5.2 Numerical Treatment
127
Vertical integration of the continuity equation gives a prognostic equation for
surface dynamic pressure, yielding:
∂q s
∂t
= −ρ o g
∂(h u)
∂ x
+
∂(h v)
∂ y
(5.5)
where q s = ρ o gη with ρ o being surface density and η being sea-surface elevation,
h is total local water depth, and . represents a vertical average.
5.1.4 Evolution of the Density Field
Seawater density depends on temperature, salinity and pressure. For simplicity, we
assume a linear dependence of seawater on temperature and salinity, ignore any
pressure effects, and assume that eddy diffusivities for heat and salt are the same. To
this end, the evolution of the density field can be described by a density-conservation
equation, given by:
∂ρ
∂t
+ Adv(ρ) = Diff(ρ)
(5.6)
where the diffusion operator is given by:
Diff(ρ) =
∂
∂ x
K h
∂ρ
∂ x
+
∂
∂ y
K h
∂ρ
∂ y
+
∂
∂z
K z
∂ρ
∂z
where K h and K z , respectively, are horizontal and vertical eddy diffusivities.
5.2 Numerical Treatment
5.2.1 The 3d Arakawa C-grid
The index triplet (i, j, k) is used as a pointer to certain grid cells of the Arakawa
C-grid (Fig. 5.1), where Δx is the grid spacing in the x-direction, Δy is the grid
spacing in the y-direction, and Δz is the grid spacing in the vertical direction. Note
that the i index runs opposite to the z-coordinate. For convenience, we locate the
Cartesian coordinate system such that the x-axis points to the east, the y-axis to
the north, and the z-axis upward. Grid points of scalars (pressure, density, Eulerian
concentration, eddy viscosity, etc.) are centred between velocity grid points. Within
each grid cell, the u-grid point is located to the east, the v-grid point to the north,
and the w-grid point above with respect to the scalar grid point.
127
Vertical integration of the continuity equation gives a prognostic equation for
surface dynamic pressure, yielding:
∂q s
∂t
= −ρ o g
∂(h u)
∂ x
+
∂(h v)
∂ y
(5.5)
where q s = ρ o gη with ρ o being surface density and η being sea-surface elevation,
h is total local water depth, and . represents a vertical average.
5.1.4 Evolution of the Density Field
Seawater density depends on temperature, salinity and pressure. For simplicity, we
assume a linear dependence of seawater on temperature and salinity, ignore any
pressure effects, and assume that eddy diffusivities for heat and salt are the same. To
this end, the evolution of the density field can be described by a density-conservation
equation, given by:
∂ρ
∂t
+ Adv(ρ) = Diff(ρ)
(5.6)
where the diffusion operator is given by:
Diff(ρ) =
∂
∂ x
K h
∂ρ
∂ x
+
∂
∂ y
K h
∂ρ
∂ y
+
∂
∂z
K z
∂ρ
∂z
where K h and K z , respectively, are horizontal and vertical eddy diffusivities.
5.2 Numerical Treatment
5.2.1 The 3d Arakawa C-grid
The index triplet (i, j, k) is used as a pointer to certain grid cells of the Arakawa
C-grid (Fig. 5.1), where Δx is the grid spacing in the x-direction, Δy is the grid
spacing in the y-direction, and Δz is the grid spacing in the vertical direction. Note
that the i index runs opposite to the z-coordinate. For convenience, we locate the
Cartesian coordinate system such that the x-axis points to the east, the y-axis to
the north, and the z-axis upward. Grid points of scalars (pressure, density, Eulerian
concentration, eddy viscosity, etc.) are centred between velocity grid points. Within
each grid cell, the u-grid point is located to the east, the v-grid point to the north,
and the w-grid point above with respect to the scalar grid point.