For "j" from the surface up to the ith level, the value h*(i) - h(j) is summed up, represented by the crosshatched areas on the graph. Curiously to me, the area in the lower right is actually negative area because of the direction of subtraction. This means that where h(j) is in excess of the h* at the level of interest, where a parcel rising undiluted from j to i would have positive buoyancy, is being counted as negative. Then the positive area in the upper left is found. The total area is summed and the ratio between the [h(sfc) - h*(i)] buoyancy (marked with a sharp squiggly line) and that total area is the entrainment rate (also multiplying by a factor representing the distance between i and the surface).
This is like considering a level and looking down, saying to yourself, "Well, if a parcel came up here from the surface it would have this much buoyancy if it didn't entrain any air at all." But say it entrained some h from each level j on its way up here. If the net sum is small, the ratio of the max buoyancy to the net area is large and we're saying the environment is so favorable to convection that our parcel could entrain such an amount and still make it to this level just as it loses buoyancy. If, however, the net sum of the areas is getting smaller and smaller as higher and higher up level i's are being considered, the entrainment rate is getting larger and larger, until the net sum of the areas crosses zero and becomes just as small but negative, then the entrainment rate is large and negative, which doesn't make sense.
Why must the calculation be done this way? Steve has already told me there are other ways... hmm... more to come.
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