ε in the ocean general circulation model

This protocol is extracted from research article:

Estimation of Basin-scale turbulence distribution in the North Pacific Ocean using CTD-attached thermistor measurements

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Sci Rep**,
Jan 13, 2021;
DOI:
10.1038/s41598-020-80029-2

Estimation of Basin-scale turbulence distribution in the North Pacific Ocean using CTD-attached thermistor measurements

Procedure

The turbulent energy dissipation data used in the OGCM, which were referred to as TideNF^{2}, were compared with the observational data obtained in this study. The model turbulent energy dissipation originally consisted of two types of horizontally 2-D (depth-integrated) data. First, is the energy conversion rate from barotropic to baroclinic (internal) tides (*Ec*(*x*,*y*)), representing the local generation of internal tides, and second, is the local energy dissipation of the internal waves, (*Ed*(*x*,*y*)). Both datasets were obtained from a 3-D high-resolution (1/15°) model forced by tides^{3} and used after being multiplied by 1.5, given that the global baroclinic conversion rate at the limit of zero grid spacing was approximately 1.5 times larger than the grid spacing of 1/15°^{2}. A 3-D distribution of energy dissipation rates *ε*_{TideNF} was constructed using three components: (1) the near-field component, *ε*_{NEAR}, which represents the local dissipation close to the generation site of the tide-generated internal waves over the rough topography, (2) the far-field component, *ε*_{FAR,} which represents the dissipation of propagating internal waves away from the generation site, and (3) the background component, *ε*_{BACK}, which represents the dissipation other than (1) and (2):

The dissipations of the three components were expressed as:

where *ρ* represents the sea water density (kg m^{−3}), *N* represents the buoyancy frequency (s^{−1}), *K*_{BACK} represents the background diapycnal diffusivity, and Γ represents the mixing efficiency. *K*_{BACK} = 10^{–5} m^{2} s^{−1} and Γ = 0.2 in the TideNF model^{2}. In this study, *N* was computed using the observed density data corresponding to each CTD cast.

*F*(*z*) represents the vertical structures of the near- and far-field components as in a previous study^{2}:

where *z* is the depth (m), *H* is the bottom depth (m), and *h* is the decay scale from the bottom (m) of the near-field dissipation. This exponential decay from the bottom topography of the near-field dissipation was based on a previous modelling^{25} and observational study^{24}. The far-field component was not based on the observations, and for simplicity, it was assumed to be vertically uniform^{2}.

The horizontally variables, *E*_{NEAR} and *E*_{FAR} represent the depth-integrated energy dissipation in the near- and far-field, respectively. They were derived as follows:

where *Ec* represents the energy conversion rate from the barotropic tide to the baroclinic internal tide, *Ed* represents the depth-integrated energy dissipation in each water column (the sum of *E*_{NEAR} and *E*_{FAR}), and *q* represents the ratio of local dissipation to the generated baroclinic energy (it was set to the constant value, *q* = 0.33)^{2,24}. *Ec* and *Ed* were calculated numerically^{3} using 3-D Navier–Stokes equations under hydrostatic and Boussinesq approximations as follows:

where *g* represents acceleration due to gravity, ${\rho}^{\prime}$ represents the deviation of sea water density from the basic field associated with the baroclinic tide motions, *w*_{s} represents the vertical velocity resulting from the interaction between the barotropic tidal flow and the bottom topography, and the overbar denotes the time average. *u*′, *v*′, and *p*′ represent the eastward and northward velocities and the pressure perturbations associated with baroclinic tidal motions, respectively.

In this study, the distribution of the energy dissipation rate was examined given that diapycnal diffusivity depends on buoyancy frequency, mixing efficiency, and energy dissipation rate, and these three factors can be different in the model and in the observations. Examples of the spatial distribution of *ε*_{TideNF} using the observed buoyancy frequency field are shown in Fig. 5. *ε*_{NEAR} was large close to the rough bottom topography, which is characterised by a large baroclinic energy. Farther from the bottom, *ε*_{TideNF} was dominated by *ε*_{FAR} and *ε*_{BACK}. Additionally, *ε*_{FAR} was assumed to be vertically uniform, and it depends only on the horizontally variable dissipation of remotely generated internal waves. *ε*_{BACK} was large in the upper ocean because it is proportional to *N*^{2}. Accordingly, *ε*_{BACK} accounted for more than 20% of all the dissipation rates in the upper 1000-m level.

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