Chapter2

Question 1

A linear set of equations does not contain

Answer 1

products of the dependent variables.

Question 2

non-linear set of equation contains

Answer 2

products of dependent variables

Question 3

he momentum equations are definitely

Answer 3

nonlinear, since the advective terms look like

Question 4

A linear set of equations support

Answer 4

linear waves

Question 5

nonlinear set of equations supports

Answer 5

nonlinear waves

Question 6

Linear waves and nonlinear waves behave

Answer 6

very differently

Question 7

Linear waves and nonlinear waves behave very differently! The primary difference is that

Answer 7

inear waves do not interact with one another, and can’t exchange energy!

Question 8

Any interference between the two waves is strictly

Answer 8

linear, meaning at a given point, the effect of the wave is just the sum of the effects of the two waves.

Question 9

Nonlinear waves interact and may

Answer 9

exchange energy!

Question 10

Nonlinear waves are much more complex, and more difficult to study, than are linear waves. Unfortunately, the governing equations are highly

Answer 10

non-linear (due to the advective terms), and therefore, atmospheric waves are nonlinear.

Question 11

The governing equations are nonlinear. In order to study the properties of atmospheric waves we

Answer 11

“linearize” the governing equations, and then study the linear waves supported by these equations.

Question 12

By studying these linear waves we hope to learn some information about

Answer 12

the waves and their relevance.

Question 13

In order to linearize the equations we use the

Answer 13

perturbation method.

Question 14

In order to linearize the equations we use the perturbation method. We start by

Answer 14

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.

Question 15

Another assumption is that the basic state must

Answer 15

satisfy the equations of motion when the perturbations are zero.

Question 16

A third critical assumption for the perturbation method is that the perturbations much be

Answer 16

small so that products of perturbations can be neglected.

Question 17

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

Answer 17

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.

Question 18

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

Answer 18

perturbation density

Question 19

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

Answer 19

Boussinesq approximation

Question 20

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

Answer 20

he reference density is allowed to vary spatially, whereas in the Boussinesq approximation the reference density is assumed to be a true constant.

Question 21

these waves are non-dispersive ie:

Answer 21

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

Question 22

any non sinusoidal disturbances can be described

Answer 22

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

Question 23

wave motions where c varies with k (different wave number components will have

Answer 23

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.