Energy and pseudoenergy flux in the internal wave field generated by tidal flow over topography

Abstract

The mechanical energy and pseudoenergy budgets in the internal wave field generated by tidal flow over topography is considered using a nonlinear, two-dimensional numerical model. The Boussinesq and rigid lid approximations are made, viscosity and diffusion are ignored and the flow is treated as incompressible. Both ridge and bank edge topographies are considered. The nonlinear energy equation and an equation for pseudoenergy (kinetic energy plus available potential energy) are satisfied to within less than 1%. For a uniform stratification (constant buoyancy frequency N) the available potential energy density is identical to the linear potential energy density $\frac{1}{2}(g^2/N^2){\tilde{\rho }}_d^2$ where ${\tilde{\rho }}_d$ is the density perturbation. For weak tidal flow over a ridge in the deep ocean, using a uniform stratification, the generated waves are small, approximately 2% of the water depth, and the traditional expression for the energy flux, $〈\tilde{u}{\tilde{p}}_d〉$ accurately gives the pseudoenergy flux. For a case with strong tidal flow across a bank edge, using a non-uniform stratification, large internal solitary waves are generated. In this case, the linear form of the potential energy is very different from the available potential energy and the traditional energy flux term $〈\tilde{u}{\tilde{p}}_d〉$ accounts for only half of the pseudoenergy flux. Fluxes of kinetic and available potential energy are comparable to the traditional energy flux term and hence must be included when estimating energy fluxes in the internal wave field.

Introduction

Over the past decade, there has been a resurgent interest in estimating the energy content of internal waves in the ocean generated by tidal flow over topography. This has been primarily driven by an improved appreciation of the role they play in mixing the abyssal ocean, and by the need for better estimates of both the amount and the distribution of this mixing. This increased interest has been triggered by a few key recent developments. Munk and Wunsch (1998) and Wunsch and Ferrari (2004) discussed the importance of mixing in the deep ocean in driving the meridional overturning circulation. They estimated that, of the roughly 2 TW of power needed, approximately 40–50% is provided through the conversion of barotropic to baroclinic tidal energy in the deep ocean by tide-topography interaction, with the remainder coming from the winds. Their estimates were based on Egbert's estimate that 0.9 TW of energy is lost from the barotropic tide (Egbert, 1997) in the deep ocean. This estimate, which was subsequently refined and updated (Egbert and Ray, 2000, Egbert and Ray, 2001), was based on TOPEX/Poseidon data, the time series having only just become long enough to isolate the small sea surface displacements associated with radiating semi-diurnal $(M_2)$ internal tides. At the same time direct observations of mixing above the western flank of the Mid-Atlantic Ridge showed a fort-nightly modulation of mixing which was strongly enhanced over rough bottom topography, suggesting a connection with the tides (Ledwell et al., 2000).

These developments have motivated several theoretical (Khatiwala, 2003, Balmforth et al., 2002; Llewellyn-Smith and Young, 2002) and numerical (Holloway and Merrifield, 1999, Holloway and Merrifield, 2003; Merrifield et al., 2001, Niwa and Hibiya, 2001, Niwa and Hibiya, 2004, Munroe and Lamb, 2005, Lamb, 2004, Gerkema et al., 2006, Legg and Huijts, 2006) studies of deep ocean internal wave generation by tidal flow over topography. Several recent field programs have estimated the energy flux in the internal wave field for both small amplitude linear waves in the deep ocean (Kunze et al., 2002, Althaus et al., 2003, St. Laurent and Nash, 2004; Nash et al., 2005) and for large nonlinear internal solitary-like waves (ISWs) in coastal regions (Jeans and Sherwin, 2001, Lien et al., 2005, Klymak et al., 2006). Energy flux in surface and internal tides in Knight Inlet were investigated observationally by Marsden and Greenwood (1994) and numerically by Stacey and Pond (2005). Inciteful reviews of much of the recent work can be found in St. Laurent and Garrett (2002) and Garrett and Kunze (2006). One goal of this work has been an improved understanding of the energy flux in internal waves generated by the interaction of the barotropic tide with a variety of topographic features. The energy flux in nearly periodic internal waves is usually equated with the vertical integral of $〈{\stackrel{\Rightarrow }{u}}^{\prime }{p}^{\prime }〉$ where the brackets denote a time average, and ${\stackrel{\Rightarrow }{u}}^{\prime }$ and $p^{\prime }$ are the baroclinic horizontal velocity and pressure perturbations. Often only the hydrostatic part of the pressure perturbation is included. Kunze et al. (2002) were the first to consider the vertical structure of the linear energy flux $〈{\stackrel{\Rightarrow }{u}}^{\prime }{p}^{\prime }〉$, for which the vertical average of $p^{\prime }$ is important. By considering the vertical structure, rather than the vertical integral as is usually done, vertically propagating internal wave beams could be detected. This was done more clearly by Althaus et al. (2003) who investigated internal wave generation at the Mendocino Escarpment. Techniques for estimating the linear internal energy flux from sparse ocean observations are explored in Nash et al. (2005).

When discussing energy budgets a distinction must be made between potential energy and available potential energy (APE) (Holliday and McIntyre, 1981, Shepherd, 1993, Winters et al., 1995). The latter is the potential energy that is available for conversion to other forms of mechanical energy and hence for mixing. It is defined by adiabatically sorting the density field to minimize the potential energy. This sorted, or relaxed state, ${\overline{\rho }}_r$ is horizontally uniform. The APE is the difference in potential energy between the original state and this relaxed state. In discussing the energetics of a wave field, it has long been recognized that of greatest interest is the energy budget for the kinetic and APE, called the pseudoenergy (Shepherd, 1993). In particular, it is the flux of kinetic energy and APE that is of physical significance. For small amplitude waves $〈u^{\prime }{p}^{\prime }〉$ gives the leading-order contribution to the pseudoenergy flux, hence the importance of this flux term. In a closed domain the APE is well defined, however, in an unbounded domain the calculation of APE can be difficult because, due to data limitations, it must often be calculated in a finite domain. This yields an estimate of the APE which is sensitive to the length of the domain used. For a uniform stratification with no background flow, ${\lambda }_n(\omega )$, the wavelength of a linear mode-n wave of frequency $\omega$, satisfies the relation ${\lambda }_n(\omega )={\lambda }_1(\omega )/n$. For periodic tidal flow over topography linear theory predicts that most of the energy is in waves of tidal frequency. In this case, if the higher harmonics are ignored all the waves have the same frequency and in principle the wave field could be spatially periodic with period ${\lambda }_1$, enabling the APE to be well defined and easily computed. In practice this would not be the case because both wave–wave and wave–mean flow interactions and diffusive and viscous effects would modify the initially uniform stratification and mean flow field. When the stratification is non-uniform the computation of APE is even more problematic because ${\lambda }_1/{\lambda }_n$ is no longer an integer, hence any multi-modal wave field cannot be periodic.

Equations for a family of APE densities, with conservation laws for the corresponding pseudoenergy, have been derived (Shepherd, 1993, Winters et al., 1995, Scotti et al., 2006). These conservation laws are obtained from a suitable combination of the conservation laws for mass and mechanical energy. Shepherd (1993) showed that the need to combine these two conservation laws can be attributed to the non-canonical Hamiltonian structure of the Eulerian equations of motion. The non-uniqueness of the pseudoenergy arises because it is formulated in terms of a reference density state. One should use the adiabatically sorted density ${\overline{\rho }}_r$ to uniquely define the APE, however in practice it is often more practical, or, in the case of open systems or when dealing with field observations, necessary to use an approximate reference density, i.e., in numerical simulations one could use the initial undisturbed density field.

Often the energy content in ISWs is estimated using KdV-type weakly nonlinear theory (e.g., Osborne and Burch, 1980; Sandstrom and Elliott, 1984, Sandstrom et al., 1989). Jeans and Sherwin (2001) estimated the energy in ISWs on the Portuguese Shelf by computing the kinetic and APE in the waves from velocity and density measurements. Chang et al. (2006) used ADCP measurements of the velocity field in the South China Sea to determine the energy content in high-frequency ISWs. Linear theory was used to estimate the potential energy spectrum from the velocity spectra and the energy content of the ISWs was then estimated by integrating the two spectra over a high-frequency band that they associated with ISWs. Klymak et al. (2006) estimated the energy in ISWs observed in the South China Sea using fully nonlinear expressions for the APE by sorting the observed rest state to find ${\overline{\rho }}_r$, as was also done by Jeans and Sherwin (2001) and Moum et al. (2006). Both the density and velocity fields used for the calculation were based on fitting a KdV model of the wave to the observed density field in part of the water column. Brickman and Loder (1993) used similar methods to estimate the kinetic energy in ISWs on Georges Bank. They approximated the APE using the linear approximation $\frac{1}{2}(g^2/N^2){{\rho }^{\prime }}^2$ where ${\rho }^{\prime }$ is the density perturbation. All of these authors estimated the energy flux as the product of the propagation speed and the wave energy.

In addition to calculating the energy flux as above, Chang et al. (2006) also estimated the energy flux using the vertical integral of ${\stackrel{\Rightarrow }{u}}^{\prime }{p}^{\prime }$ following Nash et al. (2005). This method yielded energy fluxes of between 50% and 90% of that obtained via the first method, depending on location, with the difference decreasing for higher flux values. That this method is inappropriate will be illustrated in Section 4 of this paper. Other authors have pointed out the need to use more accurate methods which include the nonlinear contributions for the energy flux (Venayagamoorthy and Fringer, 2005, Venayagamoorthy and Fringer, 2006, Moum et al., 2006). Moum et al. (2006) used highly resolved velocity measurements from a bottom lander to calculate the energy flux in ISWs on the Oregon Shelf. They calculated the energy transport in the waves using the full nonlinear expression for the energy flux, $\mathit{uE}+\mathit{up}$ where E is the sum of the kinetic and APE and p is the total pressure. The pressure–velocity energy flux was found to include important contributions from non-hydrostatic effects and from the surface displacement. The energy flux term was found to make a significant contribution to the total energy flux, highlighting the nonlinearity of the waves. They also confirmed that the vertically integrated energy flux is equal to the product of the energy and the propagation speed.

The purpose of this paper is to investigate the energy budgets for the total mechanical energy and for the pseudoenergy in high-resolution non-hydrostatic numerical simulations of internal wave generation by tidal flow over topography in order to determine the relative importance of various flux terms. Two cases are considered. The first considers weak tidal flow over a deep ocean ridge using a uniform stratification (constant buoyancy frequency N). The second case is for strong tidal flow across a bank edge using a vertically varying buoyancy frequency $N(z)$. A uniform stratification is usually used in theoretical studies because analytic solutions of the linear internal wave generation problem are available (Bell, 1975a, Bell, 1975b; Khatiwala, 2003). In addition, the constant buoyancy frequency case is distinctive because the approximation of the energy used in linear theory, $E_{\ell \mathit{in}}=\frac{1}{2}\stackrel{\Rightarrow }{u}·\stackrel{\Rightarrow }{u}+\frac{1}{2}{g}^2{{\rho }^{\prime }}^2/N^2$, is conserved by the fully nonlinear equations only if the buoyancy frequency is constant, in which case it is identical to the pseudoenergy. For non-uniform stratifications ISWs are generated if the tidal forcing and topographic amplitude are sufficiently large. These waves are inherently non-hydrostatic. This is often the case along the continental shelf where ISWs are frequently observed. Recent estimations of energy fluxes in large ISWs have used both linear (e.g., Chang et al., 2006) and nonlinear methods (e.g., Jeans and Sherwin, 2001, Moum et al., 2006, Klymak et al., 2006). The results reported here show that nonlinear methods must be used to accurately estimate energy fluxes in highly nonlinear solitary-like waves as was argued by Moum et al. (2006). Venayagamoorthy and Fringer (2005) reached the same conclusion for a very different situation than those considered here. They investigated the energy flux in nonlinear, non-hydrostatic numerical simulations of highly nonlinear shoaling internal waves in a linearly stratified fluid using a slightly different decomposition of the fluxes from that used in the present paper. In their simulations boluses formed at the bottom on the shelf after the waves had shoaled.

Several approximations are made in this paper. The flow is considered to be incompressible, inviscid and non-diffusive. The Boussinesq, rigid lid and traditional and f-plane approximations are also made. This greatly simplifies the energy budget as effects such as work done by compression, complications due to the nonlinear equation of state, air–sea interaction, and conversion of mechanical energy to internal energy are ignored. Wunsch and Ferrari (2004) discuss the role of these processes in the oceanic energy budget. This simplification allows us to isolate the dominant process, namely the conversion of barotropic tidal energy into baroclinic energy. The rigid lid approximation is made because the theoretical results are compared with numerical simulations done using a numerical model with a rigid lid. In the real ocean the deformation of the free surface by barotropic and baroclinic tides and other internal waves would need to taken into account (Moum et al., 2006).

The paper is organized as follows. In Section 2 the energy equations are discussed, both for total mechanical energy and for pseudoenergy. For the numerical simulations presented here the APE is defined relative to the undisturbed initial stratification. The linear problem is also examined and compared with the nonlinear case. This leads to a nonlinear equation for the linear energy ${\tilde{E}}_{\ell \mathit{in}}$ which shows that the linear energy is conserved only when the background stratification is uniform, in which case the linear energy is identical to the pseudoenergy. At the end of Section 2 two theoretical examples are discussed. The first is an exact nonlinear solution consisting of a single mode-n periodic wave. The second is a linear superposition of several modes, all with the same frequency. This example is relevant for the internal wave generation problem where the vast majority of the energy resides in the internal tide. In Section 3 the numerical model used for the simulations is described. Results of the numerical model are discussed in Section 4. The mechanical energy equation and the pseudoenergy equation are found to be satisfied to within less that 1% for the deep ocean case with uniform stratification. The APE computed by integrating the APE density is almost identical to that obtained by sorting the density field, and the flux of pseudoenergy is given by $〈u^{\prime }{p}^{\prime }〉$ to high accuracy. The results are quite different for strong tidal flow over bank edge with non-uniform stratification. In this case large amplitude internal solitary wave trains are generated. For these strongly nonlinear waves, fluxes of kinetic and APE make up about half the total energy flux and hence must be included in any estimates of the energy flux. For this more energetic case the mechanical energy and pseudoenergy budgets are satisfied to within 1% and 2%. The results are summarized in Section 5.