68
4 Long Waves in a Channel
The truncation error becomes reasonably small if we resolve the wavelength by
more than 10 grid points (Fig. 4.2). In other words, when using finit differences,
only waves with a wavelength greater than tenfold the grid spacing are resolved
accurately. Similar conclusion can be drawn for time step requirements to resolve a
given wave period.
4.2 Long Surface Gravity Waves
4.2.1 Extraction of Individual Processes
The Navier–Stokes equations describe a great variety of processes that can occur
simultaneously in fluid on different time scales and lengthscales. Nevertheless,
under certain assumptions, we can extract individual processes from these equations
to study them in isolation from other processes. For instance, waves of a period short
compared with the inertial period are unaffected by the Coriolis force and we can
ignore the Coriolis force for such waves, which simplifie the governing equations.
By making certain assumptions, we will progressively learn more about a variety of
physical processes existing in fluids
4.2.2 Shallow-Water Processes
From scaling considerations, it can be shown that the hydrostatic relation holds for
processes of horizontal lengthscale exceeding by far the vertical lengthscale. These
processes are referred to as shallow-water processes, even if they occur in the atmosphere or in deep portions of the ocean. It is the ratio between horizontal and vertical
lengthscales that matters here!
4.2.3 The Shallow-Water Model
We consider a flui layer of uniform density with a freely moving surface to
study long surface waves of a wavelength long compared with the flui depth
(Fig. 4.3). We assume wave periods short compared with the inertial period, so
that the Coriolis force can be neglected, and we simply ignore frictional effects
to first-orde approximation. We neglect the nonlinear terms (advection of momentum), which implies that the phase speed of waves exceeds by far the speed of water
parcels. As another simplification we consider waves that propagate exclusively
along a channel aligned with the x-direction and being void of variations in the
y-direction.
4 Long Waves in a Channel
The truncation error becomes reasonably small if we resolve the wavelength by
more than 10 grid points (Fig. 4.2). In other words, when using finit differences,
only waves with a wavelength greater than tenfold the grid spacing are resolved
accurately. Similar conclusion can be drawn for time step requirements to resolve a
given wave period.
4.2 Long Surface Gravity Waves
4.2.1 Extraction of Individual Processes
The Navier–Stokes equations describe a great variety of processes that can occur
simultaneously in fluid on different time scales and lengthscales. Nevertheless,
under certain assumptions, we can extract individual processes from these equations
to study them in isolation from other processes. For instance, waves of a period short
compared with the inertial period are unaffected by the Coriolis force and we can
ignore the Coriolis force for such waves, which simplifie the governing equations.
By making certain assumptions, we will progressively learn more about a variety of
physical processes existing in fluids
4.2.2 Shallow-Water Processes
From scaling considerations, it can be shown that the hydrostatic relation holds for
processes of horizontal lengthscale exceeding by far the vertical lengthscale. These
processes are referred to as shallow-water processes, even if they occur in the atmosphere or in deep portions of the ocean. It is the ratio between horizontal and vertical
lengthscales that matters here!
4.2.3 The Shallow-Water Model
We consider a flui layer of uniform density with a freely moving surface to
study long surface waves of a wavelength long compared with the flui depth
(Fig. 4.3). We assume wave periods short compared with the inertial period, so
that the Coriolis force can be neglected, and we simply ignore frictional effects
to first-orde approximation. We neglect the nonlinear terms (advection of momentum), which implies that the phase speed of waves exceeds by far the speed of water
parcels. As another simplification we consider waves that propagate exclusively
along a channel aligned with the x-direction and being void of variations in the
y-direction.