Chapter2 Flashcards
A linear set of equations does not contain
products of the dependent variables.
non-linear set of equation contains
products of dependent variables
he momentum equations are definitely
nonlinear, since the advective terms look like
A linear set of equations support
linear waves
nonlinear set of equations supports
nonlinear waves
Linear waves and nonlinear waves behave
very differently
Linear waves and nonlinear waves behave very differently! The primary difference is that
inear waves do not interact with one another, and can’t exchange energy!
Any interference between the two waves is strictly
linear, meaning at a given point, the effect of the wave is just the sum of the effects of the two waves.
Nonlinear waves interact and may
exchange energy!
Nonlinear waves are much more complex, and more difficult to study, than are linear waves. Unfortunately, the governing equations are highly
non-linear (due to the advective terms), and therefore, atmospheric waves are nonlinear.
The governing equations are nonlinear. In order to study the properties of atmospheric waves we
“linearize” the governing equations, and then study the linear waves supported by these equations.
By studying these linear waves we hope to learn some information about
the waves and their relevance.
In order to linearize the equations we use the
perturbation method.
In order to linearize the equations we use the perturbation method. We start by
dividing all the dependent variables into two parts. The first part is known as the basic state, and is assumed to be either constant, or only a function of the spatial coordinates.
The second part is the perturbation, and is allowed to vary with time, and in all three space directions.
Another assumption is that the basic state must
satisfy the equations of motion when the perturbations are zero.
A third critical assumption for the perturbation method is that the perturbations much be
small so that products of perturbations can be neglected.
Another assumption is that the basic state must satisfy the equations of motion when the perturbations are zero.
A third critical assumption for the perturbation method is that the perturbations much be small so that products of perturbations can be neglected. (Do not confuse this procedure with Reynolds averaging. Although the two procedures may look similar, they are really very different.)
We then take
the divided dependent variables, substitute them into the equations, and multiply everything out. Since we can ignore terms that are the products of two perturbations, any such term can be crossed out.
Note that what we’ve done is to use the basic state density everywhere except in the buoyancy term (the term involving g), where we used the
perturbation density
Note that what we’ve done is to use the basic state density everywhere except in the buoyancy term (the term involving g), where we used the perturbation density. This is essentially the
Boussinesq approximation
Note that what we’ve done is to use the basic state density everywhere except in the buoyancy term (the term involving g), where we used the perturbation density. This is essentially the Boussinesq approximation, the difference being that
he reference density is allowed to vary spatially, whereas in the Boussinesq approximation the reference density is assumed to be a true constant.
these waves are non-dispersive ie:
their phase speed is independednt of wavenumber. thus all waves of any wavenumber, propagate at the same speed (in wither direction), which means that non-sinusoidal distrurbances propagate without change od shape
any non sinusoidal disturbances can be described
a sum of components of different wavenumber, if all these waves propagate at the same speed, so will the disturbance its self, and its shape will not change
wave motions where c varies with k (different wave number components will have
surface waves on deep water, internal gravity waves, rossby waves
(different speeds, the phenomena is known as wave dispersion) - the way they interfere with one another will change with time so the shape of the disturbance will change.
