Downward Moving Peak

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If we use the above wave conditions and assume that they travel through a region of uniform

fluid such that N=μ=1 and using the fact that Holton and Lindzen showed that the mean acceleration

of the wind field that is only effected by eddy interactions and wave momentum flux can produce a

QBO we get the follow relation, equation (8):



With the above conditions as well as perturbing the mean wind field such that:



We can rewrite equation (8) such that, equation (9):



If we define a mean streamfunction such that, equation (10):



We can rewrite equation (9) as, equation (11):



If we assume now that there is no viscosity we can easily see that the streamfunction is, equation (12):



So as time goes on there is an amplitude change at all levels of the mean velocity field which also

decreases with height.


(Plumb, 1977)


This figure shows how the mean streamfunction, now called ϕ, changes in time. The vertical axis is still z but in

Plumb, 1977 they use scaled coordinates. The difference between (a) and (b) is simply that they set the

mean streamfunction to be equal to 0 at that level to show how the evolution of the mean wind field

doesn't depend on what goes on above it.


(Plumb, 1977)

Here we can see how the viscosity term, the 2nd term on the right hand side, of equation (11) affects the QBO.

As we can see from the top panel when we assume no viscosity the lower westerly flow never breaks apart and the easterly

flow can't move lower stopping the downward motion and not causing a cycle. The top panel also

indicates that no critical levels are formed which shows that the wave themselves can't form critical

levels if none existed before. The middle and lower panels show that even with a small amount of

viscosity the periodicity easily and Plumb 1977 claims that after an initial transient phase the

oscillations become regular and independent of the initial conditions. What we also see is that the

duration and descent time seems to be equal for both the westerly and easterly mean wind flows. We

know from observations that isn't the case so we can see what other properties might lead to this

discrepancy.



From this figure we can see that the top panel show a westerly propagating downward much quicker

than the easterly flow. The duration of the westerly flow is much longer at the surface than the easterly

flow. This comes about because the westerly flux's magnitude is double that of the easterly flux's

magnitude. This fact leads to easterly waves having a much more difficult time setting up an easterly

flow and have that flow descend.

But we know that the atmosphere isn't a very good Boussunesq fluid in terms of the mean density. In

the real atmosphere the mean density changes greatly. This changes equation (8) into, equation (13):



And assuming an isothermal atmosphere we can get equation (13) to be, equation (14):



Where H is the scale height of the atmosphere. According to the Holton-Lindzen model the only waves

that you need to create a QBO are equatorial trapped Kevin waves, which propagate westerly and

Rossby-gravity waves, which propagate easterly. With these two kinds of waves we need to figured out

their attenuation rates, which are given below, equation (15).



Where β is the equatorial beta parameter.

Using equations (14) and (15) we can get the follows plots.


(Plumb, 1977)

The only in both plots the thermal dissipation rate is a constant above 13 km and has a linear relation

below that level. The only difference between the two plots is that the top has an artificial semi-annual

oscillation.

Why is there only a QBO in the equatorial belt?

Main Page Attenuation Rate Equatorial Location