# Physical parameters

## Ices shelf enhanced friction

The increase of friction below permanent ice shelf (such as Weddell Sea ice shelf) has been reworked to allow easier generation of frictional members (ensemble data assimilation)
Implicit zero-flux boundary condition:

The zero-flux boundary condition (rigid boundaries) is easily implicitely enforced in the continuity equation. Its implementation in the momentum equation is a bit more tricky, and has been left aside for a while. In the latest T-UGOm version, it has been generalized to improve currents solutions in coastal areas.
 
## Linear friction (still underway)

In some shallow ocean regions, the presence of fine sediments and/or mud flow can dramatically change the physics of  bottom friction. In such locations (such as the Northern Amazon River estuary, see Yoann Lebars PhD report)), a linear parameterisation can be more adequate than the usual quadratic one. To comply with those peculiar regions, a new linear, regionally constrained parameterisation has been added to the T-UGOm spectral friction procedures.

> *Some additional adjustement are needed to avoid solution discontinuities due to the sudden change in friction (smoothed by horizontal diffusion in time-stepping simulation, but left abrupt in spectral simulation).*


## $S_2$ atmospheric surface pressure forcing

The radiational contribution (i.e. the ocean response to atmospheric pressure $S_2$ tide) represents about 15% of the total $S_2$ ocean tide (astronomical plus radiational). Separation of the astronomical and radiational component in a $S_2$ ocean tide observation is quite tricky, even if “smoothness” hypothesis can be used to deduce the $S_2$ astronomic tides from priorly known semi-diurnal constituents. As assimilated data are a measure of the total $S_2$ ocean tide, it was decided to add the $S_2$ (surface) pressure tide to the forcing of the prior $S_2$ ocean tide simulation, so the prior $S_2$ solution get more consistent with the assimilated data and closer to the (forthcoming) optimal one.
Equivalent Brünt-Vaissala frequency for the first vertical mode

![S2 atmospheric tide](../_static/S2-atm-2010.jpg)

*Figure:* $S_2$ atmospheric tide (in Pa), 2010 analysis

Presently, 10 years of ECMWF DCDA fields have harmonically analysed to extract a global surface pressure $S_2$ tide. (equations…) Preliminary tests have shown that std error is dived by a factor 2 when taking atmospheric pressure in to account in the $S_2$ tidal forcing.

## Latitude-dependent gravitational acceleration

Due to Earth rotation and geometry, the gravitational acceleration $g$ varies with latitude. In combination, the equatorial bulge and the effects of the earths inertia mean that sea-level gravitational acceleration increases from about 9.780 $m·s^{−2}$ at the Equator to about 9.832 $m·s^{−2}$ at the poles, so an object will weigh about 0.5% more at the poles than at the Equator.

This point was raised to us by Claire Perigaud some years ago, but was not really pushed forward as the tidal model errors were largely dominated by others terms. However, because of the present reduced error budget in FES2012/T-UGOm (about 9% of $M_2$ amplitude), it was thought that the implementation of the latitude-dependent gravitational constant was worth a trial.

Preliminary tests were successful, with a gain of 10% in $M_2$ simulations.


## Internal wave drag parameters

The internal tide drag parameters such as Brünt-Vaissala frequency and  characteristic wavenumber (tidal excursion) have been explored to get closer to local conditions. The Brünt-Vaissala frequency must be representative of the local dominant vertical mode (usually the first one), and various attempt have been done from climatology or model simulation outputs (solving the classical Sturm-Liouville problem for vertical eigenmode).

$$
\tau_1 = c_1 \rho_0 \frac{\kappa}{\omega} \left[ \left(N^2_b -\omega^2 \right) \left( \omega^2 - f^2 \right) \right]^{\frac{1}{2}} T^2 \mathbf{U}
$$

$$
\tau_2 = c_2 \rho_0 \frac{\kappa^{-1}}{\omega} \left[ \left(N^2 -\omega^2 \right) \left( \omega^2 - f^2 \right) \right]^{\frac{1}{2}} \left( \nabla H \cdot \mathbf{U} \right) \nabla H 
$$

where:

- $c_1$ a tunning coefficient,
- $\rho_0$ the depth-averaged ocean potential density,
- $\kappa$ the typical topography horizontal wavenumber,
- $T$ the typical topography vertical dimension,
- $\omega$ the tidal contituent frequency,
- $N_b$ the Brünt-Vaissala frequency at the ocean bottom,
- $N$ the depth-averaged Brünt-Vaissala frequency,
- $f$ the Coriolis parameter,
- $H$ the bathymetry,
- $\mathbf{U}=(u,v)$ the barotropic velocity

![Brunt-Vaissala frequency](../_static/N2-mode1.jpg)

*Figure:* Equivalent Brünt-Vaissala frequency for the first vertical mode (computed from WOA2005)

## Internal wave drag above critical latitude

In a linear case, internal tides can not propagate above critical latitudes (Gill, Ocean-Atmosphere Dynamics). Non-linearities might occasionally allow for partially breaking this rule, but numerical experiments tend to show that such exceptions are … quite exceptionals. Consequently, energy transfer between barotropic tides and baroclinic tides were assumed to be negligibles. However, reduced propagation means only reduced energy transfer, so the internal wave drag has been modified to keep a minimum energy transfer even above critical latitude.

## Generic spectral energy estimates and budget

Tidal energy examination is a powerful way to analyse and validate tidal solution. The historical LGP1xLGP1 routines have been reformed to be able to deal with all type of discretisations. Basic diagnostics are exactly computed (integrated) over the elements and then projected on LGP1 discretisation for display purposes.