4
1 Introduction
0 = −
∂ P
∂z
− ρg
(1.5)
The hydrostatic approximation forms the basis of the so-called shallow-water
model and it is employed in so-called hydrostatic level models, being frequently
applied by fluid modellers. In contrast, models developed in this volume are based
on the full equations (Eq. 1.1) to enable the predictions of both hydrostatic and nonhydrostatic processes, a method called nonhydrostatic modelling. Nonhydrostatic
processes include those in which horizontal and vertical scales are of similar order
of magnitude.
1.1.5 The Stability Frequency
The degree of density stratification in the ocean can be characterised by the so-called
Brunt-V¨ ais¨ al¨ a frequency N , defined by:
N
2
= −
g
ρ o
∂ρ
∂z
(1.6)
where ρ o is mean density. This frequency, which appears as a characteristic scale
for many stratified processes, is referred to as stability frequency in the following.
1.2 Numerical Methods
1.2.1 Finite Differences
Models developed in this book are based on finite-difference versions of the NavierStokes equations. The basis is that dynamic variables (velocity components, sea
level, density, and dynamic pressure) are calculated at certain discrete grid points in
space. This requires an accurate representation of both gradients (the first derivative)
and curvature (the second derivative) of the spatial distribution of a variable. These
derivatives can be approximated from Taylor series. The first spatial derivative of a
variable f , for instance, with respect to a space coordinate x can be approximated
in three different ways:
• ∂ f /∂ x ≈ ( f k+1 − f k )/Δx (forward difference)
• ∂ f /∂ x ≈ ( f k − f k−1 )/Δx (backward difference)
• ∂ f /∂ x ≈ ( f k+1 − f k−1 )/(2Δx) (centred difference)
where Δx is the spacing between adjacent grid points, called grid spacing, and the
index k points to a certain grid cell along the x-axis. A finite-difference representation of the second spatial derivative of a function f is:
1 Introduction
0 = −
∂ P
∂z
− ρg
(1.5)
The hydrostatic approximation forms the basis of the so-called shallow-water
model and it is employed in so-called hydrostatic level models, being frequently
applied by fluid modellers. In contrast, models developed in this volume are based
on the full equations (Eq. 1.1) to enable the predictions of both hydrostatic and nonhydrostatic processes, a method called nonhydrostatic modelling. Nonhydrostatic
processes include those in which horizontal and vertical scales are of similar order
of magnitude.
1.1.5 The Stability Frequency
The degree of density stratification in the ocean can be characterised by the so-called
Brunt-V¨ ais¨ al¨ a frequency N , defined by:
N
2
= −
g
ρ o
∂ρ
∂z
(1.6)
where ρ o is mean density. This frequency, which appears as a characteristic scale
for many stratified processes, is referred to as stability frequency in the following.
1.2 Numerical Methods
1.2.1 Finite Differences
Models developed in this book are based on finite-difference versions of the NavierStokes equations. The basis is that dynamic variables (velocity components, sea
level, density, and dynamic pressure) are calculated at certain discrete grid points in
space. This requires an accurate representation of both gradients (the first derivative)
and curvature (the second derivative) of the spatial distribution of a variable. These
derivatives can be approximated from Taylor series. The first spatial derivative of a
variable f , for instance, with respect to a space coordinate x can be approximated
in three different ways:
• ∂ f /∂ x ≈ ( f k+1 − f k )/Δx (forward difference)
• ∂ f /∂ x ≈ ( f k − f k−1 )/Δx (backward difference)
• ∂ f /∂ x ≈ ( f k+1 − f k−1 )/(2Δx) (centred difference)
where Δx is the spacing between adjacent grid points, called grid spacing, and the
index k points to a certain grid cell along the x-axis. A finite-difference representation of the second spatial derivative of a function f is: