Abstract

The energy transfers associated with internal tide (IT) generation by a semi-diurnal surface tidal wave impinging on a supercritical meridionally uniform deep ocean ridge on the f-plane, and subsequent IT-propagation are analysed using the Boussinesq, free-surface, terrain-following ocean model Symphonie. The energy diagnostics are explicitly based on the numerical formulation of the governing equations, permitting a globally conservative, high-precision analysis of all physical and numerical/artificial energy transfers in a sub-domain with open lateral boundaries. The net primary energy balances are quantified using a moving average of length two tidal periods in a simplified control simulation using a single time-step, minimal diffusion, and a no-slip sea floor. This provides the basis for analysis of enhanced vertical and horizontal diffusion and a free-slip bottom boundary condition. After a four tidal period spin-up, the tidally averaged (net) primary energy balance in the generation region, extending ±20 km from the ridge crest, shows that the surface tidal wave loses approximately C = 720 W/m or 0.3% of the mean surface tidal energy flux (2.506 × 10 5 W/m) in traversing the ridge. This corresponds mainly to the barotropic-to-baroclinic energy conversion due to stratified flow interaction with sloping topography. Combined with a normalised net advective flux of baroclinic potential energy of 0.9 × C this causes a net local baroclinic potential energy gain of 0.72 × C and a conversion into baroclinic kinetic energy through the baroclinic buoyancy term of 1.18 × C. Tidally averaged, about 1.14 × C is radiated into the abyssal ocean through the total baroclinic flux of internal pressure associated with the IT- and background density field. This total baroclinic pressure flux is therefore not only determined by the classic linear surface-to-internal tide conversion, but also by the net advection of baroclinic (background) potential energy, indicating the importance of local processes other than linear IT-motion. In the propagation region (PR), integrated over the areas between 20 and 40 km from the ridge crest, the barotropic and baroclinic tide are decoupled. The net incoming total baroclinic pressure flux is balanced by local potential energy gain and outward baroclinic flux of potential energy associated with the total baroclinic density. The primary net energy balances are robust to changes in the vertical diffusion coefficient, whereas relatively weak horizontal diffusion significantly reduces the outward IT energy flux. Diapycnal mixing due to vertical diffusion causes an available potential energy loss of about 1% of the total domain-averaged potential energy gain, which matches k m - 1 k m ρ 0 K V N 2 to within 0.5%, for k m linearly distributed grid-levels and constant background density ρ 0, vertical diffusivity ( K V ) and buoyancy frequency ( N).

Highlights

  • The world’s oceans dissipate about 3.6 TW of the tidal energy in the earth–moon–sun system, of which 2.54 TW is associated with the semi-diurnal (M2) surface tide (Cartwright and Ray, 1991; Egbert and Ray, 2001)

  • Where the barotropic potential energy density purely associated with surface motion is EP 1⁄4 q0gz and the baroclinic part is given by eEP 1⁄4 gzq~, which includes the background stratification and density anomalies associated with internal wave motion

  • We analysed the energetics of internal tide generation by stratified barotropic tidal flow impinging on a supercritical Gaussian ridge and its subsequent propagation, using the energy-conserving, Boussinesq, hydrostatic, free-surface ocean model Symphonie

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Summary

Introduction

The world’s oceans dissipate about 3.6 TW of the tidal energy in the earth–moon–sun system, of which 2.54 TW is associated with the semi-diurnal (M2) surface tide (Cartwright and Ray, 1991; Egbert and Ray, 2001). Carter et al (2008) used POM to estimate the M2 internal tide energetics around the Hawaiian ridge, employing the sum of kinetic and linearised available potential energy, which is appropriate for linear stratification and smallamplitude internal waves, discretised on the horizontal and vertical mid-point of the model grid-cells They concluded that local dissipation is a non-negligible factor in the IT generation zone, they observed an error in both the barotropic and baroclinic global energy balance on the order of 10% of the primary energy conversions.

Numerical simulation of the internal tide
Model setup
Parameter space
Velocity decomposition
Internal tide field
Energy evolution equations of a water column
Energetics of internal tide generation and propagation
Exchange between barotropic and baroclinic wave energy
D F hF X 1 i hF
Generation and propagation region
Diapycnal mixing
Global energy balance
Instantaneous and local energy transfers in GR and PR
Primary energy transfers in the generation region
Vertical and horizontal diffusion and bottom friction
Diffusive energy transfers
5.75 Â 10À3
Discussion and conclusion
Basic energy balance of internal tide generation
Basic energy balance of internal tide propagation
Numerical evaluation of energy transfers
Model equations
Boundary conditions
Decomposition of the pressure work rate
Energy equations in Oxyz-coordinates
Local transfers and boundary fluxes of energy
F E0K FX1
Findings
Coriolis effect

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