If most of jet streams are westerly, that is to say, they blow from west to east, why does the tropical/equatorial jet stream blow from east to west?
Looking at the annual climatology of the 200 hPa winds as shown below from NCAR NCEP reanalysis it is noted that the mid latitudes have by and large westerly winds (for a planet that rotates west to east). On the other hand averaged over a latitude circle(zonal average) the low latitude tropics and equator have by and large easterly winds. The reference Annual cycle of equatorial east-west circulation over the Indian and Pacific Oceans argues that tropical upper level divergence over the Central Pacific is the reason for the upper level equatorial easterlies present over the Indian Ocean and West Pacific Ocean. On the other hand annual mean upper level equatorial winds over the East Pacific and east Atlantic are by and large westerlies(if one were to perform zonal regional averaging).
The upper level equatorial easterlies never quite approach the definition of a "jet" on annual scales except during the boreal summer period. So one needs to look at seasonal scales to explain the formation of the so called Tropical Easterly Jet Stream.
From these two pioneering papers Koteswaram TEJ and Raghavan K TEJ 1973 (available behind a paywall, reproduced here - Origin of Tropical Easterly Jet), there exists at the upper levels an anticyclone over the Indian subcontinent, also known as the Tibetan anticyclone. The Tropical Easterly Jet(TEJ) is believed to be connected with the Tibetan anticyclone. The Tibetan anticyclone is present in the upper troposphere (100 to 150 hPa).
In the above image derived from NCEP NCAR reanalysis data the June through September climatology of the 100 hPa geopotential height is presented and one can clearly see the Tibetan anticyclone as part of the broader monsoon anticyclone.
So the location of the Tibetan anticyclone is in accordance with seasonal shifts. In winter the Tibetan anticyclone is further south (around 10 N) and during summer it moves to warmer latitudes.
Lower levels of the anticyclone is tilted equatorward during the boreal summer and poleward during the winter. However the upper levels of the anticyclone are tilted poleward during all seasons.
Koteswaran and Raghavan have calculated the zonal component of the equatorward outflow from the anticyclone and there appears to be no comparison with the upper level easterlies associated with the Indian Summer Monsoon(ISM). But there appears to be strong comparison with the winds derived from thermal gradient over the Tibetan Plateau. So it appears that the winds are originating due to thermal reasons alone. The TEJ is to a large extent influenced by the temperature distribution in the upper levels as a result of vertical motion associated with ISM. The Tibetan plateau comes under the influence of both sensible and latent heat and leading to the formation of a surface heat low
Given the climatology of the observed surface pressure(this surface pressure has not been extrapolated to sea level pressure) during the June through September period this data indicates that the mean value is around 500 hPa it might be helpful to look at the temperature of the 500 hPa climatology.
Why should one want to look at the 500 hPa temperature surface ? it is because 500 hPa is the average surface pressure over the Tibetan Plateau(which is an elevated plateau). Normally the 500 hPa pressure surface over sea levels is found at a height of 5 kms. Once again one can clearly see from the 500 hPa temperature surface that the Tibetan plateau provides for an elevated heat source.
Krishnamurti in the book Monsoon Models offers a detailed explanation
the southerly outflow of the subtropical jet provides a mechanism where eddy kinetic energy is converted into kinetic energy of the mean flow and all this happens above 250 hPa. Stratospheric easterly winds penetrate downwards into the northern branch of the TEJ. Above all the conversion of available potential energy into kinetic energy above 250 hPa in the region of increasing pressure gradient caused by the proximity of the Tibetan anticyclone as well as the Coriolis force convert the diabatic heat into kinetic energy of the TEJ.
The above image provides a visual representation of the explanation provided in the quoted text i.e. southerly outflow of subtropical jet + Coriolis force.
I'm guessing here, but there are two things that seem to mechanically form the organized concentrated air rivers that are jet streams. I've despaired to find no good meterologist site or link that goes beyond grand vague hand waving gestures that mostly over-explain Coriolis Forces and irrelevant complexities. Here's how I think the actual organized streams are formed.
We need to be careful about postulating that different temperatures cause the actual streams. I don't think that's a useful perspective. Certainly, different temperatures are required to set up organized Polar, Ferrel and Hadley cells, but the heart of the explanation for the actual stream formation is looking at only momentum and drag.
If the surface wind of a Ferrel Cell (not the somewhat opposite flowing high altitude wind of the same Ferrel Cell) blows northward, it's earth-based radius decreases. It's becomes closer to the spin axis of the earth. That means that its relative velocity is always more eastward than that of the air it is joining. That's the Coriolis Force. Ok, so we get northeastern lower winds throughout the band of that Ferrel Cell, progressively becoming more eastwardly farther north. But why the sudden discontinuous increase in velocity at the northern edge of the Ferrel Cell that produces a jet stream? Well. Because all throughout the transit that those low level winds negotiated on their northward trek within that Ferrel Cell, they had to gently push more eastward for every inch north they moved against the drag of the atmosphere they encountered. Their acceleration to the east was limited by atmospheric drag.
But suddenly these northward winds are stopped from going further north when they encounter the Polar cell and are forced to deflect directly eastward. They are no longer dragged down by the slower rotating atmosphere north of them because they are moving only eastward. This means that the atmosphere that would normally drag down these northeastward tending winds is mostly made of their neighbor Ferrel Cell winds, that had been northeastwardly parallel to them for a while, that are similarly suddenly deflected due east by encountering the obstruction of the Polar Cell. Now, at the northern border of the Ferrel Cell, these winds all line up moving east as they begin to rise up towards the top of the Ferrel Cell and turn around back to the south to continue their cycle in the Ferrel circulation Cell, their eastward momentums now all lined up together fully aligned with each other along the lines of latitude (parallel to the equator) and pushing forth with their excess momentum in line concentrated in a much smaller area (than the circumferal distance of the Polar Cell and Ferrel Cell border). The eastward change in momentum has been concentrated at this border by having the Ferrel Cell winds line up trying to accelerate each other because they are forced to suddenly turn east. A visual picture of a similar idea is five people thrashing through a forest line abreast - like the low level northeasterly Ferrel Cell winds. Then, suddenly they turn 90º and have a lot less thrashing to do to get through the forest because of their alignment - like those Ferrel Cell winds when they turn to line up behind each other at the circulation cell boundary. The atmosphere these winds encounter that would normally limit their eastward acceleration is now made up of similar Ferrel Cell winds trying to accelerate eastward during the big turn back to the south. The result is a reduction in atmospheric drag in relatively short distance. Boom: Jet stream.
You may wonder why it would be that at the northern border of, say, the Ferrel Cell where there is no longer a northward component of velocity would still provide an eastward force to prople the jet stream since the eastward component of the Coriolis Force disappears when the northward velocity goes to zero. It's because of lag. The eastward pressures are not fully equalized instantly when the air stops moving northeastward and is deflected due east.
If all this is the case, then why does the subtropical jet also move east? Only air that is approaching a cell border can produce a jet; not air that is retreating from such a border because of the mechanism described above. So only the upper air of the Hadley cell can contribute to a jet stream at the border between the Ferrel and Hadley cells.
Finally, why doesn't the southern flowing air in the upper reaches of the Ferrel Cell create a westward moving jet? If you examine the profile view of the circulation cells, notice the altitude where the reversal from north/south occurs. In the both the Polar and Hadley cells, due to the odd sloped frontal border of the various cells, the reversals that should spawn westward blowing jet streams happen at low altitude where too many disturbances prevent organized flow. Remember even the low altitude portion of the Ferrel Cell can't form a jet stream until it has shifted up into the high region of the cell at the turn reversal. Furthermore, the Ferrel Cell runs backwards. That is, it is thermally indirect, so not driven by disparities in surface heating like the Polar and Hadley Cells, hence it is weaker, driven by its neighboring cells, which are thermally direct. Occaionally, a west flowing subtropical jet forms from these normally turbulent regions if conditions are just right, but not normally.
It should be remembered that on a planetary scale these jets are rather weak, broken, discontinuous, and easily disrupted. For example, if global warming reduces the sharpness of the temperature variation between the Polar Cell and the Ferrel Cell, that border becomes less defined and can undulate down into the central US in what the news calls a Polar Vortex.
Alas, I can't back any of this up because I've not seen it anywhere. This could all be wrong, but it seems straight forward enough to be plausible.
Are you asking about the energy from the sun causing the circulation cells to circulate? If so, that's the easy part. I sound like I know what I'm talking about, but I'm not really a reliable source. But here's my analysis:
Sun heats the earth at the equator at steep angles, thus providing a lot of heat to the air at the equator. This air expands and grows like a mountain over the equator. Since the air in the mountain has the same mass as before (same number of molecules), its weight on the ground would be the same. But it is not. The mountain of expanded air in the high atmosphere above the equator spills downhill northward because the top of the less expanded atmosphere to the north is lower. But as this downhill falling air moves north its time spent at cool altitudes and the less direct rays of the sun at higher latitudes cause it cool and become more dense. It becomes more dense than the air it falls over on the north edge of the Hadley Cell, so it sinks down. Why does it move south to complete the cycle? Because the continually rising column of air encircling the equator causes a constant low pressure condition on the ground as it rises up and spills off the mountain. If the air spills off the equatorial mountain of air, the mountain is lighter than it was before. That the mountain got high to start with would not change its weight or the pressure at the equator. The pressure at the equator is always low because the rising mountain of expanding air is forever spilling downhill to the north and causing fewer molecules and atoms of atmosphere to occupy that mountain that sits on the equator.
Note, the the Polar Cell functions just like the Hadley Cell. But you might notice that the Ferrel Cell runs backwards against the dynamics I just described. That's because both the Hadley Cell and the Polar Cells are "thermally direct," or run the right way. The Ferrel cell is not run by the dynamics I just described because it is "thermally indirect," powered and supported by the boundary effects of bumping up against the Hadley and Polar Cells. I don't have a good grasp on the thermally indirect cell being driven by its neighbor cells, but I have a feeling it has to do with atmospheric lifting at the north edge of the Ferrel Cell caused by the Polar cell ( and the converse at the border with the Hadley Cell), perhaps encouraging upper atmospheric southern downhill flow, but boy that's a reach.