MARCH 2009 L A M B A N D N G U Y E N 559 Calculating Energy Flux in Internal Solitary Waves with an Application to Reflectance 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 signif- icant 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 inves- tigated. 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 Ó 2009 American Meteorological Society Unauthenticated | Downloaded 10/03/21 11:37 AM UTC 560 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 39 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) ÀH ÀH 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- ÀH 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 instanta- neous contributions from up but that the accumulated ›u ›u ›u 1 ›p 2 flux from this term was small. 1 u 1 w 5 À 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 À À 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 ÀH 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 $ Á [(E 1 p)~u] À m$ Á (u Á $u) À kg$ Á (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 À f À gk (9) in open systems (Scotti et al.