Calculating Energy Flux in Internal Solitary Waves with an Application to Reﬂectance

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KEVIN G. LAMB AND VAN T. NGUYEN Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada

(Manuscript received 19 July 2007, in ﬁnal form 6 August 2008)

ABSTRACT

The energetics of internal solitary waves (ISWs) in continuous, quasi-two-layer stratifications are explored using fully nonlinear, nonhydrostatic numerical simulations. The kinetic energy of an internal solitary wave is always greater than the available potential energy, by as much as 30% for the stratifications considered. Because of different spatial distributions of the kinetic and available potential energy densities, however, the fluxes are quite different. The available potential energy flux is found to always exceed the kinetic energy flux, by as much as a factor of 5. The sizes of the various fluxes in the wave pseudoenergy (kinetic plus available potential energy) equation are compared, showing that, while the linear flux term (velocity– pressure perturbation) dominates the fluxes, the fluxes of available potential and kinetic energy are significant for large ISWs. Past work on estimating the reflectance (ratio of reflected to incident pseudoenergy flux) associated with internal solitary waves incident on a linearly sloping bottom in laboratory experiments and numerical simulations has incorrectly assumed that the available potential energy flux was equal to the kinetic energy flux. Hence, the sensitivity of reflectance estimates to the way the flux is calculated is investigated. For these low Reynolds number situations, it is found that a correct account of the available potential energy flux reduces the reflectance by as much as 0.1 when the pycnocline is close to the surface.

1. Introduction flux is equal to the total pseudoenergy (kinetic plus available potential) multiplied by the propagation speed The fate of internal solitary waves incident on a slope of the waves. Energy fluxes in observed waves have is a topic of considerable importance for the coastal often been estimated by this method, assuming that they ocean and for lakes where these waves are often highly have permanent form. Some authors have used first- energetic (Sandstrom and Elliott 1984; Jeans and order weakly nonlinear (KdV) theory to estimate the wave Sherwin 2001; Klymak and Moum 2003; Carter et al. energies (Sandstrom and Elliott 1984), while others have 2005; Boegman et al. 2005a). As the waves shoal, they estimated the available potential energy and assumed break, resulting in energy dissipation, mixing, and sed- equipartition of energy to infer the kinetic energy iment resuspension. The particulars of these processes (Bogucki and Garrett 1993; Michallet and Ivey 1999). depend on many factors including the stratification, Some have independently estimated both the kinetic wave amplitude, bottom slope and roughness, and angle and available potential energy from velocity and den- of incidence of the waves. One quantity of interest is the sity observations (Jeans and Sherwin 2001; Klymak reflectance R, which is the ratio of the reflected and et al. 2006; Moum et al. 2007). The latter authors con- incident pseudoenergy fluxes, the pseudoenergy being sistently found that the kinetic energy in the waves is the sum of the kinetic and available potential energy larger than the available potential energy, which is (Shepherd 1993). To calculate the reflectance, the en- consistent with the fact that this is always the case in ergy fluxes associated with the incident and reflected exact, fully nonlinear and nonhydrostatic internal soli- waves are required. For waves of permanent form, for tary waves (Turkington et al. 1991). In contrast, Scotti example, internal solitary waves, the total pseudoenergy et al. (2006), who estimated the kinetic and available potential energy in waves observed in Massachusetts Bay, found that the available potential energy exceeded Corresponding author address: Kevin G. Lamb, Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L the kinetic energy. This could be associated with the 3G1, Canada. effects of shoaling, a process that can convert kinetic E-mail: [email protected] energy to potential energy (Lamb 2002; see also below).

DOI: 10.1175/2008JPO3882.1

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For evolving wave fields a different approach is comparable in size to the first term (Lamb 2007). Some needed. Using the mechanical energy equation (kinetic authors (e.g., Helfrich 1992; Bourgault and Kelley 2007) plus potential energy), the energy flux through a vertical have assumedÐ that the available potential energy flux line, assuming two-dimensional flow for simplicity, has APEf Ð5 uEa dz is equal to the kinetic energy flux the form Kf 5 uEk dz, resulting in the estimate ð ð 0 0 (up 1 uEk 1 urgz) dz, (1) 2 2 upd 1 rou(u 1 w ) dz, (4) H H where u is the horizontal velocity, r is the fluid density, p where ro is the reference density and w is the vertical is the pressure, and Ek is the kinetic energy density, H is velocity [Helfrich (1992) further assumes the contribu- the water depth, and the surface is at z 5 0. Viscous and tion from w is negligible]. In fact, these two fluxes can be diffusive terms have been ignored. The flux terms are quite different, as discussed below. Because of this, the often separated into a number of other terms by split- parameterization of the reflectance of internal solitary ting the density and pressure into the sum of back- waves presented by Bourgault and Kelley (2007) needs ground values and a perturbation (or disturbance) via to be reconsidered. p 5 p1 pd and r 5 r 1 rd. The pressure perturbation The outline of the paper is as follows: The theoretical can then be further subdivided into a part in hydrostatic foundations of available potential energy and pseu- h balance with the density perturbation, pd, and a non- doenergy densities are presented in section 2. In section nh hydrostatic perturbation, pd (Venayagamoorthy and 3 the numerical model and its initialization is described. Fringer 2005, 2006; Moum et al. 2007). This results in The properties of exact internal solitary waves in a ð 0 continuous quasi-two-layer stratification with a thin h nh (up1 upw 1 upd 1 uEk 1 urgz 1 urdgz) dz. (2) pycnocline are discussed in section 4. The spatial dis- H tribution of the kinetic and available potential energy The largest instantaneous contributions to the vertically densities and their fluxes are compared. Reflectance integrated flux in general come from the terms up1 urgz values, based on low Reynolds number simulations, as they are linear in the perturbation. These terms, which similar to those undertaken by Bourgault and Kelley, were ignored by Moum et al. (2007), make no net con- are presented in section 5, and the results are summa- tribution in a periodic wave field for which u has a time rized and discussed in section 6. average of zero. They are, however, the dominant term for wave fields for which u has a nonzero mean, such as in an internal solitary wave train (Lamb 2007). In nu- 2. Available potential energy and pseudoenergy flux merical simulations of periodic waves in a linearly We consider a Newtonian fluid in two dimensions stratified fluid impinging on a shelf, Venayagamoorthy under the Boussinesq approximation for which the and Fringer (2005) found that the energy flux in boluses governing equations are propagating up onto the shelf had significant instantaneous contributions from up but that the accumulated ›u ›u ›u 1 ›p 2 flux from this term was small. 1 u 1 w 5 1 n= u, (5) ›t ›x ›z r0 ›x Because the available potential energy is of more interest than the potential energy (Hebert 1988; Winters ›w ›w ›w 1 ›p rg 1 u 1 w 5 1 n=2w, (6) et al. 1995), another approach is warranted. The pseu- ›t ›x ›z r0 ›z r0 doenergy flux through a vertical line is ›r ›r ›r 2 ð 1 u 1 w 5 k= r, (7) 0 ›t ›x ›z (up 1 uE 1 uE ) dz (3) d k a ›u ›w H 1 5 0. (8) ›x ›z (see below), where Ea is the available potential energy density (Scotti et al. 2006; Lamb 2007). The pseudo- Here (u, w) is the velocity vector, (x, z) are the hori- energy flux is dominated by the term upd, which is zontal and vertical coordinates, g is the gravitational quadratic in wave amplitude. This is the only term that acceleration, p and r are the pressure and density fields, appears in linear theory. The remaining terms are cubic r0 is the reference density, n 5 m/r0 is the kinematic in wave amplitude and hence negligible for small am- viscosity (m being the viscosity), and k is the diffusivity. plitude waves. For internal solitary waves they are The bottom boundary is at z 52H(x) and the fluid has

Unauthenticated | Downloaded 10/03/21 11:37 AM UTC MARCH 2009 L A M B A N D N G U Y E N 561 a rigid lid at z 5 0. From these equations the mechanical difference in potential energies of the perturbed and energy equation relaxed states, the latter being the background poten-

tial energy, BPEL. Since the relaxed state depends on ›E 1 $ [(E 1 p)~u] m$ (u $u) kg$ (z$r) the region in which the sorting is done, so does the ›t available potential energy. This is a particular problem ›r 5 n=2E f gk (9) in open systems (Scotti et al. 2006), where eddies and k ›z other features may be present in the far field, but is well defined in a closed system such as a tank in the is easily derived. Here E 5 Ek 1 Ep is the mechanical energy density per unit volume in which laboratory. A further difficulty with the above expression is that it is a global one. Fortunately, an available r0 2 2 potential energy density can be defined that gives in- Ek 5 (u 1 w ) (10) 2 formation about the spatial distribution of available potential energy, is positive definite, and, together with is the kinetic energy density, the kinetic energy, satisfies a simple conservation law

Ep 5 rgz (11) (Holliday and McIntyre 1981; Shepherd 1993; Scotti et al. 2006; Lamb 2007). is the potential energy density, and Let r(z, t) be a reference density profile, possibly time dependent, with corresponding hydrostatic pres- f 5 2meijeij (12) sure p(z, t). Assume that r(z, t) is invertible with inverse z*(r, t). An available potential energy density can be is the viscous dissipation rate per unit volume, where defined as 1 ›u ›uj e 5 i 1 (13) ij Ea(x, z, t) 5 Q(r(x, z, t), z, t), (15) 2 ›xj ›xi where is the strain rate tensor with (u1, u2, u3) 5 (u,0,w). Because not all potential energy is available for con- version into kinetic energy and ultimately into heat, Q(r, z, t) 5 rgz rgz (r, t) 1 p(z, t) p(z (r, t), t), ð of more interest is the available potential energy z(r,t) 0 0 (APE). The APE in a finite region of horizontal length 5 g (r(z , t) r) dz , ðz L 5 xr 2 xl is r 5 g (z z(s, t)) ds. (16) xðr ð0 r(z,t)

APEL 5 (r rr) gz dz dx. (14)

xl H(x) If the background density is not invertible, as would occur if in the presence of a surface mixed layer, one can

Here rr(z, t) is the adiabatically rearranged density field work with material surfaces to define Ea (Lamb 2007, obtained by sorting the density field to minimize the 2008). potential energy of the fluid. The resulting horizon- Using this to express the potential energy density tally uniform density field is stably stratified. The vol- Ep in terms of the available potential energy density ume of fluid in any density interval (r1, r2) in the region Ea, the mechanical energy equation. (9) can be re- xl # x # xr is preserved by the sorting. APEL is the written as

ð z(r, t) › 2 ›r ›r 0 0 (Ek 1 Ea) 1 $ [u(Ek 1 Ea 1 pd)] 5 = (nEk 1 kEa) f 2kg 1 m$ (u $u) 1 g (z ,t) dz ›t ›z z ›t ›r ›z 1 kg (z, t) 1 kg (r, t), (17) ›z ›r

where pd 5 p p is the pressure perturbation relative (Shepherd 1993). Here Ea is the density of a functional to the reference pressure. This equation can be shown to constraint used in a variational method for the numer- be identical to expression (2.6) in Scotti et al. (2006). ical solution of the Dubreil–Jacotin–Long equation

Epseudo 5 Ek 1 Ea is called the pseudoenergy density used to obtain exact, fully nonlinear internal solitary

Unauthenticated | Downloaded 10/03/21 11:37 AM UTC 562 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 39 waves (Turkington et al. 1991; Lamb 2002). It is the › (E 1 E ) 1 $ [u(E 1 E 1 p )] 5 0, (19) work per unit volume required to move a fluid particle ›t k a k a d from its reference height z* to its current height z, against the buoyancy force in the undisturbed fluid. For showing that the pseudoenergy is conserved. Integrating a monotonically decreasing rest stratification r, it can (19) over a fixed Eulerian volume xl # x # xr and with z spanning the water depth gives easily be deduced that Ea is positive definite. If the reference density r(z, t) is the sorted reference d xl density r (z, t) using the region x # x # x , the integral Epseudo 5 (Kf 1 APEf 1 W) j , (20) r l r dt xr of Ea over the domain [xl, xr] 3 [2H(x), 0] is identical to APEL given in (14) (Holliday and McIntyre 1981; Lamb where Epseudo is the integrated value of the pseudo- 2008). Sorting the density field is computationally ex- energy density, pensive and can be difficult to do in an open system. ð When the domain is unbounded in the horizontal and 0 K 5 uE dz, the density field is horizontally uniform outside a region f k ðH(x) of finite length, the far- field density field can be used as 0 the reference density in the definition of Ea. The far- APEf 5 uEa dz (21) field density cannot be used in (14) if the domain has H(x) finite length (Hebert 1988; Lamb 2008). Henceforth, by are the kinetic and available potential energy fluxes, and available potential energy, the value denoted by APE, ð we mean the available potential energy in an un- 0 bounded domain, computed by integrating Ea using the W 5 upd dz (22) H(x) far-field density as the reference density. APEL refers to the available potential energy in a finite domain of is the linear energy flux, or, more accurately, the rate length L, obtained by sorting the density field and cal- work is done by the pressure perturbation. The total culating the difference between the potential energy energy flux through a horizontal location x is E 5 K 1 and background potential energy. f f APE 1 W. These are the three fluxes that we consider For a linear reference stratification with constant f in the following. buoyancy frequency N, 1 g2 3. Numerical model E 5 r2. (18) a 2 2 d N The governing equations (1)–(4) were solved using This expression also gives the leading-order contribu- a numerical model (Lamb 1994) that uses a rigid lid at tion to the APE density for small amplitude waves z 5 0. For our reflectance calculations we use approxi- (Holliday and McIntyre 1981; Shepherd 1993). Ex- mately the same geometry used in the laboratory expression (18) is often referred to as the potential energy periments of Michallet and Ivey (1999) and in the density for linear waves (e.g., Gill 1982). numerical simulations of Bourgault and Kelley (2007). As (18) shows, Ea is quadratic in the wave disturbance. The experiments were done in a small wave tank with a Hence, in (17), the flux upd is quadratic in the disturbance deep water depth of 15 cm and a linear slope at the right amplitude while the remaining fluxes uEk and uEa are end of the tank that went right to the surface. Thus, in cubic (assuming no background flow). The quantity upd our numerical simulations the bottom boundary is at dominates the flux terms for small amplitude waves. It has z 52H 1 h(x), where H 5 15 cm is the deep water received the most attention in the literature. For internal depth. Because the model uses sigma coordinates, the solitary waves the kinetic and available potential energy sloping bottom was leveled off to a water depth of 1 cm fluxes can be comparable to the traditional energy flux before it reached the surface. No-slip boundaries were (Scotti et al. 2006; Lamb 2007). For calculations of the used along the bottom and the two solid end walls, and a reflectance associated with large internal solitary waves, free-slip condition was employed along the surface. omission of the kinetic and available potential energy Zero flux conditions were used for the density field on fluxes can result in significant errors because, being cubic all boundaries. No attempt was made to parameterize in the wave amplitude, upon reflection they decrease the effects of sidewall friction that is present in labora- proportionally more than the linear energy flux does. tory simulations. Bourgault and Kelley compared sim- If the effects of viscosity and diffusivity can be ig- ulations with and without a parameterization of sidewall nored, and the reference density state is independent of friction. Thus, our simulations are not directly compa- time, the pseudoenergy equation simplifies to rable to the laboratory experiments.

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FIG. 1. Typical initial state for a numerical simulation: Contour lines are density contours. Bottom slope is 0.214.

In our simulations, the bottom boundary has the form To duplicate the experimental conditions, simulations of a linear slope with smoothed corners, given by were done using a kinematic viscosity n 5 1026 m2 s21 and diffusivity k 5 1028 m2 s21. Increasing the diffu- 27 2 21 z 5 0.15 1 s[itanh(x, xr l2, d) itanh(x, xr l1, d)], sivity to 10 m s gave similar results even though the (23) pycnocline thickened appreciably during the model simulations. The smaller diffusivity, appropriate for model- where ing a salt-stratified tank, resulted in minimal thickening ð of the pycnocline ahead of the reflected waves. This 1 x x0 a itanh(x, a, d) 5 1 1 tanh dx0, made use of the initial density profile as the reference 2 d no‘ hi density appropriate for the calculation of Ea in the re- 1 x a 5 x a 1 d ln 2 cosh (24) flected waves. The simulations were repeated with a 2 d smaller kinematic viscosity of 1027 m2 s21 so as to assess the sensitivity of the reflectance values to the Reynolds is a function that smoothly changes from 0 to a constant number. slope of 1 at x 5 a over a characteristic distance d. Thus, For the majority of the cases, a horizontal resolution in (23), the bottom slope is s, x is the location of the r of 2.5 mm was used in the horizontal. In the vertical 120 right boundary, l 5 0.01/s is the distance from the right 1 uniformly spaced grid points were used, giving a vertical wall where the bottom levels off, and l 5 0.14/s 1 l is 2 1 resolution of 1.25 mm in the deep water. These resolu- the distance from the right wall where the sloping bot- tions are identical to those used by Bourgault and tom begins. All terms in these expressions, apart from s, Kelley in the deep water. Use of sigma coordinates have units in meters. implies higher vertical resolution over the slope. Tests The background density was taken as were done with higher resolutions to ensure that the 1 z zpyc results were accurate. r(z) 5 1.02 0.02 tanh . (25) Each simulation was initialized with an exact, fully r0 dpyc nonlinear single solitary wave propagating toward the This corresponds to a density ratio of 1.04 between the slope (Lamb 2002). Figure 1 shows a typical initial lower and upper layers in the laboratory experiments. configuration. Under the Boussinesq approximation, as is used here, it is only the density difference that matters. Changing the 4. Properties of internal solitary waves density difference changes the time scale and scales the kinetic and potential energy. Dynamically it is one way In the absence of rotation, first-order weakly nonlin- to change the Reynolds number and, hence, is equiva- ear theory predicts that the kinetic and available po- lent to changing the viscosity and diffusivity. We used a tential energy in internal solitary waves are equal. The pycnocline thickness of dpyc 5 3 mm. For simplicity, we assumption of equipartition has often been used to es- used a reference density of 1000 kg m21 rather than the timate the energy in internal solitary waves (Helfrich density at the center of the pycnocline. 1992; Bogucki and Garrett 1993; Michallet and Ivey

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FIG. 2. Exact internal solitary waves for zpyc 523 cm: (a), (b) The maximal isopycnal and the surface velocity u(x,0) for a sequence of waves; (c) propagation speed c and maximum wave-induced current at the surface as a function of wave amplitude; and (d) available potential energy and kinetic energy as a function of wave amplitude. Waves shown in (a) and (b) correspond to waves 1 (smallest), 3, 4, 5, 6, 9, 12, and 15 in (c) and (d).

1999; Boegman et al. 2005a,b). Using a variational for- 23 cm are narrower and were calculated in a smaller mulation Turkington et al. (1991) proved that for exact, horizontal domain than that used for the waves for the fully nonlinear internal solitary waves the kinetic energy lower pycnocline. Both sets of results are plotted on the is always larger than the available potential energy in an same horizontal and vertical scales for ease of com- infinitely long domain, becoming identical in the weakly parison. In panels (a) the maximal isopycnals, defined to nonlinear limit. It is of obvious interest to determine be the isopycnal undergoing the maximum vertical dis- how accurate the assumption of equipartition is. placement, are shown. For small waves, the isopycnal We have calculated a sequence of waves for different undergoing the maximum vertical displacement is slightly pycnocline depths to determine how wave properties, below the center of the pycnocline. As the wave am- including the ratio of kinetic to available potential en- plitude increases, the maximal isopycnal moves up, ergy RKA, vary with wave amplitude and with depth of eventually being slightly above the center. The hori- the pycnocline. zontal velocity along the surface u(x, 0) is shown in (b). Figures 2 and 3 show results for stratifications with The wave propagation speed and maximum horizontal zpyc 523 and 25 cm, respectively. The waves for zpyc 5 current are shown in (c), while the kinetic and available

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FIG. 3. As in Fig. 2, but for zpyc 525 cm. Waves shown in (a) and (b) correspond to waves 2 (smallest), 4, 5, 6, 7,10, 13, and 16 in (c) and (d). potential energy are shown in (d). For the variational values as high as 1.5 [the originally reported value of 2 procedure used to compute the waves, the available was revised in Moum et al. (2007)] for shoaling waves of potential energy is the specified parameter, which de- elevation; however, these waves had trapped cores con- termines the wave amplitude (Turkington et al. 1991; taining fluid denser than the surrounding water. Lamb 2002). Figure 5 shows the horizontal velocity field and den-

In Fig. 4, RKA is plotted as a function of the wave sity contours for a typical internal solitary wave using amplitude hmax for pycnocline depths of 1–6 cm in 1-cm zpyc 523 cm. Figure 6 shows the energy density and increments. Values are nondimensionalized by the water energy flux density fields for the same wave. The wave, depth H. Results for two different pycnocline thicknesses, with amplitude 3.2 cm, is propagating rightward. Be- 0.3 and 0.6 cm, are shown for comparison. For each cause the upper layer is significantly thinner than the stratification, RKA increases at first, reaches a maximum, lower layer and the vertically integrated velocity is zero, and then decreases. As the pycnocline moves away from the positive wave-induced currents in the upper layer the middepth, the maximum value of RKA increases, are stronger than those in the lower layer where they are reaching a maximum of 1.3 for (zpyc, dpyc) 5 (21, 0.6) cm. in the opposite direction. Because of this, the kinetic Ratios of 1.4 were reported by Klymak et al. (2006) in the energy density Ek is larger above the pycnocline South China Sea. Klymak and Moum (2003) reported (Fig. 6a); however, there is still a significant contribution

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FIG. 5. Internal solitary wave of amplitude 3.24 cm for zpyc 523 cm and dpyc 5 0.3 cm. Black contour lines are density contours; colors are the horizontal velocity (cm s21) and white regions are velocities with magnitude less the 1 mm s21.

ferent amplitudes (4.1, 3.2, and 2.2 cm) since for a given APE the waves get smaller and broader as the undisturbed pycnocline approaches the middepth. In all cases, the maximum of the vertically integrated kinetic FIG. 4. Ratio of total kinetic and available potential energies as a function of the nondimensional wave amplitude for different and available potential energy densities is similar, with pycnocline depths. The amplitude is nondimensionalized by the the available potential energy decaying more rapidly water depth H. Solid curve: nondimensional pycnocline thickness with x. The largest difference between the two occurs dpyc/H 5 0.02; dotted curve: dpyc/H 5 0.04. For each pair of curves for the case with the pycnocline closest to the surface. the center of the pycnocline z /H is indicated, corresponding to pyc Integrating in the horizontal shows that the total kinetic zpyc 521, 22, 23, 24, 25, and 26 cm when H 5 15 cm. energy is about 25% higher than the total available potential energy for this case. The difference decreases to the total kinetic energy from below the pycnocline as the undisturbed pycnocline moves toward the mid- due to the greater thickness of the lower layer. The depth. Much larger differences can be seen in the in- available potential energy density Ea (Fig. 6b), on the tegrated energy fluxes. The maximum of APEf is 1.6, other hand, is confined to lie in and above the pycno- 2.7, and 7.1 times larger than the maximum of Kf, while cline. This is because only fluid that passes through the the horizontally integrated available potential energy density gradient of the undisturbed density field as it is fluxes are 1.3, 2.1, and 4.8 times larger than the corre- displaced vertically makes a contribution to Ea. The dif- sponding integrated kinetic energy fluxes. The large ferences in the spatial distribution of Ek and Ea have an increase in this ratio as zpyc ! 2H/2 is easily under- important consequence when the energy flux densities stood. As the pycnocline moves toward the middepth, uEk and uEa are considered. The kinetic energy flux the wave-induced horizontal velocities above and be- density is positive above the pycnocline and negative low the pycnocline become more and more antisym- below it (Fig. 6c). Integrating vertically over the wave metric about the middepth. As a result, the kinetic leads to some cancellation. The available potential en- energy density becomes more symmetric across the ergy flux density, in contrast, is largely positive as it is pycnocline. Hence, as zpyc ! 2H/2, the kinetic energy predominantly confined to the region above the dis- flux density becomes more antisymmetric about the placed pycnocline where u is positive (Fig. 6d). As a re- middepth and its vertically integrated value, Kf,de- sult, the vertically integrated available potential energy creases rapidly. The strong asymmetry of the available

flux APEf is much larger than the vertically integrated potential energy flux density, which is always confined kinetic energy flux Kf. to lie above and within the pycnocline, is independent This is illustrated in Fig. 7. Here the vertically inte- of the pycnocline depth. Thus, APEf does not approach grated kinetic and available potential energy densities zero as rapidly (it does get smaller because the wave (left column) and the kinetic and available potential amplitude decreases). energy fluxes, Kf and APEf (right column), are com- These considerations show that equating the available pared for three different waves, each with different potential energy flux APEf with the kinetic energy flux pycnocline depths (1, 3, and 5 cm). The three waves Kf can potentially lead to large errors. This is consid- have similar available potential energies but quite dif- ered in the next section.

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FIG. 6. Kinetic and available potential energy density fields, same wave as in Fig. 5: (a) kinetic 23 23 energy density Ek (J m ); (b) available potential energy density Ea (J m ); (c) kinetic energy density 22 21 22 21 flux uEk (J m s ); and (d) available potential energy density flux uEa (J m s ).

5. Shoaling behavior and reflectance slope s. For stratifications with pycnoclines above the middepth, the solitary waves are waves of depression. The shoaling behavior of internal solitary waves has As the pycnocline approaches the middepth, the waves been investigated for four stratifications, all with d 5 pyc get smaller in amplitude and broader. For z 525 cm, 3 mm, differing only in the depth of the pycnocline, pyc the initial wave is much longer than those for the higher these being 2, 3, 5, and 12 cm. To calculate the incident pycnoclines. It has a much lower maximum slope. The and reflected energy fluxes we use time series of the initial wave for z 5212 cm is a reflection about the vertically integrated fluxes at a location near the base of pyc middepth of that for zpyc 523 cm. z 5 the slope where possible. For some cases using pyc Four examples of flux time series are shown in Figs. 9 25 and 212 cm, the incident and reflected waves and 10 using two different pycnocline depths and two overlap at the base of the slope due to its proximity to different bottom slopes. Results from two cases using the breaking site. For these cases, the fluxes were cal- zpyc 523 cm, with an initial wave amplitude of 3.2 cm, culated at two locations farther away and the fluxes at are shown in the top panels, while the results from two the base of the slope were estimated by linear extrap- cases using zpyc 525 cm, with an initial amplitude of 2.2 olation. This is justified below. For most cases, the initial cm, are shown in the bottom panels. The bottom slopes 21 waves had an initial APE of about 0.032 J m regard- used are 0.214 (Fig. 9) and 0.069 (Fig. 10). Positive less of the pycnocline depth, the exception being one values correspond to incident waves propagating to the broad-crested solitary wave using zpyc 525 cm. The right, while negative values correspond to reflected isopycnals at the center of the pycnocline for each of the waves propagating to the left. For the smaller slope the initial waves are shown in Fig. 8. For this plot we have tank is 4 m long, while for the steeper slope it is 3 m shifted the location of the waves to x 5 0. In the simu- long. The initial wave enters the region over the slope in lations the initial location depends on the bottom the first 20 s. The reflected wave passes by at times t

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FIG. 7. (left) Vertically integrated kinetic energy (solid) and available potential energy (dotted) densities for three different waves using pycnocline depths of (a),(b) 1; (c),(d) 3; and (e),(f) 5 cm. Wave amplitudes and available potential energies are 4.1, 3.2, and 2.2 cm and 0.047, 0.053, and 0.053 J m21, respectively. (right) The corresponding vertically integrated kinetic energy ﬂux (solid) and available potential energy ﬂux (dotted) densities.

between 20 and 50 s for the steep slope and between 30 this stratiﬁcation Kf is much smaller than APEf in the and 70 s for the gentler slope. initial wave and in the reﬂected wave and, indeed, is

Consider the first case (zpyc 523 cm, s 5 0.214) almost negligible (cf. Figs. 7d,f). The linear energy flux shown in Fig. 9a. The largest contribution to the total W is more dominant as well. This is a consequence of energy flux Ef comes from W, the linear energy flux the smaller wave amplitudes associated with the re- term, for both the incident and reflected waves. For the duced distance of the pycnocline from the middepth. initial incident wave the available potential energy flux For the cases using the gentler slope, the reflected wave

APEf is much larger than the kinetic energy flux Kf, is much smaller in amplitude and is broad with multiple while for the smaller reflected waves Kf and APEf are small amplitude oscillations superimposed. Both Kf and similar. The reflected wave has a few oscillations, sug- APEf are negligible (Fig. 10). gesting a wave form that will evolve into several solitary Density contour plots illustrating the breaking and waves. By the time the reflected wave has reflected off reflection process are shown for three case in Figs. 11–13 the end wall and returned (between 55 and 70 s) it has The first two figures show the same wave shoaling over almost separated into two solitary waves. The time se- two different slopes. For these two cases, the pycnocline ries at this point is not used because, due to the mixing depth is 3 cm and the initial wave amplitude is 3.2 cm. that occurred during the first breaking event, the pyc- As the wave shoals, it steepens at the back. For the steep nocline is now much thicker near the slope. slope case (s 5 0.214), Fig. 11, the back of the wave rises A similar sequence of events occurs for the second above the undisturbed pycnocline depth (not shown) case (zpyc 525 cm, s 5 0.214) shown in Fig. 9b. For before breaking backward (Fig. 11b). This form of wave

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tered. A bolus forms, which moves up the slope while decreasing in amplitude. A small, long wave of elevation propagates away from the slope (Figs. 13b–d). Trailing it is an intrusion that is led by a small amplitude mode-2 wave front. This has negligible energy, a consequence of its small amplitude and propagation speed. Figure 14 shows the evolution of the kinetic and available potential energy (top) and the pseudoenergy and background potential energy (bottom; note these energies are plotted between 0.03 and 0.08 J m21)for three different cases. The ﬁrst case (Figs. 14a,d) is

for Case A, which has an ISW of depression for zpyc 5 20.03 m and a bottom slope of 0.214 shown earlier in Fig. 11. The second case (Figs. 14b,e) is identical except that the viscosity has been decreased by a factor of 10 to n 5 1027 m2 s21. The last case (Figs. 14c,f) is for a

FIG. 8. Initial waves for the four stratifications. The isopycnal at wave of elevation using zpyc 520.12 and s 5 0.214, the center of the pycnocline is plotted in each case. (top to bottom) with n 5 1026 m2 s21. As pointed out previously, the 21 The solid curves are initial waves with an APE of 0.032 J m for kinetic energy is larger than the available potential 52 2 2 2 zpyc 2, 3, 5, and 12 cm; the dashed curve shows the initial energy in the initial wave. As the ISW propagates flat-crested wave used in one simulation. toward the slope, the KE and APE slowly decrease approximately linearly in time at the same rate due breaking was referred to as a collapsing breaker by to viscous dissipation. This linear decrease, together Boegman et al. (2005a), who also illustrated several with the nearly constant propagation speed, is used to other forms of breaking shoaling waves. The formation justify the linear extrapolation used to estimate the of the waves of elevation begins before the wave rea- energy fluxes in cases for which the incident and re- ches the turning point, the location at which the pyc- flected waves were not well separated at the base of nocline is at middepth (water depth 6 cm). Beyond the the slope. For Case A, the average decay rate of the turning point solitary waves of depression can no longer kinetic and available potential energy over the first exist. Given enough time, waves of depression trans- 12 s are 0.17 and 0.14 mJ m21 s21, respectively. The form into a train of waves of elevation (Grimshaw et al. decay is about 5 times smaller in the lower viscosity 2004). In this case, the wave amplitude is slightly larger case, rather than 10 times smaller, suggesting that than the upper-layer depth and wave breaking occurs numerical viscosity is playing a more important role before the turning point is reached. The reflected waves for the low viscosity case. As the wave shoals, it slows have the form of a wave train (Figs. 11c and 11d). An down, and kinetic energy is converted to APE. For the intrusion follows the leading mode-one waves (x $ 0.7 waves of depression the maximum value of APE is in Fig. 11). The time series of the energy fluxes at the 3.4 and 4.5 times larger than the KE for the large and base of the slope for this case were presented in Fig. 9a. small viscosities, respectively. The APE reaches its Henceforth, we refer to this case as Case A. We explore peak value at t 5 18.5 s, just before the time of Fig. 11b. the shoaling process for this case in greater detail below. This is the time when breaking commences. For the For the gentler slopes, with s 5 0.069 (Fig. 12), the wave of elevation the decay rate of the kinetic and breaking occurs farther behind in a train of waves of available potential energy are 0.29 and 0.24 mJ m21 s21, elevation that form behind the initial wave of depres- respectively, about 70% higher than for the wave of sion (Fig. 12b). The corresponding energy flux time depression in Case A (see below). Almost all of the series at the base of the slope are shown in Fig. 10a. kinetic energy is converted to APE with the maximum When the pycnocline is below the middepth, the inci- value of APE/KE equal to 13. This illustrates one key dent ISW is a wave of elevation and the character of the difference in the shoaling behaviors of waves of de- breaking is completely different (Fig. 13). For this expression and elevation. ample, the pycnocline depth is 3 cm above the bottom The lower panels in Fig. 14 show the evolution of the and, by symmetry, the initial wave is a reflection about pseudoenergy and the background potential energy the middepth of that in the previous two cases. In this (BPE). The latter is calculated by sorting the density case, as the wave shoals, a turning point is not encoun- field in the closed domain, which for these cases has

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FIG. 9. Time series of the vertically integrated pressure-velocity ﬂux (solid), available po-

tential energy flux (dotted), and kinetic energy flux (dashed): (a) Case with zpyc 523 cm, initial wave amplitude 3.2 cm, and Iribarren number 0.7, and (b) case with zpyc 525 cm, initial wave amplitude 2.2 cm, and Iribarren number 1.3. Shelf slope 0.214 in both cases. length 3 m. The BPE depends on the coordinate system, and no-shear along the upper boundary, account for the so we set its initial value to be equal to the initial higher energy dissipation for the wave of elevation. pseudoenergy for the sake of comparison. Consider Case The mixing efficiency is the ratio of the increase in A, shown in Fig. 14d. Before the wave starts shoaling BPE to the loss in pseudoenergy. For the waves of de- over the sloping bottom at t 5 14 s, the pseudoenergy pression the mixing efficiencies between t 5 20 and 35 s slowly decreases linearly in time due to viscous dissi- are 0.2 and 0.27 for n 5 1026 and n 5 1027 m2 s21, pation. It drops rapidly as breaking commences at t 5 19 respectively. For the wave of elevation the mixing effi- s. The BPE is virtually constant until t 5 21 s, slightly ciency between t 5 18 and 25 s is 0.1. These values are after the commencement of breaking. Similar behavior sensitive to the time interval. For example, for the first is observed in the lower viscosity case except that the case the mixing efficiency between t 5 16 and 35 s is initial gradual decay in the pseudoenergy is greatly re- 0.13. Since much of the dissipation occurs outside of the duced. For the shoaling ISW of elevation the BPE rises stratified layer, the mixing efficiency in the pycnocline much earlier, indicative of a different breaking process. itself would be larger. Michallet and Ivey (1999) esti- As discussed above, the initial decay rate is much larger mated the mixing efficiencies in their laboratory ex- for the wave of elevation than for the wave of elevation periments. They used the energy loss over the time shown in Fig. 14d for Case A. These two cases have interval spanning the arrival of the incident and re- initial waves that are reflections of each other about the flected waves at the base of the slope, while the increase middepth. For the wave of depression the largest wave- in the background potential energy was that over the induced flow is in the upper layer, while for the wave duration of the experiment. They justified this by as- of elevation it is in the lower layer. Differences in suming that most of the mixing would occur during the boundary conditions, no-slip along the lower boundary first breaking event. For their case 16, which is closest to

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FIG. 10. As in Fig. 9, but for a shelf slope of 0.069: Iribarren numbers are 0.23 and 0.41.

our Case A (slightly smaller wave, same pycnocline in the potential energy and background potential en- depth, larger density ratio), they estimated a mixing ergy. The available potential energy computed in the efficiency of about 0.2, which is remarkably similar to finite domain is smaller than that in an infinite domain the value in our numerical simulation. Since the tank because the pycnocline in the sorted density field is used in the laboratory had sidewall friction, which below that in the far-field density profile (see Lamb would increase viscous dissipation due to the presence 2008). The difference between the two is almost con- of boundary layers along the side walls, one could ex- stant with time. pect lower mixing efficiencies in the laboratory experi- As mentioned above, for a shoaling wave of depres- ments because less energy would arrive at the breaking sion the pseudoenergy starts to decrease rapidly before location. Mixing during secondary breaking events the BPE increases, that is, before any mixing occurs. would tend to increase their estimate. The cause of this is illustrated in Fig. 17, which shows Figure 15 compares the background density field r(z) plots of the horizontal velocity and vorticity fields and with the sorted density profile at t 5 0 and at t 5 40 s, the viscous dissipation rate for Case A at t 5 19 s. well after the first breaking event, for Case A. Only part Density contours are overlain. The horizontal velocity of the water column, centered on the pycnocline, is field shows the presence of a strong jet of fluid being shown. The pycnocline is lower in the sorted density squeezed out beneath the descending pycnocline as the profile since the initial wave is a wave of depression. wave reaches the boundary. At the trough of the wave a After the wave breaking event the pynocline has large adverse pressure gradient results in boundary thickened noticeably, as expected due to the increase in layer separation and the creation of a vortex (Fig. 17b). BPE. Figure 16 compares the evolution of the APE with Such vortices have been observed in laboratory exper-

APEL for Case A. APEL is the available potential en- iments (Boegman and Ivey 2009). The bottom panel ergy in the closed domain of length 3 m obtained by shows that the dissipation rate f is large in the ﬂuid jet sorting the density ﬁeld and calculating the difference and in the vortex. This accounts for the drop in energy

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FIG. 11. Shoaling wave: Shelf slope 0.214, pycnocline depth 3 cm, Iribarren number 0.7, and initial wave amplitude 3.2 cm at (a) t 5 17 s, (b) t 5 19 s, (c) t 5 28 s, and (d) t 5 32 s.

beginning before wave breaking fully develops. The the sloping wave front, there is no adverse pressure boundary layer separation and vortex formation seems gradient at this location. The ejection of fluid beneath to play an important role in the breaking process, at the wave front results in the creation of a small de- least at these Reynolds numbers. In simulations using a pression at the back of the sloping wave front. Associ- shoaling broad, flat-crested wave (see Fig. 8) breaking ated with this is a weak adverse pressure gradient that, occurs at the front of the wave, not at the rear. This is presumably, results in the boundary layer under the illustrated in Fig. 18, which shows density contours and wave front separating. A large vortex does not appear in the vorticity field at the front part of a broad flat-crested this case. Instead, oscillations appear in the boundary wave interacting with the slope at three different times. layer, indicative of a flow instability, and the wave The wave is about 2 m long with a flat crest approxi- starts to break (Fig. 18b). The reflected wave leaves the mately 1 m in length. A bottom boundary layer with slope well before the back of the incident wave arrives, negative vorticity is present under the wave due to a and the superposition of the incident and reflected leftward wave-induced current, which extends beyond waves results in the absence of any overturning at the the left side of the plot. As the sloping front of the wave rear of the wave. This has the consequence of restricting (x between 0.95 and 1.3 m in Fig. 18a) impinges on the the wave breaking to a small patch of the wave slope, fluid between the pycnocline and the boundary is (Fig. 18c). squeezed out, intensifying the flow. Because the wave- The total incident and reflected fluxes were estimated induced current is not decelerated immediately behind by integrating the flux time series over an appropriate

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FIG. 12. As in Fig. 11, but for shelf slope 0.069 and Iribarren number 0.23 at (a) t 5 19s, (b) t 5 28 s, (c) t 5 32 s, and (d) t 5 44 s.

time interval. Michallet and Ivey (1999) estimated lecular viscosity and it ignores the large change in the the available potential energy flux from the time series sorted density rr(z, t) that occurs in the tank due to the of the iscopycnal displacements by estimating the mixing that occurs when the wave breaks on the slope. available potential energy in the wave using the ex- The former effect is small for the diffusivity and times pression of Bogucki and Garrett (1993) for a two-layer considered here. The mixing has a big impact on the fluid, assuming equipartition of kinetic and available thickness of the pycnocline in the vicinity of the slope: potential energy, and multiplying the total energy by the However, we take the point of view that in the open propagation speed. This approach is only valid for ocean the reflected wave will be propagating into a waves of permanent form and hence is not generally region where mixing over the slope has not had a valid for the reflected waves. Helfrich (1992) and chance to modify the stratification. Hence, the far-field Bourgault and Kelley (2007) integrated the kinetic en- density field is the appropriate choice for the reference ergy flux times series but assumed that the available density. potential energy flux was equal to the kinetic energy Figure 19 shows the reflectances as a function of the flux. As discussed earlier, this is far from the case (see Iribarren number (Boegman et al. 2005a; Bourgault and

Fig. 7). Here we calculate Kf from the known velocity Kelley 2007) ﬁelds, while APE is calculated using the initial density f s proﬁle as the reference density. This ignores the fact j 5 1/2 , (26) that the initial pycnocline slowly thickens due to mo- jjao/Lw

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FIG. 13. As in Fig. 11, but for shelf slope 0.069, pycnocline depth 12 cm, Iribarren number 0.23, and initial wave amplitude 3.2 cm at (a) t 5 13 s, (b) t 5 17 s, (c) t 5 30 s, and (d) t 5 44 s.

where s is the shelf slope, ao is the wave amplitude, and 212 cm, all with dpyc 520.3 cm. The first three sets Lw is the length of the wave defined by of cases have the pycnocline above the middepth and ð the initial ISWs are waves of depression. The cases with ‘ 1 zpyc 5212 cm have the pycnocline below the middepth Lw 5 hc(x)dx (27) a0 ‘ and the ISWs are waves of elevation. By symmetry, the initial waves for this case are reflections about the in which hc(x) is the vertical displacement of the center middepth of those for zpyc 523 cm. Thus, they have of the pycnocline. This isopycnal is not necessarily the the same Iribarren numbers. The breaking process is isopycnal undergoing maximal displacement (see Figs. 2 very different for the waves of elevation and depression, and 3). Using the isopycnal undergoing maximal dis- as illustrated in Figs. 11 and 13. Five runs were done placement would be tedious and would not change the for each stratification using the same initial wave but result significantly. Results using a kinematic viscosity different bottom slopes (0.3, 0.214, 0.169, 0.1, and n 5 1026 m2 s21, appropriate for simulations of the 0.069). laboratory experiments, are shown with a solid diamond The reflectances incorrectly calculated using the time in Fig. 19. Results from higher Reynolds number sim- integral of W 1 2Kf are indicated by the 3 symbols for ulations, with n decreased by a factor of 10, are shown the low Reynolds number cases (n 5 1026 m2 s21). with open diamonds. These should be compared with the solid diamonds, the For the reflectance calculations, four different strati- difference between the two being the error made in

ﬁcations were considered, zpyc 522, 23, 25, and assuming APEf is equal to Kf. The incorrectly calculated

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FIG. 14. Energy evolution for three different cases: (a)–(c) Evolution of the total kinetic (solid curves) and available potential energy (dashed curves); (bottom) (d)–(f) the total energy (solid curves) along with the back- 26 2 21 ground potential energy (dashed curves). (a), (d) Case A: zpyc 523m,s 5 0.214, n 5 10 m s . (b), (e) Same as 28 2 21 (a), (d) but with reduced by a factor of 10; (c), (f) same as (a), (c) but for zpyc 5212 cm. In all cases k 5 10 m s .

values are consistently higher than the correct reflec- numerical simulations, which included a variety of tances, the maximum difference being about 0.1 for stratifications and density ratios (hence Reynolds some cases using zpyc 522 cm. For the cases with zpyc 5 numbers). 25 cm, APEf and Kf are very different (see Fig. 7), so The simulations of laboratory tank experiments us- one may expect the largest differences between the two ing n 5 1026 m2 s21 are low Reynolds number simu- methods to occur for this case. However, because wave lations, with Re of O(5000) using a typical velocity of 5 21 amplitudes decrease as zpyc approaches the middepth, cm s and a length scale of 10 cm. To investigate the so does APEf /W and Kf /W. As a consequence, the sensitivity to the Reynolds number the simulations energy flux is more and more dominated by the linear were repeated using a viscosity 10 times smaller (open energy flux W and errors in estimating APEf become diamonds). The reflectances are always larger for the insignificant. higher Reynolds number simulations. Part of this can The parameterization proposed by Bourgault and be attributed to lower energy loss as the wave propa- Kelley (2007), gates between the bottom of the slope and the location where the wave breaks (see Fig. 14). If the difference R 5 1 ej/0.78, (28) was solely due to this, one would expect the increase in the reflectance to decrease as the Iribarren number isindicatedinthefigurebythesolidcurveasaref- increases (i.e., as the slope increases) since the distance erence. This parameterization was obtained by fitting between the bottom of the slope and the breaking region 2j a function of the form 1 2 e /l to the results of their decreases. This is what occurs for pycnocline depths of

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FIG. 16. Comparison of the evolution of APE (dashed), the available potential energy in an inﬁnite domain, with APEL (dashed–dotted), the available potential energy in a closed domain obtained by sorting the density ﬁeld. Same case (Case A) depicted in Figs. 14a,d.

FIG. 15. Density profiles for Case A (zpyc 523 cm, s 5 0.214): The background density (solid), the initial sorted density (dashed), mimicking laboratory experiments, have been consid- and the sorted density profile at t 5 40 s (dashed–dotted). ered using idealized, continuous, quasi-two-layer stratifications consisting of a single thin pycnocline at different depths. 2 and 3 cm; however, the behavior is different for the The energetics were explored using an equation for other two pycnocline depths. The reason for this is not the wave pseudoenergy density: the sum of kinetic and clear. One would further expect these differences to available potential energy densities. Here the available decrease as the pycnocline moves downward, as this potential energy is defined to be the value in an un- decreases the distance from the base of the slope to the bounded domain. The ratio, RKA, of the kinetic energy breaking site. This trend is observed for the first three to available potential energy in the solitary waves is cases, the exception being the stratification with the always larger than one and, for the stratifications con- pycnocline lying below the middepth. sidered here, was found to have a maximum value of Two cases were run at double the resolution (zpyc 5 just under 1.3, with the value increasing as the pycno- 20.03 m, j 5 0.7 and zpyc 520.05 m, j 5 1.3) for both cline moves away from the middepth. Because of dif- viscosities. The Iribarren numbers and reflectances ferent spatial distributions of the kinetic energy density changed between 1% and 2%, indicating little sensi- Ek and the available potential energy density Ea, the tivity to the resolution. kinetic and available potential energy fluxes are quite

different. The available potential energy ﬂux APEf was consistently larger than the kinetic energy ﬂux K . Their 6. Summary f ratio increased as the pynocline approached the mid- Internal solitary-like waves are often highly energetic depth, with a maximum value of seven for the cases features of the coastal ocean, making an accurate de- considered here. The increase is associated with the fact termination of their energy important. This can be dif- that the kinetic energy is distributed throughout the

ficult in the ocean environment where they are never water column; hence, there is cancellation of uEk when isolated features. They are affected by tides, currents, integrating over the water column since u changes sign eddies, horizontally and temporally varying stratifica- across the pycnocline and Ek is positive everywhere. In tion, and other phenomena. They rarely propagate in contrast, the available potential energy density is con- water of constant depth and are affected by the earth’s fined to the region in and above/below the pycnocline rotation. Some of the difficulties in making accurate in an internal solitary wave of depression/elevation. measurements of these waves are discussed in Moum Hence, uEa is largely positive and little cancellation and Smyth (2006), Scotti et al. (2006), and Moum et al. occurs when integrating over the water column. As the (2007). Here, via numerical computations, the ener- pycnocline approaches the middepth, the horizontal getics of internal solitary waves in an idealized setting, velocity field becomes more and more antisymmetric

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FIG. 17. Velocity, vorticity, and dissipation rate for Case A, shoaling wave with zpyc 520.03 m, s 5 0.214, (n, k) 5 (1026,1028)m2 s21 [same case as in Fig. 14(a), (d)]: (a) Horizontal velocity (cm s21), white indicates vorticity values less than 2 mm s21 in magnitude; (b) vorticity (s21), white indicates vorticity values less than 0.2 s21 in magnitude; (c) energy dissipation rate (mJ s21 m23).

and the kinetic energy density more symmetric about both APEf and Kf make a smaller contribution to the the middepth. Hence, the vertical integral Kf decreases total flux. Hence, errors in calculating APEf have a rapidly. While both Kf and APEf decrease due to de- smaller effect. Indeed, for the cases with the pycnocline creasing wave amplitudes (for fixed APE), APEf /KEf closest to the middepth, zpyc 525 cm, both APEf and increases. The ratio of the total flux (integrated in Kf make a negligible contribution to the total flux, as time or space across the wave) had a maximum of about illustrated in Fig. 19c. Although the total APE flux is 5. This shows that approximating the available potential five times larger than the kinetic energy flux, because of energy flux by the kinetic energy flux is inappropriate. the much smaller wave amplitudes, APEf and Kf are For other stratifications, for example, an exponential both negligible compared with W, so a large error in density profile, Kf, can be larger than APEf. estimating APEf has little impact (see Figs. 9b and 10b). Calculating the available potential energy flux cor- Because correctly estimating the available potential rectly decreases predicted reflectances by a maximum of energy flux results in only a small change in the reflec- 0.1 relative to the values obtained using the approxi- tances, the parameterization suggested by Bourgault mate method used by Bourgault and Kelley (2007). One and Kelley (2007) remains a useful estimate of the re- can expect larger differences in the reflectance for larger flectances for low Re typical of those in the laboratory

ISWs. Since APEf /KEf increases as the pycnocline moves experiments of Michallet and Ivey (1999). Some sensi- toward the middepth, one may expect that the error tivity to the pycnocline depth is apparent, and the pa- made in assuming APEf 5 Kf should increase: However, rameterization appears to be less useful as zpyc it decreases because the wave amplitudes decrease and approaches the middepth. This could be associated with

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FIG. 18. Vorticity (colors) and density contours (black lines) for a shoaling broad solitary wave with zpyc 520.05 m, s 5 0.214, (n, k) 5 (1026,1028)m2 s21.(a)t 5 19 s, (b) t 5 24 s, (c) t 5 26 s: White indicates vorticity values less than 0.2 s21 in magnitude.

the increased width of the waves, resulting in breaking the slope and the breaking location, in which case affecting a smaller volume of the wave. Indeed, for the sensitivity to Re could decrease significantly as Re is broad wave illustrated in Fig. 18 the reflectance is esti- increased further. Because of the sensitivity to Re, mated to be about 0.98 (obtained by extrapolating from the application of the parameterization suggested by 1 m away from the base of the slope). The Iribarren Bourgault and Kelley (2007) to oceanographic situa- number is 1.7, so the reflectance is higher than the es- tions, with Reynolds numbers several orders of mag- timated value of 0.93 obtained by interpolating the nitude higher, should be used with caution. Further values plotted in Fig. 19c. The result for the same bot- investigation of higher Reynolds number cases is re- tom slope plotted in Fig. 19 is just below 0.9 for an Iri- quired. Sensitivity of the reflectance R to other factors, barren number of about 1.27. These two waves have such as bottom roughness and angle of incidence also similar maximum slopes. The difference between the needs to be explored. Simulations of mixing in pro- results for these two simulations suggests that the Iri- gressive waves on a thin pycnocline by Fringer and barren number may not be the best choice for param- Street (2003) have demonstrated that mixing processes eterizing the reflectance. are different in two- and three-dimensional simula- These simulations of laboratory tank experiments tions, so the reflectances and mixing efficiencies in had Re of O(5000). Increasing Re by a factor of 10, still three-dimensional simulations are possibly different well below oceanographic values, increased the re- than in two-dimensional ones: hence, this also needs to flectances, in some cases significantly. Part of this is be explored. We have also not considered the sensi- due to a decrease in energy loss between the base of tivity of R to the wave amplitude.

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FIG. 19. Reﬂectance as a function of the Iribarren number j: Solid and open diamonds are results

calculated using the available potential energy flux APEf; plus signs are reflectances calculated by assuming the available potential energy flux is equal to the kinetic energy flux; solid curve is parameterization of Bourgault and Kelley. Cases with solid diamonds and plus signs used (n, k) 5 (1026, 28 2 21 10 )m s while those with open diamonds had a viscosity 10 times smaller. (a) Cases with zpyc 522 cm. (b) Cases with zpyc 523 cm. (c) Cases with zpyc 525 cm. (d) Cases with zpyc 5212 cm.

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