Disentangling discrete and continuous spectra of tidally forced internal waves in shear flow

Abstract

Generation of internal waves driven by barotropic tides over seafloor topography is a central issue in developing mixing and wave drag parameterizations for ocean circulation models. Traditional analytical approaches estimate the energy conversion rate from barotropic tides to internal waves using a modal expansion of the wave field. However, this framework becomes inadequate if a background shear flow is present, as singular solutions associated with critical levels emerge. To uncover the distinct roles of regular eigenmodes and singular solutions in tidal energy conversion, this study analytically investigates wave generation over a localized small topography in the presence of shear flow without Coriolis force. Applying horizontal Fourier and temporal Laplace transforms, we identify regions in the topographic wavenumber and forcing frequency space where unbounded energy growth occurs. These regions coincide with the spectrum of an operator governing free wave propagation and consist of discrete and continuous parts, which correspond to regular eigenmodes and singular solutions, respectively. Asymptotic evaluation of the Fourier integral reveals that the far-field response comprises standing wave trains linked to the discrete spectrum and evolving wave packets associated with the continuous spectrum. While the velocity amplitudes of the wave packets decay, their vertical velocity gradients grow during propagation, potentially leading to wave breaking. Finally, we derive a formula for the net barotropic-to-baroclinic energy conversion rate, extending the classical one by incorporating the contributions from both the discrete and continuous spectra.

Figures (9)

• Figure 1: The situation considered in this study. Internal gravity waves are generated over a spatially localized small bottom topography forced by an oscillatory barotropic flow in the presence of a sheared background current.
• Figure 2: Integration contour of the inverse Laplace transform for a positive $k$. We shift the contour originally defined in the upper half of the complex plane in the negative direction along the imaginary axis. There exist an infinite number of poles, which involve the eigenvalues of $\mathsfbi{M}_k$, i.e., the discrete spectrum $\varSigma^d_k$, and the forcing frequency $\omega_n$. An element of the discrete spectrum, $\sigma \in \varSigma^d_k$, is either in a range $\sigma < k U(0)$ or $\sigma > k U(1)$, while the forcing frequency $\omega_n$ may be located anywhere on the real axis. There also exist three branch points on the real axis. Two branch points $kU(0)$ and $kU(1)$ originate from the factors $\gamma^\pm(0)$ and $\gamma^\pm(1)$ and the intermediate point $kU(z)$ from $\gamma^\pm(z)$, which are involved in the solutions of the Taylor--Goldstein equation, \ref{['eq:A01']}. Since the location of the intermediate point depends on $z$ and thus ranges from $k U(0)$ to $k U(1)$, the continuous spectrum, $\varSigma^c_k$, is composed of $\sigma \in \left[ k U(0), kU(1) \right]$.
• Figure 3: (a) The spectrum of an operator $\mathsfbi{M}_k$ for a constant shear and stratification case, $N = 1$ and $U = 0.15 z$. Blue and red curves correspond to $D(k, \sigma) = 0$ and represent the dispersion relations of vertically standing eigenmodes. The magenta region represents the continuous spectrum. (b-d) Spatial structures of stationary solutions for each horizontal wavenumber are demonstrated by drawing the real part of $A_1(z; k, \sigma) e^{{\rm i} k x}$.
• Figure 4: Integration contour of the Fourier integral \ref{['eq:Fourier_transform']} for positive $x$. The contour originally defined on the real axis is shifted over a finite distance on the complex plane along the imaginary axis. For a wavenumber $k = k_r + {\rm i} k_i$, the exponential factor in the integrand becomes $e^{{\rm i} k x} = e^{{\rm i} k_r x - k_i x}$. Therefore, for a large $x$, contributions from the line segments in the upper half of the complex plane become exponentially small, and the integration is dominated by those around the poles and branch points, which approach the real axis in the inviscid limit $r \to 0$. For negative $x$, the integration contour is shifted in the opposite direction. Note that the branch point around which the solution \ref{['eq:algebraic_solutions']} is evaluated is $k' = (\omega_n + \textrm{i} r) / U(z)$. Currently, we are considering the case when $U>0$ for any $z \in [0, 1]$. For $U < 0$, the corresponding branch point exists in the lower half-plane.
• Figure 5: The spectrum of an operator $\mathsfbi{M}_k$ for a constant shear and stratification case, $N = 1$ and $U = 0.05 + 0.15 z$. On the dispersion curves of discrete modes, there exist points where group velocity $c^g_j$ vanishes.