2.b. Power laws in Insulator Metal Transition. Power law behavior has been reported near the insulator metal transition in several systems. The temperature coefficient of the electrical resistivity (TCR) as a function of a dopant concentration. ns, of these systems changes sign at a critical concentration, nc. When TCR is negative the resistivity decreases with increasing temperature as in an insulator. When TCR is positive the behavior is thought to be typical of metals. In 2D (two-dimensional) systems the resistivity is actually a sheet resistance expressed in units of h/e2, where h is the Planck’s constant and e is the electron charge. For such 2D sheet resistance the resistivity is expressed in terms of ohms/square or R/. The sheet resistivity, r, is not dependent on the area of the sheet. The resistivity of these 2D systems is found to collapse into a simple universal power law with the resistivity, r, being related to the temperature ,T by a power law with r = f[½(ns-nc)½/nc]Tb. The interesting aspect is that the power law exponent change sign at the critical value, nc, at which TCR changes sign which is conventionally regarded as an insulator-metal transition (IMT).
3.1. Some technical earthquake terms. One requires to familiarize one-self with technical terms used in earthquakes before one goes into the technical aspects of earthquakes. For instance, until now I thought that the only technical term used for the magnitude of earthquakes is the Richter scale. It turns out it is not so in modern times. There could be a considerable error.
The Richter magnitude scale, ML,
accurately reflects the amount of seismic energy released by an earthquake up
to about ML 6.5, but for increasingly larger earthquakes, the
Richter scale progressively underestimates. For instance an earthquake of
magnitude 7.5 in the old Richter scale would be about 9 in the modern scale.
The scale used more recently is the moment magnitude, MW, which is
related to the seismic moment, M0 of an earthquake. The seismic
moment is defined as M0 = DAm where D is the average displacement over the entire
fault surface, A is the area of the fault surface, and m is the average shear rigidity of
the faulted rocks. The value of D is estimated from observed surface
displacements or from displacements on the fault plane reconstructed from
instrumental or geodetic modeling. A is derived from the length multiplied by
the estimated depth of the ruptured fault plane, as revealed by surface
rupture, aftershock patterns, or geodetic data. The seismic moment is more
directly related to the amount of energy released. In the Kanamori relationship
first obtained for southern California, the moment magnitude, MW =
2/3logM0 – 10.7. The moment magnitude is linear with the Richter
scale for ML £ 5.0. Above this value the MW value is usually reported. MW
is used to describe great earthquakes because of absence of saturation effects
in this scale. Moment release is a measure of the seismic energy release.
The slopes and curvature in the hilly or mountain ranges due to plate collisions is called an orogenic belt. Very little details are available about such collisional orogeny especially around Nepal and the Uttarakhand which forms the region between the Central thrust and the front of the thrust belt.
The areas covered by water and resulting sediments changed with tectonic activity Cenozoic tectonic evolution consisted of four stages. The tectonic/sedimentary evolution history of the Tibetan Plateau and its surrounding mountains underwent repeated periods of uplift (Molnar and Chen, 1983). If the uplifts around the Himalayas are due to the impact of a giant meteorite, one could imagine that the earth to the north of what is now the Himalayas drained out of water/sediments to what it is now. The resultant dry desert left behind gave the conditions that finally resulted in monsoons in the sub continent, say, 10 to 7 million years ago or fifty million years after the uplift began.
Tremendous technological improvement in satellite positioning and communication that lets us use the Global Positioning System (GPS) in everyday life has been exploited to measure sub mm surface movements on Earth. The results of such studies are given in Fig 10, without going into the details of the measurement process. The image on the left of Fig 10 gives the elevation, while that on the right gives the scale of the surface movements. I would not know how to analyze further these diagrams. Simply from a visual examination, the topography of the movements does not seem to be inconsistent (double negatives are always suspicious) with the formation of giant impact basins. The rapid movement in the Himalayas would support the model of the Indian plate colliding with the Eurasian plate. On the other hand, I do not know if the rapid compressive movement in the Himalayas takes place simply to compensate for the expanding volume changes due to the slower tectonic movement elsewhere in the plateau.
Having written perhaps a little excessively and inconclusively (there are no definitive statements even from experts, anyway) on the formation of the Himalayas, I still cannot resist adding a little bit about the terrain maps of the Himalayas. As a first step I was looking for direction of plate uplifts in the terrain maps using www.maps.google.com. The idea was that if I look at the terrain of the highest peaks, I could find features marking the beginning of the uplift, borrowing literally from Aurobindo’s quotes:- Wherever thou seest a great end, be sure of a great beginning”. As per common geological perceptions high mountain ranges are found in suture zones where two continental plates have joined through collision. In the most naïve sense one could then expect that the folds in these mountain ranges would be in a direction perpendicular to the direction of the drift of the plate.
I could not find believable (even for me) evidence regarding such uplifts. The features in Fig 11 around some of the largest peaks in the Himalayas seem to indicate that the relief features are not initiated by the uniform uplift of a plate due to subduction of another plate below it. It is possible that after the plate was thrust up and stressed it cracked giving rise to typical crack features (such as that in dried clay, see Fig II, bottom right, inset). These cracks could form, for example, the basic topography of the valleys. The continuing uplift on a cracked terrain then forms the features due to peaks. This simple picture would then indicate that the tectonic uplift does not have uniform, unidirectional features expected from an uniform uplift of a plate due to subduction of another plate below it.
It is possible that after the plate was thrust up and stressed, it cracked giving rise to typical crack features (such as that in dried clay, see Fig 11, bottom right, inset). These cracks could form, for example, the basic topography of the valleys. The continuing uplift on a cracked terrain then forms the features due to peaks. The tectonic uplift would not then have uniform, unidirectional features.
the other hand, the features of the ridges formed seem to be one of bifurcating
flow. In the case of such bifurcating flow models, one could imagine that the
flow initiated with a build-up of stress or volume due to a result of various
(perhaps competing) uplifts. It could have been uplift at a point because of
some earlier earthquake event. One such example is shown in bottom right of Fig
11 (taken from Google Terrain features) immediately to the north east of Mount
Everest. The extent of bifurcation
increases, in this case, as the slope increases from west to east (left to
right). A painting “Bifurcation, 2003” by Michael Vandermeer using fluid
dynamics and chaotic bifurcation is shown in the left below while a simulated
river network is shown at the right.
It is perhaps possible that potholes that are created when they “…are teamed with the steep rise or descent of flyovers and connecting roads” as in Pune (bottom right) could also be fractals, which may throw some insights into the Himalayas if not the peculiar nature of Indian roads which always get affected by rains.
It turns out
that most large peaks of mountainous terrains show (Fig 11, bottom right) signs
of bifurcation similar to those found in propagating cracks, or lava flows or
4. d. The problem with river fans
Before one treats the Himalayas as sand piles one probably requires a justification for finding the huge volume of sandy soil (or clay) to fill up the huge fountains. Part of the answer could come by not looking at the top of the mountains but at the bottom of the sea.
It seems that one of the important but less discussed geological features due to Himalayan erosion is what they call the Bengal Fan, in the Bay of Bengal. The area of the corresponding Indus Fan in the Arabian Sea is nearly half that of the Bengal Fan. More soil seemed to have drained out to the east than west! The Bengal Fan is thought to be derived from the Ganges-Brahmaputra river system delta to well south of the Equator. If one looks at the relief features in Figs 9 amd 10, one is immediately struck by what seems to be deep mountain ridges around the present Arunachal Pradesh, Myanamar, Yunnan regions. Such ridges could have been caused by the flow of water from the Tibetan plateau --- where the large basin-like features now are --- through what are now the Mekong, Salween and Yangtze rivers into what is now the Bay of Bengal.
The Bengal Fan is roughly three million square kilometers in area, which is roughly the size of India itself. The average thickness of this fan is estimated to be between 15-20 kilometeres! It extends to south of the Equator. The total volume of the soil in the Bengal Fan is then ~ 45-60 milllion cubic kilometers!!! This is a humongous volume. This volume is much more than the volume marked out (Fig 9 right) by the slab in red of dimension 1500 x 600 x 5 km3 ~ 4,5 million cubic kilometers. This amount could be increased three times to nearly 15 cubic kilometers if one includes the whole of the Tibetan plateau.
There is still a missing term of about 30 cubic kilometers!!!
I guess there must be various ways to account for this volume --- if the estimate of Bengal Fan sediment is correct. One way to account for missing volume would be to assume that the Himalayas were much higher. Continuous erosion reduced this height and contributed to the volume of the Bengal Fan sediment, with additional contribution from, what is currently, the Indo-Gangetic plain.