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

This study analytically demonstrates that in the presence of shear flow, tidal energy conversion over topography involves both discrete regular eigenmodes and continuous singular solutions, with the latter forming evolving wave packets that can lead to breaking and necessitating an extended formula for net energy conversion.

The Gist

The Big Picture: The Ocean's Hidden Engine

Imagine the ocean as a giant, multi-layered cake. The top layer is warm and light; the bottom layers are cold and heavy. This layering is called stratification.

Now, imagine the moon and sun pulling on this cake, creating a massive, slow-moving tide that flows horizontally across the entire ocean floor. This is the barotropic tide. When this giant tide hits a bump on the ocean floor (a seamount or a ridge), it acts like a hand pushing up on a trampoline. It lifts the water layers, creating ripples that travel inside the ocean, not just on the surface. These are internal waves.

These waves are crucial. When they eventually crash and break (like surf on a beach, but underwater), they mix the water. This mixing brings nutrients up from the deep and heat down from the surface, driving the global climate system.

The Problem: The "Shear" Complication

For decades, scientists have tried to calculate exactly how much energy is transferred from the big tide to these internal waves. They used a method that worked great when the water was still or moving uniformly.

But in the real ocean, water doesn't just move; it shears. Imagine a deck of cards. If you push the top card, it moves fast. The bottom card stays still. The cards in between move at different speeds. This is shear flow.

When internal waves travel through this "sliding deck of cards," things get weird. The old math breaks down because the waves interact with the layers moving at different speeds in a way that creates "singularities"—mathematical points where the energy seems to blow up to infinity.

The Solution: Two Types of Waves

The authors of this paper (Onuki and Venaille) decided to untangle this mess. They realized that when a wave hits a shear flow, the energy doesn't just go into one type of wave. It splits into two distinct families:

1. The "Standing Choir" (Discrete Spectrum)

• The Analogy: Imagine a guitar string. When you pluck it, it vibrates at specific, distinct notes (frequencies). These are the Discrete Spectrum waves.
• What they do: These waves are like the "standing modes" of the ocean. They form stable, repeating patterns that travel away from the bump on the ocean floor. They are predictable and easy to count, just like the notes on a guitar.
• The Paper's Finding: These waves carry energy away in steady, organized trains.

2. The "Shifting Fog" (Continuous Spectrum)

• The Analogy: Now imagine a thick fog rolling over a hill. It doesn't have a single shape or a single "note." It's a continuous, shifting mass that changes as it moves. This is the Continuous Spectrum.
• What they do: These waves are associated with the "critical level"—the specific depth where the wave's speed matches the speed of the current.
• The Surprise: The paper discovered something counter-intuitive. As these "foggy" waves travel away from the bump:
  • Their height (amplitude) gets smaller (they fade out).
  • But their steepness (vertical gradient) gets sharper.
• The Metaphor: Think of a long, gentle wave that stretches out. As it stretches, it gets flatter, but if you look closely at the slope, it becomes incredibly steep, like a cliff. This steepening is dangerous because it leads to wave breaking (turbulence), even though the wave itself looks like it's dying out.

The "Magic Formula"

The main goal of the paper was to write a new "recipe" (formula) to calculate how much energy is converted from the big tide into these internal waves.

• Old Recipe: Only counted the "Standing Choir" (Discrete waves). It ignored the "Shifting Fog."
• New Recipe: Counts both.
  • It adds up the energy from the stable, guitar-string waves.
  • It also adds up the energy from the shifting, steepening fog waves.

The authors found that in many real-world ocean scenarios (especially with strong currents), the "Shifting Fog" (Continuous Spectrum) contributes significantly to the mixing. If you ignore it, you might be underestimating how much the ocean mixes.

Why Does This Matter?

1. Climate Models: To predict climate change, we need to know how much heat and carbon the ocean absorbs. This depends on mixing. If our models miss the "foggy" waves, they are missing a key piece of the puzzle.
2. Wave Breaking: The paper explains why waves break in sheared currents. It's not just because they get too tall; it's because the shear makes them get incredibly steep as they travel, eventually snapping like a dry twig.
3. Better Predictions: By separating these two types of waves, scientists can now build better computer models to simulate the ocean, leading to more accurate weather and climate forecasts.

Summary in One Sentence

This paper reveals that when ocean tides hit underwater mountains in a sheared current, they don't just create simple, repeating waves; they also create a second, invisible type of wave that gets steeper and steeper as it travels, and we must count both types to understand how the ocean mixes and drives our climate.

Technical Summary

1. Problem Statement

The generation of internal gravity waves by barotropic tides interacting with seafloor topography is a primary driver of oceanic mixing and a critical component for parameterizing wave drag in global circulation models.

• The Gap: Traditional analytical approaches rely on a modal expansion of the wave field, assuming a static or uniform background flow. This framework decomposes the wave field into a discrete set of standing eigenmodes. However, this approach becomes inadequate when a vertically sheared background flow is present.
• The Challenge: In the presence of shear, the governing Taylor–Goldstein equation admits not only regular eigenmodes (discrete spectrum) but also singular solutions associated with critical levels (where the wave phase speed matches the background flow speed). These singularities form a continuous spectrum. Previous studies largely ignored the continuous spectrum's contribution to energy conversion, focusing only on discrete modes.
• Objective: The study aims to analytically investigate wave generation over localized small topography in a sheared flow (neglecting Coriolis force for tractability) to:
  1. Disentangle the roles of regular eigenmodes and singular solutions.
  2. Derive a unified formula for the net barotropic-to-baroclinic energy conversion rate that accounts for both discrete and continuous spectral contributions.

2. Methodology

The authors employ a linearized, inviscid, two-dimensional Boussinesq model with a small-amplitude topographic assumption.

• Mathematical Framework:

  • Transforms: The problem is solved using a horizontal Fourier transform (spatial) and a temporal Laplace transform. This converts the initial-boundary value problem into an algebraic equation involving the resolvent of a linear wave operator, $\mathbf{M}_k$.
  • Spectral Analysis: The spectrum of the operator $\mathbf{M}_k$ is identified as the set of singularities of the resolvent. The authors rigorously demonstrate that this spectrum consists of two distinct parts:
    1. Discrete Spectrum ($\Sigma^d_k$): Isolated poles corresponding to regular eigenmodes (standing waves).
    2. Continuous Spectrum ($\Sigma^c_k$): A branch cut in the complex plane corresponding to singular solutions at critical levels ($kU(z) = \sigma$).
  • Asymptotic Evaluation: The inverse Fourier integral is evaluated asymptotically for large distances ($|x| \to \infty$) from the topography. The contour is deformed to separate contributions from poles (discrete) and branch cuts (continuous).

• Energetics:

  • The study utilizes wave-activity diagnostics (pseudomomentum and pseudoenergy) to formulate a consistent energy budget. This is necessary because, in shear flow, wave energy alone is not conserved due to exchange with the mean flow.
  • The net energy conversion rate is derived by analyzing the work done by the bottom topography and the interaction between the wave field and the mean flow.

3. Key Contributions and Results

A. Spectral Decomposition and Wave Structure

The study provides a clear physical interpretation of the two spectral components in the far field:

1. Discrete Spectrum (Regular Eigenmodes):
   • Corresponds to poles in the resolvent.
   • Generates standing wave trains that propagate away from the topography with constant amplitude (in the inviscid limit).
   • These are the traditional internal tides observed in non-sheared models.
2. Continuous Spectrum (Singular Solutions):
   • Corresponds to branch cuts in the resolvent.
   • Generates evolving wave packets rather than standing waves.
   • Asymptotic Behavior: While the velocity amplitudes of these packets decay algebraically with distance ($|w| \propto x^{-3/2}$, $|u| \propto x^{-1/2}$), their vertical velocity gradients grow ($|\partial_z u| \propto x^{1/2}$).
   • Physical Implication: This growth in vertical shear suggests that even if the wave amplitude decays, the wave packets are prone to wave breaking and turbulence generation far from the source, a mechanism missed by discrete-mode-only models.

B. Energy Conversion Rate Formula

The authors derive a comprehensive formula for the time-averaged barotropic-to-baroclinic energy conversion rate ($P$):
$$P = \sum_{n} \left[ \sum_{k_j \in K^d} (\omega_n - U k_j) \chi_j |\hat{\psi}|^2 + \int_{K^c} (\omega_n - U k) R(k) |\hat{\psi}|^2 dk \right]$$

• First Term (Discrete): Summation over discrete modes, weighted by the action production coefficient $\chi_j$ (related to group velocity).
• Second Term (Continuous): Integration over the continuous spectrum, weighted by the coefficient $R(k)$ (related to the imaginary part of the vertical structure function at the boundary).
• Key Finding: The continuous spectrum contribution is non-negligible. In specific configurations, the tidal and steady-flow contributions to the continuous spectrum can have opposite signs, leading to strong cancellations or enhancements that are invisible to standard modal analysis.

C. Critical Level Dynamics

• For localized topography, the critical level is not a single fixed altitude but varies with the horizontal wavenumber.
• Unlike the sinusoidal topography case where waves focus at a specific critical level, localized forcing creates a continuum of critical levels. This prevents wave accumulation at a single altitude but leads to a cumulative growth of vertical shear across the domain.

4. Significance and Implications

• Parameterization Improvement: The derived formula extends classical energy conversion estimates (e.g., Khatiwala 2003) by explicitly including the continuous spectrum. This is crucial for improving mixing and wave-drag parameterizations in ocean circulation models, particularly in regions with strong background shear (e.g., western boundary currents like the Kuroshio).
• Mechanism for Mixing: The finding that continuous-spectrum wave packets develop growing vertical gradients despite decaying amplitudes provides a theoretical basis for dissipation far from the topography. This helps explain observed energy dissipation downstream of seamounts in sheared currents.
• Theoretical Foundation: The work establishes a rigorous link between the resolvent formalism of wave-mean flow interaction and the physical energetics of tidal generation. It clarifies that the "missing" energy in shear flows is often hidden in the continuous spectrum.
• Limitations and Future Work: The study assumes 2D flow, no rotation (Coriolis), and small topography. The authors note that extending this to include rotation (which introduces inertial frequencies and potential instability) and finite-amplitude topography is a necessary next step.

Conclusion

Onuki and Venaille successfully demonstrate that the classical discrete-mode approach is insufficient for describing internal wave generation in sheared flows. By rigorously accounting for the continuous spectrum associated with critical levels, they reveal a dual nature of the wave field: persistent standing modes and decaying packets with growing shear. Their new energy conversion formula offers a more complete and physically accurate tool for modeling ocean mixing and wave drag in realistic, sheared ocean environments.