is calculated using stratification from the global World Ocean Circulation Experiment dataset, interpolated to the Smith and Sandwell grid.

The Oregon continental slope, where observations of internal wave energy are described in Nash et al. (2007) and Kelly et al. (2012), is an example of a complicated slope shape, consisting of a supercritical slope near the bottom, an extended region of near-critical slope, and a supercritical slope region near the top of the slope (similar to a hybrid of CS3000variable1 and CS3000variable2). As in simulations in this study, the region above the near-critical slope has enhanced dissipation, but this band of enhanced dissipation does not extend up along the wave characteristic because it is blocked by the presence of the upper portion of supercritical slope. Based on the piecewise slope calculations in this study (section 5b), considerably less total dissipation would be expected than for a uniform critical slope. The observational scenario is also complicated by the presence of the barotropic tide.

In the theoretical framework in which this process study was conducted, several idealizations were made, including uniform stratification and 2D flow. Companion simulations with variable stratification showed that, as found by Hall et al. (2013), for smoothly varying stratification, WKB stretching of the vertical coordinate can be used to apply the same scaling for dissipation/reflection and transmission, as for uniform stratification. Two simulations in 3D with axisymmetric seamount topography confirmed that the 2D theory is valid for waves normally incident on a convex topography. Obliquely incident waves, such as those observed on the Oregon continental slope (Martini et al. 2011; Kelly et al. 2012), were not considered, except at the flanks of the seamount. Chapman and Hendershott (1981) and Zikanov and Slinn (2001) describe some of the complications introduced by obliquely incident internal waves. Downstream effects of three-dimensional topography, such as the focusing of internal wave energy in the lee of seamounts and finite-length ridges, are described by Johnston and Merrifield (2003) and Johnston et al. (2003). Other focusing effects, such as by horizontally concave topography, such as canyons, are a subject for future research.

As shown in section 5b, cross-slope variations in the topographic slope have the largest impact on the results. Most of the ridges considered here have smooth shapes, with monotonically increasing topographic height all the way to the ridge crest, and a smooth transition from gentle slope through maximum slope back to gentle slope. Realistic continental slopes and ridges have more variations in slopes so that they cannot be classified simply as critical/subcritical/supercritical. For example, Nash et al. (2004) describe observations of mixing generated by internal tide scattering from a critical slope section below a steeper supercritical slope. A semianalytical model such as S. Kelly et al. (2013, unpublished manuscript) can help to identify an overall reflection efficiency of a complex slope. These simulations are intended as a preliminary survey of parameter space, and higher resolution, turbulence- and boundary layer–resolving numerical simulations, such as performed by Gayen and Sarkar (2011) for the generation problem, would be a desirable next step.

Finally, to incorporate this information about the fraction of energy deposited at the slope compared to that transmitted/reflected in a parameterization of internal wave–driven mixing for large-scale models, information about the incident internal wave field is required. Such information might be obtained from global internal wave models such as those described in Simmons (2008) and Arbic et al. (2010).

Acknowledgments

Robert Hallberg, Caroline Muller, Eric Kunze, Samuel Kelly, and three reviewers are thanked for their comments on an earlier version of this manuscript. This work is a contribution to the Internal Wave Driven Mixing Climate Process Team, funded by NOAA and NSF. Support has been provided by Award NA08OAR4320752 from the National Oceanic and Atmospheric Administration, U.S. Department of Commerce. The statements, findings, conclusions and recommendations are those of the author and do not necessarily reflect the views of the National Oceanic and Atmospheric Administration or the U.S. Department of Commerce.

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Footnotes

1

An exact solution for the upslope-propagating waves on a subcritical slope is given in Wunsch (1969) and this is shown to be equivalent to a flat-bottomed mode in the limit of shallow slope. For simplicity, the flat-bottomed modes along with WKB theory and energy flux conservation will therefore be used in this regime analysis.