NEMO-OPA

In the following we mainly refer to NEMO manual (Madec and the NEMO team,2016).

2.1.1.1 Governing equations

NEMO-OPA is a primitive equation, free-surface⁵, finite differences, 3-D ocean model, built for modelling ocean circulation at regional and global scales. It

of the analysis values from the observations y, weighted by the accuracy of the observations R(the observations error covariance matrix), plus the sum of the squared deviations of the forecast fields x^bfrom the analyses, weighted by the accuracy of the forecast B (the forecast error covariance matrix). The term three dimensional is referred to the particular form of the cost function used: J(x) = ¹₂(x − x^b)^TB⁻¹(x − x^b) +¹₂(y − h(x))^TR⁻¹(y − h(x)), where h is the relation between the analysis fields and the observations. The resulting analysis is given by x^a= min

x J(x).

5A variable η is introduced to describe the sea-surface height with respect to a reference height z = 0. This variable is solution of a prognostic equation: ^∂η_∂t = −∇ · [(H + η)UBT] +

P − E, where H is the positive ocean depth with respect to z = 0, UBT is the vertically

averaged horizontal velocity vector (see section2.2.2.1), E and P are the volume fluxes of water through the surface due to evaporation and precipitation.

In GOFS16 (and also in SURF, see section2.2) a linear free-surface formulation is adopted:

a linearised version of the previous equation is used, the vertical levels thickness is constant in time (also the one of the first level at surface) and the vertical boundary conditions (see section2.1.1.3) are applied at the fixed surface z = 0 rather than on the moving surface z = η.

34 Chapter 2. Large scale and shelf scale numerical models

solves the Navier-Stokes primitive equations under the hydrostatic⁶and Boussi-nesq⁷ approximations, along with a turbulence closure⁸ and a nonlinear equa-tion of state which couples temperature and salinity with momentum and mass conservation. The fluid is assumed to be incompressible.

The coordinate reference system chosen is a curvilinear spherical one with an orthogonal set of unit vectors (i,j,k) linked to the earth such that k is the local upward vector and (i,j) are two vectors orthogonal to k (tangent to the geopotential surfaces, which are assumed to be spherical).

The vector invariant form of the primitive equations⁹ in the (i,j,k) vector sys-tem is given by the following six equations (the momentum balance, the hy-drostatic equilibrium, the incompressibility equation, the heat and salt conser-vation equations and an equation of state):

∂U_h

• U = (u, v, w) stands for the three-dimensional velocity vector;

• U_h is the horizontal velocity vector;

• t and z are the time and vertical coordinates;

6The vertical momentum equation (2.2) is given by a balance between vertical pressure gradient and buoyancy forcing.

7Density is assumed to be constant in all the Navier-Stokes equations terms, except for the buoyancy term (right hand side of equation2.2).

8Turbulence moments for which explicit equations are not used are written as a function of moments of lower order, so as to equal the number of unknowns and equations after, for example, a Reynold’s decomposition.

9Invariant under coordinate transformations so that they can be applied uniformly in any orthogonal curvilinear coordinate system such as spherical coordinates, avoiding the explicit representation of new metric terms, thanks to the identity (U·∇)U = (∇×U)×U+¹₂∇(U²) (mitgcm.org).

2.1. Large scale modelling: GOFS16 35

• θ, S and p are potential temperature, salinity and pressure;

• f = 2Ω · k is the Coriolis parameter (Ω the Earth angular velocity);

• g the gravitational acceleration;

• ρ₀ a reference density;

• ρ the in situ density given by the modified UNESCO equation of state formula byJackett and Mcdougall (1995);

• A^lm, A^vm are the horizontal and vertical eddy viscosity coefficients;

• A^{lT ,S}, A^{vT ,S} the horizontal and vertical eddy diffusivity coefficients;

• F^U, F^T, F^S the surface forcing terms.

The second and third terms from the end in equations 2.1, 2.4, 2.5 are the parametrisations of sub-grid scale physics (e.g. turbulence) for momentum, temperature and salinity: in GOFS16 a horizontal biharmonic operator is used to represent the lateral subgrid-scale mixing for momentum, while lateral tracers mixing and vertical tracers and momentum mixing are parametrised through the Laplace operator. The same for SURF (see section 2.2), except for the use of the Laplace operator also for lateral momentum mixing.

The horizontal eddy viscosity and diffusivity coefficients (A^lm, A^{lT ,S}) are cho-sen to be constant in space and time, while, instead, the vertical eddy viscosity and diffusivity coefficients (A^vm, A^{vT ,S}), comprising all the vertical sub-grid scale physics, can be specified as a function of the local fluid properties or with a turbulence closure model. In GOFS16, A^vm and A^{vT ,S} are computed through the TKE turbulence closure scheme, based on a prognostic equation for the turbulent kinetic energy and a closure assumption for the turbulent length scales. SURF (see section2.2), instead, uses the Pacanowski-Philander mixing parametrization (Pacanowski et al.,1981), in which A^vm and A^{vT ,S} are a function of the local Richardson number R_i = N²/(∂_zU_h)² (being N the Brunt-Väisälä frequency).

2.1.1.2 Spatial discretization

The NEMO governing equations are spatially discretised in finite differences on a staggered Arakawa C-type grid (Arakawa et al,1977). Scalar quantities (sea level height, density, pressure, horizontal divergence, temperature and salinity) are located at the centre of grid cells (T-grid); while velocities (u, v and w) are shifted by half a grid to the centre of the grid faces in three different directions:

west/east for zonal velocity (U-grid), south/north for meridional velocity (V-grid), up/down for vertical one (W-grid) (see fig. 2.1).

36 Chapter 2. Large scale and shelf scale numerical models

Figure 2.1: Arrangement of variables using Arakawa C-type grid. T indicates scalar points where scalar quantities are defined; u, v and w indicate vector points where the three com-ponents of velocity are defined; f indicates vorticity points where both relative and planetary vorticities are defined. Reproduced fromMadec and the NEMO team(2016).

2.1.1.3 Boundary conditions

Surface boundary conditions At the surface, the vertical velocity, mo-mentum, salinity and heat fluxes are prescribed by

w|z=η= Dη

• η is the sea surface height with respect to a reference height z = 0. Ac-tually, in GOFS16 and SURF (see section2.2), since a linear free surface formulation is adopted (see footnote in section2.1.1.1), the conditions in equations2.7,2.8,2.9and2.10are applied at z = 0 rather than at z = η;

• τ is the wind stress;

• E and P are the volume fluxes of water through the surface due to evap-oration and precipitation;

• ^Dη_Dt is a total derivative, i.e. ^Dη_Dt = ^∂η_∂t + U_h|_z=η· ∇_hη

• c_p is the ocean specific heat;

• Q_ns is the non-penetrative part of the net surface heat flux Q (positive when received by the ocean). Q can be divided into four terms:

2.1. Large scale modelling: GOFS16 37

Q = Qsr+ Q_lw+ Q_s+ Q_e

| {z }

Qns

Q_sr is the short wave heat flux, Q_lw is the long wave portion of the net radiation received at the sea surface, Q_s the sensible heat flux and Q_e the latent heat flux.

Bottom boundary conditions At the bottom, the normal components of velocity, heat and salt fluxes are null and the friction is modelled by a quadratic function. These conditions are expressed, respectively, by

w|_z=−H= −U_h|_z=−H· ∇_h(H) (2.11)

• H is the (positive) ocean depth with respect to a reference height z = 0;

• C_b is the bottom drag coefficient;

• e_b a bottom turbulent kinetic energy due to tides, internal waves breaking and other short time scale currents.

Lateral closed boundary conditions Along the coastline (∂Ω), the normal components of heat and salt fluxes are null:

A^{lT ,S}∂(θ, S)

A no-slip condition is applied for SURF (see section 2.2), a free-slip condition for GOFS16:

(U · ˆn)|∂Ω= 0 (2.15)

for both SURF and GOFS16;

(U · ˆt)|_∂Ω= 0 (2.16)

for SURF only, where ˆn is the unit vector in the normal to the coast direction and ˆt is the unit vector in the tangential to the coast direction.

38 Chapter 2. Large scale and shelf scale numerical models