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Near-Inertial Pollard Waves Modeling the Arctic Halocline

Christian Puntini

TL;DR

This work develops an explicit nonlinear solution for near-inertial Pollard internal waves in a three-layer Arctic halocline, using rotated spherical coordinates and an $f$-plane approximation aligned with the Transpolar Drift Current. The halocline is described by a Pollard-type Lagrangian flow with trochoidal particle paths, while a uniform mean current in the overlying layer couples to the halocline motion; two dynamic boundary conditions yield a simple dispersion relation where the wave speed satisfies $c^2 = \frac{f^2}{k^2}\left(1+\frac{f^2 c_0^2}{\mathfrak{g}^2}\right)$. Linearization fails to satisfy both dynamic boundaries, illustrating the essential role of nonlinearity in this two-boundary internal-wave problem. The analysis also characterizes mean-flow properties, Stokes drift, and cross-layer mass flux, providing a rigorous analytic baseline for Arctic halocline dynamics and potential climate-relevant feedbacks related to halocline depth and ocean-ice interactions.

Abstract

We present an explicit and exact solution to the governing equations describing the vertical structure of the Arctic Ocean region centred around the North Pole. The solution describes a stratified water column with three constant-density regions: a motionless bottom layer, a middle layer - the halocline - described by nonhydrostatic, near-inertial Pollard waves, and an upper layer presenting a mean current and a wave motion associated with the one in the halocline layer.

Near-Inertial Pollard Waves Modeling the Arctic Halocline

TL;DR

This work develops an explicit nonlinear solution for near-inertial Pollard internal waves in a three-layer Arctic halocline, using rotated spherical coordinates and an -plane approximation aligned with the Transpolar Drift Current. The halocline is described by a Pollard-type Lagrangian flow with trochoidal particle paths, while a uniform mean current in the overlying layer couples to the halocline motion; two dynamic boundary conditions yield a simple dispersion relation where the wave speed satisfies . Linearization fails to satisfy both dynamic boundaries, illustrating the essential role of nonlinearity in this two-boundary internal-wave problem. The analysis also characterizes mean-flow properties, Stokes drift, and cross-layer mass flux, providing a rigorous analytic baseline for Arctic halocline dynamics and potential climate-relevant feedbacks related to halocline depth and ocean-ice interactions.

Abstract

We present an explicit and exact solution to the governing equations describing the vertical structure of the Arctic Ocean region centred around the North Pole. The solution describes a stratified water column with three constant-density regions: a motionless bottom layer, a middle layer - the halocline - described by nonhydrostatic, near-inertial Pollard waves, and an upper layer presenting a mean current and a wave motion associated with the one in the halocline layer.

Paper Structure

This paper contains 22 sections, 169 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Governing (approximated) equations
  3. Classical and rotated spherical coordinates
  4. Equations of motion
  5. Standard approximations in geophysical fluid dynamics
  6. Tangent plane approximation
  7. Traditional approximation
  8. $f$-plane approximation in the rotated spherical coordinates framework
  9. North Pole and coordinates relative to the TDC
  10. Description of the flows
  11. The deep motionless layer beneath the halocline
  12. The halocline layer $\mathscr{H}(t)$
  13. The layer $\mathscr{M}(t)$ above the halocline
  14. The upper surface of the halocline $\eta_1$
  15. The lower surface of the halocline $\eta_2$
  16. ...and 7 more sections

Figures (5)

  • Figure 1.1: Map of the Arctic Ocean. Adapted from the IBCAO map https://www.ngdc.noaa.gov/mgg/bathymetry/arctic/currentmap.html. Credit IBCAO
  • Figure 1.2: A depiction of the model we are considering
  • Figure 2.1: Left: classical spherical coordinates. Right: rotated spherical coordinate. Images from CJ2023. CC BY 4.0 (http://creativecommons.org/licenses/by/4.0)
  • Figure 2.2: Depiction of the basis at the North Pole. In reality, the angle $\gamma$ is bigger than the one depicted here (close to $90^{\circ}$). However, for graphical clarity reasons, we did it smaller ($70^{\circ}).$
  • Figure 3.1: Schematic depiction of the flow pattern, at a fixed coordinate $y$