Rolling over under the Fleet River

Does anyone have an answer for this one yet? I cannot figure it out.

All very interesting, but I was talking about what happens west of St Pauls, between there and Chancery Lane

So what exactly do you mean by 'rolling over' - east of St. Pauls the tubes are on top of each other. West of St Pauls the tubes are separate and don't cross over each other.

However, the eastbound drops about 34' vertically Chancery Lane to St Pauls and the westbound climbs about 2' vertically on the west of St Pauls to Chancery Lane.

To clarify - in several places, e.g. on the Victoria Line in the Euston area and the Northern Line in the City, the tunnels are reversed so there is right hand running - to get in that position one line must pass over the other. (This is most obvious west of White City, where it happens in the open air). Between St Pauls and Chancery Lane something similar happens, but in a vertical plane instead of a horizontal one - at St Pauls the westbound is directly above the eastbound, but at Chancery Lane it is directly below.
But somewhere in between the two stations the tunnels must be at the same level. Where does this take place?
Does one tunnel stay level and the other descend from above it to below it? Or do both descend as they travel from one station to the other in their respective directions?
Given the topography of the area, and the shallowness of the tunnels at the stations, I guess the need to pass under the valley of the Fleet River between these two stations has something to do with it.

The vertical stacking of platforms I understand - the streets above are relatively narrow and it allows a single lift shaft on one side of the road (rather than in the middle) to serve both platforms. What I don't understand is why they swap places between the two stations.

Why the need to have tunnels crossing though?

I've consulted endless online sources and my London Railway Atlas (which doesn't even recognise the change in position of the roads), and still no luck. I know a couple of people who work on that group who might know - if so, I'll let you know.

OK, a dig through CLR gradient diagrams and some twiddling around with maps and ratios gives the rough point at which the tunnels are at the same level, and it's about the point that they pass under Thameslink. I suspect that this is coincidence: the tunnels are quite deep here, far deeper than if they just needed to pass beneath Thameslink, and also deeper than they need to be to go under the Fleet. Both lower platforms at Chancery Lane and St Paul's are just over 80 ft below the surface, and the upper platforms are about 15 ft above them.

First, a bit of history.

The CLR was determined to use the 'sawtooth' profile on their lines, whereby stations are entered on a rising gradient (mostly 1.66%, or 1 in 60), and departed on a falling gradient (mostly 3.3%, or 1 in 30). This saves wear and tear on the brakes, as the rising gradient helps slow the trains. It helps accelerate trains away from stations. It also reduces the depth of the platform, and thus speeds up the journey. This principle is followed at all of their stations with two exceptions:

• Notting Hill Gate (EB) has no falling gradient, and in fact trains depart on an ascending gradient of 1.1%, or 1 in 90.
  
• Holland Park (WB) has no retarding gradient to the east, with trains approaching on a falling gradient of 1.1%
  

These two stations are configured as such because of the rise of the land forming Notting Hill, and the need to prevent Notting Hill Gate station from being too deep.

Anyhow, back to the original question. Street level at St Paul's is somewhat lower than at Chancery Lane. Having superimposed the gradient profiles, and then redrawn them to show the gradients if the lines didn't 'roll over' (i.e., one direction uses both upper platforms, and the other uses both lower platforms), the best explanation I can offer is this:

Had the two lines used the platforms in the same configuration at both stations, then it would have been very difficult to get the right gradients to the west of both Bank and St Paul's stations. Depending on which tunnel was used for which direction, this would have prevented the accelerating boost away from both these stations westbound, or removed the retardation on EB trains entering. By switching over the tunnels between the stations it allows all of the platforms at Chancery Lane, St Paul's, and Bank to have the preferred profile.

As it is, EB trains have a long descent between Chancery Lane and St Paul's, whilst WB trains remain almost on the level. They then had a relatively easy climb at 0.88% (1 in 114) to British Museum station.