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Geometric Mechanics of the Vertical Slice Model

Darryl D. Holm, Ruiao Hu, Oliver D. Street

Abstract

The goals of the present work are to: (i) investigate the dynamics of oceanic frontogenesis by taking advantage of the geometric mechanics underlying the class of Vertical Slice Models (VSMs) of ocean dynamics; and (ii) illustrate the versatility and utility of deterministic and stochastic variational approaches by deriving several variants of wave-current interaction models which describe the effects of internal waves propagating within a vertical planar slice embedded in a 3D region of constant horizontal gradient of buoyancy in the direction transverse to the vertical plane.

Geometric Mechanics of the Vertical Slice Model

Abstract

The goals of the present work are to: (i) investigate the dynamics of oceanic frontogenesis by taking advantage of the geometric mechanics underlying the class of Vertical Slice Models (VSMs) of ocean dynamics; and (ii) illustrate the versatility and utility of deterministic and stochastic variational approaches by deriving several variants of wave-current interaction models which describe the effects of internal waves propagating within a vertical planar slice embedded in a 3D region of constant horizontal gradient of buoyancy in the direction transverse to the vertical plane.
Paper Structure (27 sections, 6 theorems, 136 equations, 3 figures)

This paper contains 27 sections, 6 theorems, 136 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation of the paper:
  3. Main goals of the paper:
  4. Plan of the paper:
  5. Vertical slice models (VSMs)
  6. Lagrangian formulation of VSMs
  7. Clebsch-Lin derivation of VSMs
  8. Hamiltonian formulation of VSMs
  9. Interpretations of the Poisson matrix \ref{['Eqn: Tangled-Deterministic']}.
  10. 'Untangled' Hamiltonian form.
  11. Computational Simulations of VSMs
  12. Wave mean flow interaction (WMFI)
  13. Clebsch-Lin formulation of the EB VSM with wave dynamics
  14. Hamiltonian formulation of the EB VSM with wave dynamics.
  15. The geometry of the generalised Lagrangian mean (GLM) approach to wave-current interaction
  16. ...and 12 more sections

Key Result

Theorem 2.1

Recalling the preceding dynamical variables for the VSMs, namely the slice vector field $u_S \in \mathfrak{X}(M)$, the transverse velocity scalar $u_T \in \mathcal{F}(M)$, the in slice volume form $D\,d^2x \in \operatorname{Den}(M)$ and the potential temperature scalar $\vartheta_s \in \mathcal{F}(M subject to the following constraints derived from their definitions where $(v_S,v_T) := (\delta \p

Figures (3)

  • Figure 1: The vertical slice domain
  • Figure 2: Snapshots of the transverse velocity field $u_T$ (left) and the buoyancy field $b'_s$ (right) of the numerical simulation of the VSM \ref{['eqn:EBE buoyancy']} at days $4,7,9$ and $14$. The solutions are periodic on the lateral boundaries and their in-slice velocity ${\mathbf{u}}_S$ has no normal component on the top and bottom boundaries; so the buoyancy field is advected horizontally at the top and bottom of the domain.
  • Figure 3: Snapshots of the horizontal component of in slice velocity field ${\mathbf{u}}_S\cdot \widehat{{\mathbf{x}}}$ (left) and the pressure field $p$ (right) of the numerical simulation of the VSM \ref{['eqn:EBE buoyancy']} at day $14$.

Theorems & Definitions (23)

  • Theorem 2.1: Affine Euler-Poincaré theorem for VSMs
  • Remark 2.1
  • proof
  • Remark 2.2
  • Corollary 2.1: The Euler-Boussinesq Eady model
  • Remark 2.3: In what sense is the VSM for the Euler-Boussinesq Eady equations three dimensional?
  • Remark 2.4
  • Theorem 2.2: Kelvin-Noether theorem for the Euler-Boussinesq VSM
  • Remark 2.5: PV conservation for the VSM
  • Theorem 3.1: Kelvin-Noether theorem for the Euler-Boussinesq VSM
  • ...and 13 more