1. Introduction

Our understanding of the dynamic balance of the oceanic internal wave field is still far from complete but its major components are the following: internal waves are generated either as long near-inertial waves at the surface by changes in the atmospheric wind stress or as (mostly) long internal tides at the bottom by barotropic tidal currents; as these waves propagate away from their sources, nonlinear interactions among them (and possibly other processes like scattering at bottom topography) cascade energy to smaller scales until the waves break and dissipate their energy to turbulence (e.g., Müller and Briscoe 2000). This internal wave induced turbulence is assumed the major source for diapycnal mixing in the ocean. To understand and predict diapycnal mixing one must understand and predict internal waves. For this and other reasons, the dynamics of internal waves is studied extensively now observationally, theoretically, and numerically. The Internal Wave Action Model (IWAM) is one attempt to put all the dynamical processes into one common framework and construct a numerical model that will predict the internal wave field in response to atmospheric and tidal forcing in an environment given by an oceanic general circulation model (OGCM). In return, IWAM will provide the OGCM with dynamically consistent diapycnal diffusion coefficients as a function of space and time. A description of the IWAM model is given in Müller and Natarov (2003). IWAM emulates the WAM model for surface waves (Komen et al. 1994).

The IWAM model is a statistical model. It is based on the integration of the radiation balance equation (Müller and Olbers 1975). Its derivation requires three major assumptions: the weak interaction, the random phase and geometric optics assumption. The dynamical processes of generation, nonlinear transfer and dissipation enter the radiation balance equation through source terms S_gen, S_nl, and S_diss.

Currently, the dissipation rate S_diss is estimated from a steady-state version of the internal wave energy balance. It is assumed that the generation region at low wavenumbers is well separated from the dissipation region at high wavenumbers. In steady state, there must then be a constant flux of energy F₀ through vertical wavenumber (m) space from the generation region at small wavenumbers to the dissipation region at high wavenumbers (see Fig. 1). This flux is assumed to be maintained by nonlinear interactions among the waves. In the dissipation region m > m̃ the flux divergence must equal the dissipation rate

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Integration from m̃ to infinity

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yields

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Therefore, the dissipation rate S_diss equals the energy flux F₀ in the constant flux regime. The flux F₀ has been calculated from weak wave–wave interaction theory by McComas and Müller (1981) and from eikonal theory by Henyey et al. (1986). In these studies, the flux F₀ has been found to depend on integral properties of the internal wave spectrum, such as overall energy, shear, and strain. This led to empirical relations between the diapycnal diffusion coefficient K_ρ and integral properties of the internal waves (Gregg 1989; Henyey 1991; Wijesekera et al. 1993; Polzin et al. 1995; Sun and Kunze 1999; Gregg et al. 2003), which have been used to “measure” K_ρ. In the thermocline, these and other measurements of K_ρ suggest values of the order of 10⁻⁵ m² s⁻¹ (Gregg 1987). This is an order of magnitude smaller than the values of K_ρ inferred from fitting observed, smoothed density profiles to an advective–diffusive balance (e.g., Munk 1966). To account for this discrepancy, it has been suggested that the energy dissipation rates have to be highly nonuniform in space and time.

While this approach has been highly successful, it must be realized that it is based on a steady-state argument. The divergence of the flux must balance the dissipation rate. In any nonsteady-state situation where the wave field tries to adjust itself to changes in the forcing and the environment, it is the imbalance between these two terms that determines changes in the wave field. One needs to calculate the flux and the dissipation independently.

There is a long history of calculating and evaluating the term S_nl in the radiation balance equation that describes the nonlinear interaction among internal waves, starting with Olbers (1976) and McComas and Bretherton (1977), via Müller et al. (1986) to Lvov et al. (2004). Here we investigate the dissipation term S_diss, which does not have such history. The reason is that S_nl can be derived from the basic hydrodynamic equations under a set of well-defined assumptions. There is some dispute on how justified these assumptions are in certain circumstances and how one can or should relax them. But all arguments go back to the basic hydrodynamical equations. This is not the case for S_diss. Wave breaking is such a highly nonlinear process that S_diss cannot be derived from the hydrodynamical equations. While we know a few things about wave breaking, we do not know the pressure, stress, and buoyancy signals of wave breaking in the momentum and density equations. We cannot derive S_diss, but we need it for IWAM. So we do the best we can. We take what we do know about wave breaking: that it is due to either shear or gravitational instability, that these instabilities are caused by either chance superposition or encounter of critical layers, that breaking events are localized in space and time, that they are sparse and rare, etc. We use all this information to guess a family of dissipation functions that is consistent with this knowledge and contains a sufficient number of free parameters for eventual calibration.

Once we have arrived at such a family of physically motivated dissipation functions we study the sensitivity of solutions of the radiative balance equation with respect to the values of the free parameters. Unfortunately, we can do this only in a limited context. The presumably all-important interplay between nonlinear transfers and dissipation cannot be studied because there do not yet exist efficient algorithms to calculate S_nl in nonequilibrium situations. They are under construction as part of the IWAM effort. We thus only study the decay in space or time of a given internal wave spectrum under the action of dissipation. While this is not fully satisfying, we are able to discern the meaning of each of the free parameters. We also find the solutions to be sufficiently sensitive to the parameter values. We thus envision that we can eventually calibrate the parameter values by comparison of IWAM solutions with observations and arrive at a calibrated and validated dissipation function, usable in IWAM.

2. The radiative balance equation

The IWAM model is based on the radiative balance equation—a statistical version of wave action conservation (Müller and Olbers 1975). It describes the evolution of the wave action density spectrum n(k, x, t) in wavenumber space k, position space x, and time t and is given by

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where

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is the group velocity,

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is the rate of refraction, and the function Ω(k, x, t) is the dispersion relation for internal gravity waves. The source term S(k, x, t) describes all the processes that do not conserve wave action. It is convenient to decompose the general source term S(k, x, t) into three parts:

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where S_gen(k, x, t) accounts for processes that generate wave action, S_nl(k, x, t) for processes that redistribute wave action in k space without changing its integral over k, and the dissipation function S_diss(k, x, t) for processes that remove wave action from the internal wave field.

The derivation of the radiation balance equation requires various assumptions. The most important ones are the weak interaction, the random phase and the geometric optics assumption.

The weak interaction assumption asserts that the oceanic internal wave field is basically a linear phenomenon. The wave field consist of a superposition of sinusoidal waves with amplitude a, wavenumber vector k and frequency ω. The frequency is determined by the dispersion relation. The amplitude has a magnitude and a phase.

The random phase assumption asserts that one cannot keep track of the phase of the waves for long. They get randomized quickly by a “noisy” environment. One can only keep track of the wave energy density E ∼ |a|² or of the wave action density n = E/ω, the energy density divided by the (intrinsic) frequency. This randomization of the phase also decorrelates waves of different wavenumbers and makes their amplitudes statistically independent. Once the random phase approximation is employed, one cannot reconstruct the actual internal wave velocity and displacement fields but only their root-mean-square values.

The geometric optics approximation is a two-scale approximation. One divides the ocean into grid boxes larger than the wavelength of the longest wave considered. Within each box one performs a spatial Fourier decomposition. For each wavenumber k one then calculates the action density. Since the different wavenumber components are statistically independent the action density becomes the sum over the action density spectrum n(k). Similarly, one divides time into intervals longer than the longest wave period considered. Within each time interval, the time dependence of each wave component k is given by the dispersion relation. The grid box index and the time interval index are then turned into a slow dependence of the action density spectrum on position x and time t, n(k, x, t).

The extent to which these assumptions are reasonable is not exactly known. There are certainly situations where they do not hold; most obviously, they are not appropriate for bore-like and solitary waves, whose existence relies on deterministic phase relations.

Other quantities of interest can be inferred from n(k, x, t) as follows:

• energy density spectrum

  

  

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• energy density flux spectrum

  

  

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• vertical shear spectrum

  

  

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• and inverse Richardson number spectrum

  

  

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In the following we will use the same symbols to denote spectral densities and their integrals and write the arguments explicitly to avoid confusion. Thus, for example, E(x, t) denotes ∫ d³k E(k, x, t). We will also use different representations of k space—for example, the (ω, m, ϕ) representation, where ω is the frequency, m is the vertical wavenumber, and ϕ is the azimuthal angle. In this representation E(ω, x, t) denotes ∫ dm ∫ dϕ E(ω, m, ϕ, x, t). Often we suppress the (x, t) dependence.

It is stressed that the radiation balance equation encompasses a statistical or space–time average. The inverse Richardson number Ri⁻¹(x, t) = ∫ d³k Ri⁻¹(k, x, t) is not the inverse Richardson number at any particular point in space and time but an average over the grid box and time interval denoted by x and t. It is the overall inverse Richardson number. If our discussion requires the inverse Richardson number at a point in space and time we will refer to it as the local inverse Richardson number. This local inverse Richardson number cannot be constructed from the averaged or “overall” inverse Richardson number.

3. The dissipation function

It is generally believed that oceanic internal waves dissipate their energy mainly through wave breaking, either due to shear or gravitational instability. We do not have solutions of the basic hydrodynamic equations that fully describe the onset, growth and decay of these highly nonlinear breaking events. Most rigorous results are about the onset of instability. Our attempt to construct a dissipation function must therefore rely on empirical and heuristic arguments. It is based on the dissipation model of Garrett and Gilbert (1988, henceforth GG88), which can be summarized as

1. compute

   

   

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2. solve Ri⁻¹_m* = 1 for m*, and

3. annihilate all waves with m > m*.

The GG88 model thus produces infinitely fast decay for waves with m > m* when the overall inverse Richardson number is larger than 1 and no decay at all when it is less than 1. We modify this model by making the decay time scale finite and Richardson number dependent and we expand it by assuming that waves with vertical wavenumbers m ≤ m* are affected as well.

To arrive at the general form of our dissipation function we first follow some of the mathematical arguments put forward by the surface wave community (Komen et al. 1994). They realized that although wave breaking is a highly nonlinear process locally, it only causes slow changes in the overall spectrum if the breaking events are sufficiently sparse in space and rare in time. Thus, wave breaking is weak-in-the-mean. A formal consequence of this assumption is that S_diss(k, x, t) can be expanded as a functional power series in n (k, x, t)

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where α_i(·) are expansion coefficients that could in principle be constructed from a hydrodynamical model of wave breaking, which we do not have, and statistical considerations. The above expansion also assumes that the dissipation function is completely determined by the intrinsic dynamics of the internal wave field. No external field affects wave breaking. A zeroth order term is omitted since it would produce unphysical negative energy density spectra. Hasselmann (1974) and Snyder et al. (1992) have shown that in order to represent a dissipation function the coefficients α_j(·), (j > 1) must contain a δ-function

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and that β_j (k, k₁, . . . , k_j−1) ≥ 0. The dissipation function thus becomes quasi-linear

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with the dissipation coefficient

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Again the coefficients β_j can in principle be determined from a hydrodynamical model of wave breaking and statistical considerations.

The second step introduces the overall inverse Richardson number Ri⁻¹ into the dissipation coefficient γ. As already pointed out, internal wave breaking has evaded a satisfactory hydrodynamic description beyond the initial stages of developing out of shear or gravitational instability. Studies of the shear instability of a time-independent horizontal flow U(z) indicate that instability occurs when the actual inverse gradient Richardson number exceeds the critical value of 4:

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For time dependent U, instability has been found for all Ri_g⁻¹ (Majda 2003), with growth rates monotonically increasing with Ri_g⁻¹. For overturning to occur, the horizontal fluid velocity of the internal wave has to exceed its horizontal phase speed, U²/c² > 1. For ω² ≪ (N ² + f ²)/2, where most of the internal wave field energy resides, this condition becomes similarly U²m²/N ² > 1, where m is the vertical wavenumber of the wave. For a random Gaussian internal wave one can estimate the probability of how often, over which time interval and over which spatial volume the actual inverse gradient Richardson number exceeds the critical value 4. This probability and hence the number and intensity of breaking events depends on and increases with the overall inverse Richardson number Ri⁻¹ of the wave field (Desaubies and Smith 1982). We thus assume that the dissipation coefficient is of the form

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and increases monotonically with increasing overall inverse Richardson number.

To determine the wavenumber dependence of γ(k, Ri⁻¹) we first look at the effect of wave breaking on long waves. These waves experience the breaking events as random events localized in space, in the limit as δ-function events. The number of events and their intensity depends on the overall shear, which is determined not by the long waves but mostly by the short waves. These δ-function events affect the long waves more uniformly and do not have the tendency to smear out gradients (Munk 1981; Müller 1999). We thus assume that at low wavenumbers the dissipation coefficient γ does not depend on k. Long waves experience wave breaking as Rayleigh damping rather that scale selective damping. In a Taylor expansion of γ with respect to k we only keep the zeroth-order term. At high wavenumbers, we merge this constant into a modified GG88 behavior.

Specifically we suggest the following form of the dissipation coefficient

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where

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and

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where m* is determined by

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The coefficients (c₀, p, q) = p are adjustable model parameters. The coefficient c₀ gives the overall magnitude of dissipation. It will turn into a simple scaling factor for space and time. The parameter p determines the sensitivity of the dissipation rate to the inverse Richardson number. If p = 0 the dissipation rate is insensitive to the inverse Richardson number. The waves get dissipated at the same rate regardless of how much shear they collectively create. As p increases, the dissipation rate becomes more sensitive to the overall inverse Richardson number. As p → ∞, waves dissipate instantaneously for large Ri⁻¹ and stop losing energy for Ri⁻¹ < 1. For p > 0 the dissipation function is a nonlinear functional of the wave spectrum. The parameters c₀ and p determine the overall dissipation rate. The parameters m* and q determine its distribution in wavenumber space.

Even for a spectrum with overall inverse Richardson number Ri ⁻¹ ≤ 1, like the Garrett and Munk spectrum, wave action is dissipated due to occasional instabilities arising from chance superpositions of the waves. In this case m* = ∞ and our dissipation coefficient reduces to

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We are aware that the above arguments do not constitute a derivation of our dissipation function. Rather they are plausibility arguments to show that it is a sensible choice. Furthermore we have not specified a dissipation function but a family of dissipation functions because of the free parameters (c₀, p, q). We hope that this family contains the real dissipation function if the free parameters are properly calibrated by inverse methods. In the following we investigate whether solutions of the radiation balance equation are sufficiently sensitive to the values of the free parameters to warrant such inversion.

4. Reduced balance

We cannot yet include nonlinear wave–wave interaction in the radiation balance. Even if we assert that weak interaction theory is appropriate, we lack efficient numerical algorithms to evaluate the resulting interaction integrals for other than equilibrium or near-equilibrium situations. We will also assume for simplicity a homogeneous and stationary environment so that there is no refraction term. Furthermore, we assume the generation term to be independent of the wave spectrum. Because the dissipation function is expressed in terms of the inverse Richardson number rather than wave action it is convenient to rewrite the radiative balance equation in terms of the inverse Richardson number spectrum and to scale time and space by c₀. Thus

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where

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and

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where ∫ g(k)d³k = 1, so that G represents the magnitude of the forcing.

Equation (4.1) includes propagation in physical space, generation of wave action, and dissipation. To develop some basic understanding and intuition about the role played by the free parameters of the dissipation function we first solve this reduced radiative balance equation for simplistic balances and spectra. The more complex case of the reflection of the Garrett and Munk spectrum off a linear slope will be considered later.

5. Solutions

a. Response to constant forcing

In this section we consider the idealized case when the internal wave field is forced by sources distributed uniformly is space and time. We assume that the forcing is sufficiently weak, so that Ri⁻¹ < 1 and hence m* = ∞. The evolution of the inverse Richardson number spectrum is then described by

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Steady state is reached when

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Integration in k yields

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or

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The parameter c₀ scales the response in a trivial way. The parameter p determines the sensitivity of the solution. For p = 0 the response is linear in the forcing. As p increases, the response becomes less sensitive to the magnitude of forcing. Two orders of magnitude in forcing collapse to within a factor of two in the response for p ≈ 5 and larger. The same is true for the energy and other integrals of the inverse Richardson number spectrum.

b. Free decay of spectrum

Next we consider the free decay of a spatially homogeneous spectrum with Ri⁻¹ ≤ 1. The radiative balance equation then reduces to

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Integration over k yields

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The time evolution for the overall Ri⁻¹ is thus given by

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for p ≠ 0 and

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for p = 0, where Ri₀⁻¹ = Ri⁻¹ (t′ = 0). For p ≠ 0, the characteristic dissipation time

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depends on the overall inverse Richardson number and on (c₀, p). For p = 0, the characteristic dissipation time is τ_diss = 1/c₀ and a constant.

c. Spatial decay of a monochromatic spectrum

Next we consider an ensemble of waves that all have the same wavenumber vector k₀ = [k₀, l₀, m₀]. The inverse Richardson number spectrum then has the form

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The cutoff wavenumber is

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Assume that these waves are generated at a plane with normal vector n directed into the interior of the fluid. The steady-state radiative balance equation for both cases then becomes

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where x′_n is the scaled normal distance from the wall and υ_n (k₀) = v (k₀) · n is the normal group velocity. The solution is

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for p ≠ 0 and

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for p = 0 where Ri₀⁻¹ = Ri⁻¹ (x′_n = 0). For monochromatic wave fields with Ri₀⁻¹ > 1, the scaled distance at which the inverse Richardson number reaches its critical value is given by

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for p ≠ 0 and

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for p = 0 and is shown in Fig. 2 as a function of p for a value of Ri₀⁻¹ = 20. It decreases monotonically with p. The inverse Richardson number Ri⁻¹ is plotted versus x′_n/x′_n,crit in Fig. 3 for various values of the parameter p and Ri₀⁻¹ = 20. For p = 0 the decay is exponential. For large values of p the inverse Richardson number first decays very fast and then slowly relaxes toward the critical value of Ri⁻¹ = 1.

The energy density spectrum and energy flux density spectrum are obtained by multiplying the inverse Richardson number spectrum by a certain function of k [see Eqs. (2.2)–(2.5)]. Thus the normal energy flux decays as F_n(x′_n) = F₀/A(x′_n) where A(x′_n) is the denominator in Eq. (5.13), which is Ri₀⁻¹ at x′_n,crit. An important consequence is that the total energy flux dissipated within the critical distance is given by

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for Ri₀⁻¹ > 1, and is independent of model parameters.

d. Spatial decay of bichromatic spectrum

To demonstrate the effect of the parameter q on the solution, we now consider a bichromatic spectrum generated, as in previous section, at a wall with normal vector n

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The governing equations are now

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where Λ(k) = c̃(k; m*)/υ_n(k). After dividing the first equation by the second we obtain

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which is independent of p. This can be rewritten as

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Integrating in x′_n yields

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The same relationship holds for other quantities, in particular for the energy fluxes

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First assume Ri₁⁻¹ (0) < 1 Ri⁻¹₁ (0) < 1 and Ri₂⁻¹ (0) ≫ 1. Then m* = m₂ and = c̃ (k_1,2; m*) = 1 for both spectral components. The parameter q drops out of the problem and we have

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The first spectral component, the one with the small inverse Richardson number, also decays and it decays the faster the larger υ_n(k₂)/υ_n(k₁). Thus near-inertial internal waves that have small group velocities may be subject to significant dissipation even if their inverse Richardson number is small.

Next consider the case when both spectral components contain enough shear, so that m₂ > m₁ > m*. Then we have

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and the model parameter q enters the problem. If q → ∞ then

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and the second spectral component with larger m dissipates completely before the first one loses any energy. For given k₁ and k₂, the parameter q determines the relative decay rate and therefore the E/Ri⁻¹ ratio.

These simple analytic examples elucidate the basic roles played by each of the free parameters in our dissipation function. The parameter c₀ scales time and space in a trivial way. The larger c₀ the more rapid is the temporal and spatial decay of the spectrum. The parameter p determines the shape of this decay whereas the parameter q determines the relative decay rates of the different spectral components. We find the same to be true in the more complex example considered next.

6. Reflection off a linear slope

To consider a more complex situation, in this section we study reflection of an incoming Garrett and Munk (GM) spectrum off a linear slope. The reflection laws require that there be no normal energy density flux through the boundary. The incoming normal energy flux equals the outgoing normal energy flux. Within the framework of inviscid linearized dynamics, waves of critical frequency are reflected into waves of infinite vertical wavenumbers. This leads to infinite shear and energy density in the reflected spectrum (Eriksen 1982). We consider the specific case where the incoming spectrum is given by the Munk (1981) version, the topographic slope by γ = 0.1, the buoyancy frequency by N = 5.24 × 10⁻³ s⁻¹ and the Coriolis frequency by f = 3.14 × 10⁻⁴s⁻¹—that is, f /N = 0.06. The incoming normal energy flux density for this situation is F_in = 17.3 mW m⁻². The overall Richardson number of the incident spectrum is Ri⁻¹ = 0.2 ≪ 1. The overall inverse Richardson number of the reflected spectrum, on the other hand, is infinite due to critical reflection. Its spectrum as a function of vertical wavenumber is shown as a dashed red line in Fig. 7, and grows as m → ± ∞. Its cutoff wavenumber is m* = 52π/b = 0.13 m⁻¹ with b = 1300 m⁻¹. Since near the boundary the overall shear is dominated by contributions from the reflected waves, we neglect the contributions of the incident spectrum to overall Ri⁻¹. The problem then reduces to solving

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for reflected waves (which have υ_n(k) > 0) with the boundary condition Ri⁻¹(k; x′_n = 0) computed from the reflection laws for the linear slope. Until we reach inverse Richardson numbers of O(1) the shear is dominated by waves with m > m*. We can therefore neglect the slight drift in m* toward higher values with increasing x′_n and fix m* at its value m* = 52π/1300 m⁻¹ at x′_n = 0.

Overall, this is not a particularly realistic setup of the reflection problem. Since the shear and energy density of the reflected spectrum are infinite, one expects vigorous adjustments not only by dissipation but also by nonlinear wave–wave interactions. We expect that most of the energy dissipated at high wave numbers is replenished by wave–wave interactions. The latter process can unfortunately not be included in our analysis since, as pointed out before, there are currently no numerical algorithms available to calculate the nonlinear transfer in a highly directional spectrum. It is still a sufficiently reasonable setup for our purposes since we are interested in the sensitivity of the solution to the free parameters of our dissipation function. Because we neglect nonlinear transfers it is also not meaningful to vary the environmental parameters in order to make inferences about mixing in the ocean. We simply keep them fixed.

Introducing the new independent variable

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reduces (6.1) to a linear equation

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The equation for the energy density spectrum has the same form. The solutions are

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The dependence on p has been absorbed into ζ. Treating ζ as the independent variable one can compute E(p; ζ) and Ri⁻¹(p; ζ) without specifying p. By solving the linear problem (6.3), we can therefore compute the total flux available for mixing F_d and E(Ri⁻¹), as well as spectral signatures of dissipation for arbitrary p.

First we calculate the energy flux F as a function of Ri⁻¹. We define the energy flux available for mixing by

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that is, as all the energy that is dissipated in the volume between x′_n = 0 and x′_n,crit. GG88 computed F_d as a function of f /N and topographic slope γ. In our model F_d also depends on the model parameter q and N. The sensitivity of F_d with respect to q, with the environmental parameters fixed at the values given above, is shown in Fig. 4. The maximum F_d/F_in ∼6.8% is achieved when the parameter q is set equal to zero. The case q → ∞ is equivalent to the procedure used in GG88 and corresponds to the minimum F_d/F_in ∼5%. The differences are insignificant for all practical purposes.

Figure 5 shows the energy density versus inverse Richardson number for q = 0 and q = 2. The solutions differ considerably for the two different q values. This sensitivity encourages setting up an inverse problem for determining q from observations. The projections of Ri⁻¹(k) at x′_n,crit onto σ and μ spaces, where σ = ω/N is the nondimensionalized frequency and μ = m(b/π) is the nondimensionalized vertical wavenumber or mode number (with b = 1.3 km), are shown in Figs. 6 and 7. In frequency space, the difference between the two solutions is relatively small due to the fact that most of the dissipation occurs around the spike at the critical frequency σ_crit ≈ 0.116. The most significant difference is observed for near-inertial waves, which have small group velocities and take a very long time to traverse the highly dissipative region of high Ri⁻¹. Near-inertial waves are damped more strongly in the q = 0 case.

In vertical wavenumber space, the differences between the solutions are more pronounced (Figs. 7 and 8). The resulting Ri⁻¹(μ) spectrum is sharply peaked at μ* for q = ∞ and becomes more broadband with decreasing q. This signature in μ space should also be exploitable in inverse applications.

Figure 8 explains why the resulting overall energy density E is larger for larger q, as in Fig. 5. Unlike shear, energy is mostly contained in low vertical wavenumbers. Elimination of high μ waves (as in q = ∞ case) therefore does not diminish overall energy of the wave field as much as elimination of lower μ waves (as in the case of q = 0).

Figures 9 and 10 show the evolution of Ri⁻¹(μ) spectrum with the distance away from the wall expressed in terms of overall Ri⁻¹. Such choice of the “distance variable” allows the graph to represent solutions for arbitrary values of the model parameters c₀ and p. The solution for q = 0 shows a substantially broader Ri⁻¹(μ) spectrum than the solution for q = 2. The situation is reversed for the energy density. The energy spectrum in Fig. 11 (q = 0) is narrower than spectrum shown in Fig. 12 (q = 2).

One could produce similar contour plots of Ri⁻¹(μ) and E(μ) as a function of x′_n/x′_n,crit These plots depend additionally on the value of p. Their essence is shown in Fig. 13, which graphs the overall inverse Richardson number Ri⁻¹ as a function of x′_n/x′_n,crit for different values of p. The behavior is qualitatively similar to the solutions for a monochromatic spectrum. Large values of p produce rapid initial decay and slow approach to the critical value afterwards. Smaller values of p produce more uniform decay. Plots against the actual distance also depend on the parameter c₀, which scales the rate of decay linearly.

Overall, we find again that the parameter q determines the spectral distribution of energy or shear whereas the parameters p and c₀ determine the spatial (or temporal) distribution of these quantities.

7. Discussion and conclusions

The IWAM model is based on the integration of the radiative balance equation and requires a sink term that describes the dissipation of wave energy by wave breaking. The highly nonlinear nature of wave breaking defies any systematic derivation of this dissipation function from hydrodynamic principles. Instead, we have put forward a family of physically plausible dissipation functions that hopefully contains the real dissipation function, at least approximately. Our dissipation function is proportional to the wave spectrum with the coefficient being a nonlinear function of the overall inverse Richardson number Ri⁻¹. It contains four parameters (m*, c₀, p, q). The cutoff wavenumber m* is the vertical wavenumber at which the integral of Ri⁻¹(k) over k space reaches its critical value Ri⁻¹ = 1. The parameters c₀, p and q are treated as free parameters.

The main goal of our study was to find out whether solutions of the radiative balance equation are sufficiently sensitive to the values of these parameters so that they can eventually be calibrated by observations, solving an inverse problem. To this end the radiative balance equation has been solved for a number of idealized problems, which could be solved analytically, and for the more complex problem of wave reflection off a linear slope, which required numerical evaluation. The idealized cases provided basic insight how the free parameters affect the solutions. The parameter c₀ is a linear scaling factor that determines how rapidly the wave field decays in space or time. The parameter p determines the shape or form of this decay, whether it is fast at the beginning and then slows down or whether it is more uniform. The parameter q determines the relative decay of different wavenumber components. The solutions to the idealized cases also provided test cases to check the numerical algorithms used to solve the reflection problem.

For the reflection problem, we find similar results. While some quantities, most notably the energy flux available for mixing, do not sensitively depend on the free parameters, others do. The parameter c₀ determines the distance at which the inverse Richardson number decays to its critical value and the parameter p determines the shape or “half width” of this decay. The parameter q determines the spectral distribution in vertical wavenumber space and hence the ratio of energy to inverse Richardson number. The main result for the IWAM effort is that these dependencies are sufficiently strong so that the parameters can be calibrated by comparison with observations using inverse methods. Such inverse problems will have to include additional processes in the radiative balance equation, most notably wave–wave interactions, and will have to account for the peculiarities of the observational site.