### Abstract

We investigate theoretically and numerically the modulation of near-inertial waves by a larger-amplitude geostrophically balanced mean flow. Because the excited wave is initially trapped in the mixed layer, it projects onto a broad spectrum of vertical modes, each mode n being characterized by a Burger number, Bu_{n}, proportional to the square of the vertical scale of the mode. Using numerical simulations of the hydrostatic Boussinesq equations linearized about a prescribed balanced background flow, we show that the evolution of the wave field depends strongly on the spectrum of Bu_{n} relative to the Rossby number of the balanced flow, ∈, with smaller relative leading to smaller horizontal scales in the wave field, faster accumulation of wave amplitude in anticyclones and faster propagation of wave energy into the deep ocean. This varied behaviour of the wave may be understood by considering the dynamics in each mode separately; projecting the linearized hydrostatic Boussinesq equations onto modes yields a set of linear shallow water equations, with Bu_{n} playing the role of the reduced gravity. The wave modes fall into two asymptotic regimes, defined by the scalings Bu_{n} ∼ O(1) for low modes and Bu_{n} ∼ O(∈) for high modes. An amplitude equation derived for the former regime shows that vertical propagation is weak for low modes. The high-mode regime is the basis of the Young & Ben Jelloul (J. Mar. Res., vol. 55, 1997, pp. 735-766) theory. This theory is here extended to O(∈^{2}), from which amplitude equations for the subregimes Bu_{n} ∼ O(∈^{1/2}) and Bu_{n} ∼ O(∈^{2}) are derived. The accuracy of each approximation is demonstrated by comparing numerical solutions of the respective amplitude equation to simulations of the linearized shallow water equations in the same regime. We emphasize that since inertial wave energy and shear are distributed across vertical modes, their overall modulation is due to the collective behaviour of the wave field in each regime. A unified treatment of these regimes is a novel feature of this work.

Original language | English (US) |
---|---|

Pages (from-to) | 406-438 |

Number of pages | 33 |

Journal | Journal of Fluid Mechanics |

Volume | 817 |

DOIs | |

State | Published - Apr 25 2017 |

### Fingerprint

### Keywords

- quasi-geostrophic flows
- rotating flows
- waves in rotating fluids

### ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering

### Cite this

*Journal of Fluid Mechanics*,

*817*, 406-438. https://doi.org/10.1017/jfm.2017.124

**Near-inertial wave dispersion by geostrophic flows.** / Thomas, Jim; Smith, K. Shafer; Bühler, Oliver.

Research output: Contribution to journal › Article

*Journal of Fluid Mechanics*, vol. 817, pp. 406-438. https://doi.org/10.1017/jfm.2017.124

}

TY - JOUR

T1 - Near-inertial wave dispersion by geostrophic flows

AU - Thomas, Jim

AU - Smith, K. Shafer

AU - Bühler, Oliver

PY - 2017/4/25

Y1 - 2017/4/25

N2 - We investigate theoretically and numerically the modulation of near-inertial waves by a larger-amplitude geostrophically balanced mean flow. Because the excited wave is initially trapped in the mixed layer, it projects onto a broad spectrum of vertical modes, each mode n being characterized by a Burger number, Bun, proportional to the square of the vertical scale of the mode. Using numerical simulations of the hydrostatic Boussinesq equations linearized about a prescribed balanced background flow, we show that the evolution of the wave field depends strongly on the spectrum of Bun relative to the Rossby number of the balanced flow, ∈, with smaller relative leading to smaller horizontal scales in the wave field, faster accumulation of wave amplitude in anticyclones and faster propagation of wave energy into the deep ocean. This varied behaviour of the wave may be understood by considering the dynamics in each mode separately; projecting the linearized hydrostatic Boussinesq equations onto modes yields a set of linear shallow water equations, with Bun playing the role of the reduced gravity. The wave modes fall into two asymptotic regimes, defined by the scalings Bun ∼ O(1) for low modes and Bun ∼ O(∈) for high modes. An amplitude equation derived for the former regime shows that vertical propagation is weak for low modes. The high-mode regime is the basis of the Young & Ben Jelloul (J. Mar. Res., vol. 55, 1997, pp. 735-766) theory. This theory is here extended to O(∈2), from which amplitude equations for the subregimes Bun ∼ O(∈1/2) and Bun ∼ O(∈2) are derived. The accuracy of each approximation is demonstrated by comparing numerical solutions of the respective amplitude equation to simulations of the linearized shallow water equations in the same regime. We emphasize that since inertial wave energy and shear are distributed across vertical modes, their overall modulation is due to the collective behaviour of the wave field in each regime. A unified treatment of these regimes is a novel feature of this work.

AB - We investigate theoretically and numerically the modulation of near-inertial waves by a larger-amplitude geostrophically balanced mean flow. Because the excited wave is initially trapped in the mixed layer, it projects onto a broad spectrum of vertical modes, each mode n being characterized by a Burger number, Bun, proportional to the square of the vertical scale of the mode. Using numerical simulations of the hydrostatic Boussinesq equations linearized about a prescribed balanced background flow, we show that the evolution of the wave field depends strongly on the spectrum of Bun relative to the Rossby number of the balanced flow, ∈, with smaller relative leading to smaller horizontal scales in the wave field, faster accumulation of wave amplitude in anticyclones and faster propagation of wave energy into the deep ocean. This varied behaviour of the wave may be understood by considering the dynamics in each mode separately; projecting the linearized hydrostatic Boussinesq equations onto modes yields a set of linear shallow water equations, with Bun playing the role of the reduced gravity. The wave modes fall into two asymptotic regimes, defined by the scalings Bun ∼ O(1) for low modes and Bun ∼ O(∈) for high modes. An amplitude equation derived for the former regime shows that vertical propagation is weak for low modes. The high-mode regime is the basis of the Young & Ben Jelloul (J. Mar. Res., vol. 55, 1997, pp. 735-766) theory. This theory is here extended to O(∈2), from which amplitude equations for the subregimes Bun ∼ O(∈1/2) and Bun ∼ O(∈2) are derived. The accuracy of each approximation is demonstrated by comparing numerical solutions of the respective amplitude equation to simulations of the linearized shallow water equations in the same regime. We emphasize that since inertial wave energy and shear are distributed across vertical modes, their overall modulation is due to the collective behaviour of the wave field in each regime. A unified treatment of these regimes is a novel feature of this work.

KW - quasi-geostrophic flows

KW - rotating flows

KW - waves in rotating fluids

UR - http://www.scopus.com/inward/record.url?scp=85015926681&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85015926681&partnerID=8YFLogxK

U2 - 10.1017/jfm.2017.124

DO - 10.1017/jfm.2017.124

M3 - Article

VL - 817

SP - 406

EP - 438

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -