3 Introduction to Computational Fluid Dynamics and Ocean Modelling
83
scheme. This relationship is characterized by the Courant number
C CFL =
cct
x
.
(3.18)
For an explicit scheme for the hyperbolic equation
∂f/∂t + c∂f/∂x = 0
of the form:
f
(n+1)
m
= αf
(n)
m−1 + βf
(n)
m + γf
(n)
m+1 ; α, β, γ = const
with a constant ratio between time step and spatial resolution t//x, a necessary
condition for numerical stability is the Courant–Friedrichs–Lewy (CFL) condition
C CFL ≤ 1. The CFL condition guarantees that the FD method can propagate information (energy) at least as fast as the PDE.
3.3 Numerical Ocean Modelling
We now move away from the general discussion on numerical models and focus
on numerical ocean modelling within the framework of GCMs. It is essential that
a numerical model captures the prominent features of the physical problem, so we
will start the discussion by considering what these features are in the case of fluid
motion in ocean and large sea areas.
3.3.1 Physical Characteristics of Ocean Dynamics
The ocean is a dynamic system where water is in constant motion. Fluid motion
is driven by local pressure differences which are induced by several forcing mechanisms. The gravitational force acting on the Earth from the Moon and the Sun
creates ocean tides. Solar radiation heats the surface water which results in the water cycle of evaporation and precipitation. Local and regional temperature variations
in the atmosphere create horizontal pressure gradients, giving rise to wind which in
turn generates waves through shear stress. Rivers and glaciers provide sources for
fresh water input, and the formation of sea ice contributes to changes in salinity and
the reduction of wind effects. At the sea floor the bottom friction can be a significant
factor, as well as ground water seepage and sources of geothermal heat. In addition
to these physical processes, there are a number of chemical and biological factors
that may influence the properties of water masses.
The main function of ocean currents on the global scale is to transport warm
surface water from the equator to the poles and cold bottom water to the equator.
This global motion is called the thermohaline circulation and is often pictured as a
83
scheme. This relationship is characterized by the Courant number
C CFL =
cct
x
.
(3.18)
For an explicit scheme for the hyperbolic equation
∂f/∂t + c∂f/∂x = 0
of the form:
f
(n+1)
m
= αf
(n)
m−1 + βf
(n)
m + γf
(n)
m+1 ; α, β, γ = const
with a constant ratio between time step and spatial resolution t//x, a necessary
condition for numerical stability is the Courant–Friedrichs–Lewy (CFL) condition
C CFL ≤ 1. The CFL condition guarantees that the FD method can propagate information (energy) at least as fast as the PDE.
3.3 Numerical Ocean Modelling
We now move away from the general discussion on numerical models and focus
on numerical ocean modelling within the framework of GCMs. It is essential that
a numerical model captures the prominent features of the physical problem, so we
will start the discussion by considering what these features are in the case of fluid
motion in ocean and large sea areas.
3.3.1 Physical Characteristics of Ocean Dynamics
The ocean is a dynamic system where water is in constant motion. Fluid motion
is driven by local pressure differences which are induced by several forcing mechanisms. The gravitational force acting on the Earth from the Moon and the Sun
creates ocean tides. Solar radiation heats the surface water which results in the water cycle of evaporation and precipitation. Local and regional temperature variations
in the atmosphere create horizontal pressure gradients, giving rise to wind which in
turn generates waves through shear stress. Rivers and glaciers provide sources for
fresh water input, and the formation of sea ice contributes to changes in salinity and
the reduction of wind effects. At the sea floor the bottom friction can be a significant
factor, as well as ground water seepage and sources of geothermal heat. In addition
to these physical processes, there are a number of chemical and biological factors
that may influence the properties of water masses.
The main function of ocean currents on the global scale is to transport warm
surface water from the equator to the poles and cold bottom water to the equator.
This global motion is called the thermohaline circulation and is often pictured as a