A wave in the T field on the ground will produce a geopotential anomaly on the ground that decays in magnitude with height.

Likewise, a wave in the T field on the tropopause will produce geopotential anomalies that decay as you move down through the atmosphere.

But notice: +T’ on the tropopause gives +’ because it is like –PV’.

If the horizontal scale is smaller, the rate at which the perturbations fall off with height is greater, so these are less able to influence what happens on the other boundary.

If the horizontal scale is smaller, the rate at which the perturbations fall off with height is greater, so these are less able to influence what happens on the other boundary.

But the longer waves will have an influence all the way to the other boundary. This ground wave is moving east with time (relative to background U).

Colors: geopotential anomalies; white lines are of the anomalies of vg that they create.Dashed lines are negative (wind out of the page); solid are positive (into the page).

And this one on the tropopause is traveling west relative to the background flow, which is blowing east.

Colors: geopotential anomalies; white lines are of the anomalies of vg that they create.Dashed lines are negative (wind out of the page); solid are positive (into the page).

U to the east +Wave to west

U ~ 0 + wave to east

U to the east +Wave to west

U ~ 0 + wave to east

This whole system can move in tandem to the east…staying in phase so that the disturbance on the bottom amplifies the one on top, and the one on top amplifies the one on the bottom.

Which is exactly what the most unstable wave does. They mutually amplify each other and have a structure through the atmosphere that looks like this:

The circulation around these L and H pressure systems is shifted in phase, as we’d expect.

And the relationship between the T field (anomalies in white—positive solid, negative dashed) and the geopotential is also as we’d expect. Note that the maximum temperature perturbation is at and to the east of the ground anomalies, and at and to the west of the tropopause ones.

Maximum ascent is to the east of low pressure systems, with maximum descent behind and leading the next ridge.

This shows what we found before: this model predicts that longwaves will be unstable, but short ones will not. Shortwaves on opposite boundaries propagate independently without affecting their counterpart on the other one.

In the real atmosphere, shorter waves have a chance to be unstable too…not all disturbances are truly confined to the tropopause and ground only. So in this case, the separation between the two disturbances, p, would be smaller, so higher values of k2+l2 could be unstable after all.

But there is clearly an optimal wavelength for growth, and in our atmosphere, this corresponds to systems that are about ~1000 km—the synoptic scale.