Himalayan Instabilities: What One Wanted to Ask but was Afraid to Know.

“We use the word Himalayan Blunder yet the blunder has been in Himalayas. Ironically the salvage from humans is as uphill as the possibility (of salvage) looks downhill.” Ananth Shankar

There is a tide in the affairs of men using computers that leads them to become indifferent to hazards that do not appear on their screen over extended periods of time. One of these is the hazards of the Himalayas. The Uttarakhand devastation is no longer in our collective periscope.

Given that computers are used by every Ram, Amitabh and Murugan, this indifference is a threat that is perceived by a few. Corrective steps will have to be taken by these few.

One cartoon on the internet that I like is from a 2009 blog by one Arnaud Georgin which I have modified slightly without insulting apes. It just suggests that we are in a stage in which we require a more difficult evolution away from the computer.

This second evolution will not be of the present kind such as evolving out of the desktop to a mobile or a future mobile to implanted chip kind which motors one’s thoughts and actions for a globally bankrupting usurer’s vision.

There has been much written emotionally and screeched dramatically in the multimedia for people who would like to form instant opinions based on opinions of instantly recognizable faces such as those of film stars and cricket commentators. These “experts” would probably be the first to admit their ignorance when asked. They are not asked because the TRP --- the only measure of goodness now ---would go down. As far as the Himalayas is concerned, a serious (without TRP concerns) debate --- without wringing hands --- on it does not seem to have begun.

If one has to do something different, one is obliged to form an informed opinion to begin with. This blog is an attempt to do so with some seriousness with the proviso that serious errors may occur once one is in a position to recognize such errors. It is desirable for a remaining few that there should be quantification. In my “incorrigible” way, I have tried to put in my own inputs and models.

In order to urge an interested reader further, I cite something which is well known.  Scientific debate often rings immediately boring in many minds, with assumptions suggesting that the topic at hand is simply an academic one, and nothing important or impacting. Such is not the case for the debate over whether plate tectonics apply on the continental scale as well as on the fault scale, as more knowledge in these areas will help to broaden our understanding as well as our preparedness of earthquakes, a topical issue (Joshua Hill, Plate Tectonics apply to Continents too).

It does take considerable time to know the Himalayas. I have probably spent a considerable fraction of what is remaining in my life for this blog. I wish I had done this much earlier. I have taken some effort to do as much research as I think is relevant. It seems that geology is not a subject that has been studied as much as it should have been.

Since I will not probably write a blog on this topic again It will be nice if it is useful to one interested.

1. Himalayas and the Sandpile model

I have written something about why the Himalayas should be treated as a pile of sand in section 4c.

I had listened to Per Bak at the Centre of Theoretical Physics in Trieste just after the time he had developed his model for self-organized criticality in sandpiles. It seemed such a simple thing to do, although his forceful, if not aggressive, style made me wonder if one requires missionary zeal when one is convinced about the grand nature of one’s work. Per Bak (1948 – 2002) realized that as he piled sand on sand the slope of the sandpile becomes steeper until the slope reaches a particular angle. After this angle is reached any further piling of sand will cause the sandpile to increase its height but it will reorganize such that the critical slope is retained. This happens when the sand is dry and finely granular; the sand particles spontaneously roll of one another because of gravity until the sand pile forms a stable shape. This very simple common sense model was developed quantitatively by Per Bak. He went on to find a philosophy, describing the sandpile as a self-organizing critical state. This concept of self-organized criticality,is applicable to systems which spontaneously dissipate energy and naturally flow to a critical state. The physics of such a state may be described in all scales without tuning any single parameter. It is applicable to many natural phenomena including volcanic activity, forest fires, earthquakes, river-bank failure,

In these self-organized states there is an emergence of spontaneous fluctuations (which can be described as avalanches in the case of sandpiles) without an input from outside. Thus when a system show self-organized critical behavior, there will be a series of fluctuations  in some property of the system even though there  is a continuous input. In sandpiles, for example, there are a series of sand avalanches as sand grains are steadily added. The frequency of these avalanches depends on the size and there is a power law in the frequency-size distribution.  Because of this scaling with size, self-organized critical states have a resemblance to self-similar or fractal objects. The value of the exponent in these power laws are used to detect the nature of hierarchies in environmental studies.

Early studies on applying the sandpile model to the Himalayas focused on the frequency of landslides for various topology and rock slopes. Similar power law results have been found despite different rock formation, steepness of slopes and landslide triggering mechanisms. It is of course recognized that the Himalayan rock volume is not uniform like the sieved dry sand in laboratory studies of sandpiles. For this reason power law studies, although roughly agreeing with the sandpile models, have not been followed very rigorously. Moreover, it is difficult to follow a frequency-size behavior when the landslides are not anticipated.

There is, however, another route. In the case of sandpiles, the geometry of the pile is described by a cone which has an angle of ~35^o as shown in Fig 1, left, when the sand is dry.

The slope of such sandpiles is expected to have an angle close to 35^o. One finds that large amount of sand accompanies landslides and that the shape of some of the hills without vegetation looked like piles of sand (Fig 1, middle, right, straight lines are a guide to the eye for 35^o slope). Before erosion or when sand content is small the slopes become steeper than 35^o. The nature of erosion would depend on the rocks formed.

2. Power laws and Stretched Exponentials.

2.a. Power laws and Earthquakes. A power law relationship between two quantities gives a linear log-log relationship between the two quantities preferably over several orders of magnitudes. The higher the range of the linear relationship, the more valid is the power law. The slope of the log-log plot is a characteristic property of the relationship between the two quantities. Quite simply put, the power law relationship between two properties y and x will appear as y = cx^a, where c is a constant and a is the characteristic exponent. If we chose another variable x’ = gx where g is also a constant then y’ = cg’^ax^a º c’x^a. In this case, the slope of the logy vs logx plot will be the same as logy’ vs logx’ plot. The only difference is that there is a scaling of the power law relationship between the two variables by the constants c or c’. This will appear as a change in the intercept of the log-log plot. a is then the scaling exponent.^.

Because of this common framework, self-organizing systems are characterized by self-similarity and fractal geometries. Power laws express this self-similarity of the large and the part so that there would be more of small structures than the larger ones. The power law indicates complex systems which have different correlations between different levels of scale so that there is a hierarchy in each level of scale. Such systems cannot be described by the usual normal or exponential distributions. They are also amenable to descriptions as self-similar object or dynamics or sequences when the structure of the whole is the same as its part. Ideally, the system would be infinitely large. In most cases it is only approximately so. A typical and well-known example is said to be the  coastline which is an approximate natural fractal. If N is the number of self-similar segments of length R that a fractal object is divided into then the power law gives N  = R^-D where D is the fractal dimension.

When the coastline is obtained from geometrically self-similar features of the surrounding land when filled by water, one could expect that the landscape would also be fractal. It is well known that folds of mountains, drainage patterns, clouds, trees, leaves, bacteria cultures, roots, lungs, rivers and so on look similar on many scales.

It is fortunate that they do because we can understand these complex systems by one common approach.

Systems which share the same scaling exponent belong to the same universality class. Because of this scaling, the power law exponent will not depend on the units in which the two variables are expressed. For instance, in money making schemes of interest to most materialists, scaling laws are seen in macroeconomic rich-get-richer properties such as “the distribution of income, wealth, size of cities and firms, and the distribution of financial variables such as returns and trading volume.

In the context of this blog, we will be interested with the frequency with which an event occurs in relation to its size. The frequency of an event is a measure of the probability of the occurrence of the event and the power law is a measure of the probability distribution. In particular we will be concerned with the power law relationship (Fig 2, left, circles, from Bak et al 2002 for earthquakes in a California region) between the temporal frequency, N, of earthquakes and the magnitude, m, of the earthquake given in a logarithmic scale in the Gutenberg-Richter law log₁₀N"(M  > m) µ -bm, b ~ 0.95. The leveling off at low m values has been attributed to uncertainty in measurements.

There are dynamical processes involved in such power laws. There are other dynamical systems which may be measured repeatedly in the laboratory. Among the many available, I have chosen here --- somewhat unconventionally if not unwisely --- the way the probability of electrical transmission, the electrical conductance, of a two-dimensional sheet changes with some external field. Ideally one would have liked to see the changes in electrical conductance with time. A proper mapping in the context of electrical conductivity is to map the magnitude of noise versus frequency, f, with the technical terms being shot noise and 1/f noise. It is found that in two-dimensional electron gas (2DEG) the shot noise of the conductance does not exceed e²/h, a value which depends only on universal constants, and which is a quantum limit of conductance.  

In what follows, I examine the scaling exponent of the electrical conductance as a function of temperature in a semiconductor. For such semiconductors a barrier to electrical conduction has to be overcome, so that the electrical conductance at a particular temperature is a measure of the barrier that is overcome at that temperature.

The relationship between energy gaps in semiconductors and magnitude of earthquakes could be related by the geological term asperities that measure the roughness of a surface. Earthquakes can begin by the release of an asperity that is stuck on a fault, energy being released in the process. When asperities are not uniform the release of asperities could lead to a series of earthquakes. In the context of earthquakes an asperity is usually the contact point where two rock surfaces are in contact. In this sense an asperity may be taken as a barrier to fault release or tectonic motion. One may then find the mapping between power law plots of electrical conductivity vs temperature in semiconductors and the frequency vs magnitude plots of earthquakes as in Fig 2.

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. n_s, of these systems changes sign at a critical concentration, n_c. 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/ˆ.  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(n_s-n_c)½/n_c]T^b.  The interesting aspect is that the power law exponent change sign at the critical value, n_c, at which TCR changes sign which is conventionally regarded as an insulator-metal transition (IMT).

So far, there does not seem to be a mapping of a resistivity vs temperature power law dependence near an IMT to a frequency of vs magnitude power law plot for earthquakes. We have shown in Fig 2, right the 2D conductivity s (= 1/r) On further thought this does not seem to be too far-fetched. The resistance, R, is a measure of the resistance to current flow between two electrodes per unit time. The conductance, G (º1/R) is a measure of the frequency of current flow in time. The barrier to current flow in semiconductors at a given temperature is measured by an energy gap. In relative terms, the barrier to charge transport in semiconductors becomes effectively higher as temperature is reduced. The temperature dependence of conductance is thus a measure of the dependence of the frequency of charge transport between electrodes to barriers. The magnitude of 1/T is thus a measure of the magnitude of the gap. One expects the frequency of charge transport to increase as the gap decrease or the resistances to current transport is reduced.

If we extend the above arguments of low temperatures being equivalent to high barriers or high magnitudes of earthquakes, then the behavior in the so-called metallic compositions that have positive TCR would indicate that there could be a condition in which the frequency of large earthquakes increases as their sizes increase. This is a doomsday scenario if correct. It is, of course, unlikely if not incorrect.

What could be interesting, however, is that the exponent changes sign. As shown in Fig 2, left, the value of the exponent is nearly the same in the two regions where TCR has different signs.

2. c.Time dependence. Sequential earthquakes in time, t, such as aftershocks from earthquakes follow a power law in time where the number N(t) of earthquakes decreases with time as t^-a. Such a law was perhaps first proposed by Omori (1969) which appears in a modified form as N(t) µ (t + c)^p where c is a constant “time-offset” parameter.

When systems are not complex, relaxation times to the equilibrium state in dynamical processes are described by the Debye relaxation times. Thus the probability distribution function for the relaxation of a property, q(t), relative to its equilibrium value, q₀, ys given by an exponential decay rate for a given temperature by q(t) = q₀exp[-(t/t_e)^b] for homogeneous systems with b = 1. The term t_e is an effective relaxation time that is characteristic of the system.

Complex systems are usually heterogeneous systems which should be expected to have a hierarchy of relaxation times which would be distributed differently. In this case one would have a description given by 0 < b < 1. This is the slower Kohlrausch or stretched exponential decay function.  The value of b is thus sometimes taken as a measure of the heterogeneity, with 1/b being a heterogeneity parameter. The heterogeneity parameter is many a time an integer value (2, 3, and so on) and could mean a dependence on the dimension of the event.

The relaxation in stretched exponentials is thus thought to arise from a statistical distribution of parallel relaxation channels. It may also arise as a hierarchical sequential relaxation (in the PSAA model, Palmer, Stein, Abrahams, Anderson, Phys. Rev. Lett., 1984) in which a level must relax before another is released for relaxation. 

Just as Bak’s power law probability distribution function for relaxation systems, the stretched exponential function has found application in several areas which include several seemingly unrelated areas including earthquakes, biological extinction, economic crashes, scientific citations, biological extinction, glasses, polymers, proteins and so on. Because of the commonality of relaxation systems where power law as well as stretched exponential behavior occurs, it could seem that the power law and stretched exponential distributions apply to different limits of characteristic relaxation times.

A power law distribution is taken signify the absence of a characteristic size independently of the value of the power law exponent. It is a description of self similarity with the same proportion of smaller and larger events so that there is scale invariance. In the case of exponential decays as in stretched exponentials, there is a dependence on a characteristic scale, such as the relaxation time, t_e, in temporal events. There is no scale invariance. Despite such differences there have been attempts to include both these dependences in a common formulation of relaxation processes.

Among the systems where stretched-exponential-decays are commonly observed are glass-forming liquids below their melting point, T_m.  In this solidified glassy state such decays occur in the temperature range T_g < T < T_m, where T_g is a glass transition temperature. The glass transition temperature depends on cooling rate and viscosity of the liquid and the liquid-to-glass phase transition, if any, is not of the conventional kind. Below the glass-transition temperature the stretched exponential decay is not adequate to describe the decay to equilibrium. When T < T_g, there is a self-similar or fractal behavior in time with a power law decay. Ordinary glasses that we are familiar with have T < T_g.

Many a times, it is simpler to understand the direction in which a physical system is going is simply use a parameter that is a measure of the appropriate direction. In order to quantify such an understanding, one borrows from magnetic systems. In these systems a direction is given by a magnetic moment. These are in turn related  to the spin of elementary particles. In particular, one uses Ising spins that have only up- or down orientations to represent the magnetic moments. A spin is often represented by an arrow. The orientation of these spins relative to each other could change with time. In the glassy state of these spins (the spin-glass state) below a glass transition temperature, T_g, the individual spins with a random orientation with respect to each other. The “order parameter” of the spinglass state is a measure or the extent of randomness. At low temperatures, spin glasses are characterized by a very wide spectrum of relaxation times. In this case there could be self-similarity (fractal behaviour) in time and power law relaxation would appear.

These glasses are essentially unstable because of the inherent lurking correlations between the spins that tend to impose an ordered arrangement between the spins. One of the characteristics of spin glasses is the dynamics of the spin system as they relax to a more stable state. The way the dynamics is governed is by the way the individual spin changes its orientation with time, t. If the total spin has a value S(t) at a time t and has a value S(0) at t = 0. This is the auto correlation function. It turns out that different glassy systems have profound similarities in their relaxation properties.  The parameter of importance is the autocorrelation function as, q(t) = áS_i(t)^.S_ii(0)ñ. Empirically, there is a consensus that q(t) decays as power law with respect to that, q(¥), at infinite time, t = ¥. The autocorrelation function q(¥)  > 0 when T  < Tg while q(¥) = 0 for T > T_g.  Thus it is found that [q(t) – q(¥)] = ct^-µ

One could write a generalized expression (Pickup, PRL, 2009) for the relaxation as q(t) = t^-xexp[-(t/t_e)^b] that combines the power law with the stretched exponential. When t_e ®¥, one would be left only with the power law. This would perhaps require that the power law relaxations and Kohlrausch or stretched-exponential relaxations are parallel processes with different relaxation times, t_PL and t_e. In this model, the power law relaxation becomes operative when t_PL becomes less than t_e, at temperatures well below the glass transition temperature.

Just as there is a stretched exponential decay in time one could also have a stretched exponential growth in time. There does not seem to be many reports on a stretched exponential growth. Among the few that I could find straightaway from the internet is a report on the stretched exponential growth of photo-induced statics (within experimental time) defect population in optically thick films. Stretched exponential growth is also found in the time-dependence of electric birefringence near the consolute point of 2,6-lutidine and water liquid mixtures (a consolute point is more familiar as cloud point of binary liquid mixtures, when as a function of an external field, say temperature, when the liquids become immiscible and a cloudiness appears, a typical  familiar example being coconut-oil/water mixtures).. In mathematical topology (which is too complex for me) there is a “ … stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms”, whatever that means.

3. Foreshocks to forecast earthquakes

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, M_L, accurately reflects the amount of seismic energy released by an earthquake up to about M_L 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, M_W, 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, M_W = 2/3logM₀ – 10.7. The moment magnitude is linear with the Richter scale for M_L £ 5.0. Above this value the M_W value is usually reported. M_W 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.

3.2. Premonitions of Quakes. The foreboding of larger earthquakes from signals of smaller foreshock earthquakes is not built into most models of power laws. Foreshocks to main shocks are recognized only with hindsight mainly because they usually have different spatial and temporal scales besides having different magnitudes. It is therefore not easy to determine a priori whether a small grumble is a premonition to a larger rumble. This is especially so when Gutenberg-Richter frequency-magnitude relationships are used.

It is an important matter from a modeling viewpoint to decide whether one can extrapolate from the size, time, and magnitude of earthquakes whether a main-shock has appeared in the past (when the smaller shocks will be treated as aftershocks). More importantly for those who are worried about their finite lifetimes, it would be important to know whether one can predict from systematic of foreshocks the location and time of a future main shock. A breakthrough in earthquake prediction seems to have been uncovered recently (2006) by Feizer and Brodsky quantified distance-dependence of triggering of aftershocks from earthquakes. Felzer and Brodsky examined the average seismicity rate of aftershocks in California in small time windows following main events. They found an inverse distance, r, from epicenter dependence (r^-1.35) for event density (average number of events in a distance interval) over a wide range of distance and magnitude. They therefore considered the triggering of aftershocks to be dynamic in nature. The number of aftershocks (and foreshocks) changes almost exponentially with the mainshock magnitude. Although this work has its problems, it allows one to identify and classify clusters of earthquakes as mainshocks, aftershocks and foreshocks. It may be possible to classify aftershocks (or foreshocks) by a spatial scale, such as a radius from the epicenter of a main shock. One may then normalize the density of earthquakes in terms of the radius and by the total number of aftershocks (or foreshocks).

One figure that impressed (Fig 3, left) is one in a paper by Lippiello and coworkers on forecasting large earthquakes from the spatial distribution of foreshocks as a function of time. From a consideration of the linear density probability r(Dr) of aftershocks (or foreshocks) in a given time and by normalizing on the effect of the choice of a length scale they obtain a quantity R^-1 which is the inverse of the distance from the main shock within a maximum radius R_max (taken as 3 km). A high value of R^-1 indicates a large density of earthquakes. They analyzed main shocks with a magnitude M  = 4, 3, and 2. In all cases, the value of R^-1 for foreshocks grows with time while for the aftershocks decreases with time being a maximum at the time of the mainshock. These changes are usually interpreted as being exponential with time near the mainshock so that the mainshock appears as a singularity.

We find that the best fits to the decay or growth are given by a stretched exponential equation. Thus we find that the aftershock for M = 2 mainshock decay is well fitted (X symbol in Fig 3, right) as R^-1(t = t_max)exp((-t )^b) with the best fit (Fig 3, right, fitting parameters on top; best fit shown by dashed blue line for  b = 0.25) being for,b = 0.25 – 0.225. A value of b ~ 0.225 - 0.25 for the exponent suggests a dimension 1/b ~ 4 for the heterogeneity parameter.   There is a sort of “time reversal symmetry” for R^-1(t) of foreshocks and aftershocks about the time, t_max, at which R^-1 reaches a maximum value. The value of foreshock and aftershock can therefore be fitted to a common expression as R^-1(t) = R^-1(t = t_max)exp((-½(t_max – t)½/t_e)^b) with t_e ~ 1 and b ~ 0.25.

The predictive power of such a fit to foreshocks is diminished since one requires a premonition of t_max. One also requires knowledge of the dependence of b as well as t_e with the magnitude of the earthquake.  We have shown in Fig 3, right the way the calculated best fit value of R^-1(t) as b is changed (t_e­ is not changed much). It is seen that the slight asymmetry seen in the foreshock and aftershock dependence with time is towards the direction of slightly increased b for foreshocks as compared to aftershocks, Further, from the nature of the changes it is seen that b tends to increase as M increases. Since 1/b is the heterogeneity parameter this suggests that the dimensionality or heterogeneity decreases as M increases. Such a variation in b as well as t_e, adds to the uncertainty of a prediction.

Mechanisms for the generation of foreshocks do not seem to be well known. In a very recent article on  “The long precursory phase of most large interplate earthquakes” (Nature Geoscience, 2013  ), Bouchon et al, find from their analyses of earthquakes in the North pacific between 1999 and 2011 that interplate earthquakes (at plate boundaries)   are preceded by accelerating seismic activity of foreshocks while for the intraplate ones this is not so or much less frequent.

The recorded earthquake data for the 2011 Tohoku earthquake (M  ~ 9) are shown in Fig 4. The frequency of aftershock earthquakes within a radius of 50 kms from the epicenter decreases as power law with an exponent ~ 1.0 for all magnitudes of earthquakes. There is no indication of such a power law growth of foreshocks to this earthquake. This lack of long term precursory phase to the Tohoku earthquake, despite all evidences pointing to an inter-plate event, highlights the large difference in probable mechanisms of the way earthquakes cluster in time and space. 

The geological faults and asperities that drive large earthquakes are likely to have very slow dynamics and could be characterized by power law relaxations. The low magnitude earthquakes could have stretched exponential relaxations with short relaxation times.

4. Making and Breaking of Himalayas

4. a. Hill slopes and Sandpiles.

It is not straightforward to model damages to the roads carved out of the Himalayan slopes. When the soil is sandy there are landslides (Fig 5 left, 35^o slopes are shown). It is not as if there would be no more landslides once the 35^o slope is reached. It’s just that the slope will not change when there are further landslides from steeper slopes above. There are roads through rocky terrain (Fig 5, right) the strength and vulnerability of which could vary from one slope to another (Fig 5, left).

The rocks in the area of the Himalayas in Uttarakhand is mainly sedimentary rocks such as dolomite, (Ca,Mg)(CO₃), which is propably derived from limestone (mainly CaCO₃) by a low-temperature process. Fig 5, bottom right, is a road through what looks like dolomite or limestone. Dolomites dissolve quickly in low pH solutions or even in water which results in the formation of underground caves and subsequent cave-ins; A region called Gangolighat (popularized by the Kali mandir of Bengalis if not gangulys) in eastern Uttarakhand is known for its dolomite caves. What looks like solid rock (Fig 5 bottom right) may turn out to be quite vulnerable if they are dolomites.

Among the other rocks is quartzite, which is a metamorphic rock, associated with tectonic compression of the sedimentary rock, sandstone, due to increased pressure and temperature. Slate is a metamorphic rock derived from shale. Slate consists of thin-sheet-like structures (foliated structures like the rocks in Fig 5, top right?) with the sheets being perpendicular to the direction of tectonic compression. It is an aluminosilicate with compositions close to that of mica. Schist is also a flaky rock formed in the early stages of metamorphosis from clay-like material. Gneiss is a metamorphic rock which is also banded due to bands of iron-rich and iron-poor strata of a mainly silicate material.

Fig 6 compares the rock distribution (left) in the Uttarakhand region with the relief map of the region. Some immediate correspondence between the relief and the mineral distribution is seen. It should be noted that the region in the Uttrakhands where pilgrims aspire going and contractors build dams have the very vulnerable dolomite character, being very subject to erosion and cave-in. Unlike other dolomite regions such as the famous dolomite hills of Italy, the dolomite ranges in the Uttarakhand have considerable mixed character as is obvious from Fig 6, left. Such a mixture is seen not only in one mountain slope (Fig 7, left) but also within a single rock (Fig 7, right).

Erosion of quartzite would give clay/sand while erosion of dolomite would give alkaline solutions rich in Ca and Mg (which is later useful in scavenging back CO₂ from air). The well-known point is that it is the erosion of the Himalayas that give the Indo-Gangetic plains their fertile soil.

Earlier, in my treks through Nepal, I had noticed that the approach roads to the hills were very sandy-clayish and I wondered where the clay-sand came from. I thought at that time that the clay-sand came from the alluvial planes which was uplifted during the tectonic movements that made the Himalayas, Actually the term, alluvial (made of earth and sand left by rivers or floods), presupposes the existence of rivers. In the context of the Himalayas, the mighty Himalayan rivers are thought to have formed after the tectonic movement that formed the Himalayas. The landscape of lower Nepal (Fig 8, left) is consistent with sand-pile-hills being formed due to run-off of torrents of water from the Himalayas. These sandpiles are then flattened out by further piling up of soil as a result of continuous erosion. The 35^o angle at the bottom of landslides from steeper slopes appears as an ubiquitous feature in the Himalayas, such as those made by road construction in the Himalayas. The 35^o slope after a landslide is also seen (Fig 8, right) in the Lhasa landscape.

4.c. Himalayan Tectonics

The tectonic movements that gave rise to the Himalayas describes an event 65 million years ago when a part of the African continent got chipped off due to some violent seismic/cosmic events and the plate skidded away towards the Asian continent and climbed onto it. The understanding of the nature of the geology of the Himalayas that resulted from such a movement would require extensive knowledge of many things, not least of which are the geological terms. I have, perhaps fortunately, little idea of these geological terms and how they are to be used.

The formation of the Himalayas due to a siding of the Indian plate onto the Asian plate has been often schematically sketched without any really quantifiable estimate. There should have been a part of the sea that was trapped between the Indian and the Eurasian plate. A land bridge between India and the Asian mainland seems to have been established in the early Cenozoic period (the Eocene). Most of the initial impact was in the east when the Indian plate India subducted (carried under the edge of an adjoining continental or oceanic plate) under the Tibetan plate to become Tibet’s basement.

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 nearly circular features in the Tibetan plateau on top of the Himalayas are generally attributed to oroclines, or bends of the tectonic elements, when initially straighter linear elements respond to a variety of local boundary conditions, ranging from local variations in the colliding terranes and stress-fields. The changes in curvature therefore give insights into the local boundary conditions. It could also be that local curvatures set local boundary conditions including the nature of metamorphic rocks that are formed.

One such scheme in the middle of Fig 9 is really a cartoon. The Tibetan plateau is thought to be the result of the continuous thrust strong Indian plate into the weaker Eurasian lithosphere since the last 50-70 million years or so.

I have a weakness (hopefully not based simply on ignorance) for attributing circular uplifts to asteroid Impacts some of which could have happened during the Cenozoic period (See my blog “Asteroid Impact: Destruction and Creation - Shiva as Ashutosh” of Dec 5 2009) when dinosaurs (and other giants) became extinct. I could now succumb to the temptation of attributing the circular Himalayan ranges to the formation of gigantic Impact basins north of the Himalayas around the same time or earlier. To the right of Fig 9, I have compared the size of impact-basins on the moon with some of the circular features on the Tibet plateau. 

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.

Most of the qualitative descriptions are based on a two‐dimensional thin viscous sheet approximation is widely used as a description of the lithosphere (outer crust and upper mantle layer of the Earth). This layer is treated as a fluid layer in which buoyancy forces balance tectonic boundary conditions by causing a deformation.

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.

There are other difficulties for my understanding, even if we remember that I am a novice. For example, the plate movement is now thought to be as accurate as 1 mm per year and the Indian plate is said to be moving at about 1 cm per year. One would expect the plate to have moved at least by 6000 km after the collision between continental plates in the Cenozoic period (60 million years ago). This scale of movement is not apparent unless layers of pieces of sliding continents are stacked one on top of the other.

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.

On 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 river beds.

The Earth's lithosphere is sometimes treated as an elasto-plastic material (plasticine is an example) which could deform soften due to strain in a hard (brittle) or soft (ductile) manner that depends on the geometric boundary conditions. The deformation can localize on faults or in zones where there are spatially different shear direction that prevent localization. From an analysis of the topography of the dead sea region Devès et al (Earth and Planetary Science Letters, 2011) have been able to predict where deformations can localize and where they are distributed. Stress levels are an order of magnitude larger when the deformations are distributed than where they are localized at faults. For a given displacement, faults first appear in regions of low strain  and follow simple shear directions so that there can be substantial displacement without changing geometry. Localised deformation along a fault with a release of strain seems to be more common than those where strain is accumulated by distribution over a stress zone that would require later processing through earthquakes or energy-releasing hot magmatic processes.

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.

When shear directions change from place to place faults cannot extend and strains build up and get distributed. It would seem that another way to relieve the accumulated strain is to pile to form giant mountains. This seems to be what is observed in the case of the great peaks of the Himalayas (Fig 11). It may not therefore be a coincidence that the ” seismic gap” region of the Himalayas is also the regin where the high peaks are found.

Independent of how one treats the missing volume of soil/sand, there seems to have a huge volume of soil that has flowed down from the Himalayas, Such a low would have piled sand on sand (or clay upon clay) on the ever creeping, ever rumbling, ever grinding, ever breaking, tectonic plates.

 5. What we could know from the Tibetan Plateau

 The role of the Tibetan plateau in this huge sand flow would be vital. Fortunately China has been studying this aspect more and more. It seems that the Lhasa Terrane (a section of the Earth's crust that is defined by clear fault boundaries, with stratigraphic and structural properties that distinguish it from adjacent rocks) in southern Tibet has been accepted as a tectonic block rifted from Gondwana and drifted northward across the Tethyan oceans and collided with the Qiangtang Terrane to the north. Its subsequent collision with the northward moving Indian continent in the early Cenozoic period marked the closure of the Neo-Tethyan Ocean, The rocks in the Lhasa Terrane should provide an understanding of the Tethyan Ocean basins and provide models of the origin and evolution of the Tibetan Plateau. A recent (2013) article by Zhu and coworkers on “The origin and pre-Cenozoic evolution of the Tibetan Plateau” seems to have contributed new insights into the creation of the Tibetan plateau. They found that the geology of the central Lhasa subterrane and Tethyan Himalaya (the structurally highest units of the Himalayan fold and thrust belt) is similar to that of northern Australia! Their model for the building of the Tibetan plateau is given in Fig 12 (click to expand). The pkate tectonics of the Tibetan plaeau has certainly seen a lot of grinding and crushing, if it has not been pulverized further by giant asteroid impacts that I like to believe.

The problem of understanding observed seismograms is to relate them to envisioned geometric, kinematic and dynamic parameters of a model for the physical phenomenon of fracture of the Earth's lithosphere. One of these is to quantify the complexities of continental deformation. Recently, in what has been hailed as a major paper, Loveless and Meade from Harvard (“Partitioning of localized and diffuse deformation in the Tibetan Plateau from joint inversions of geologic and geodetic observations”, 2011) have looked micro-plate rotation rates that are calculated from slip rates of the kind seen in Fig 10. These authors also looked at the deformation in the greater Tibetan plateau region using a potency rate. Quite simply, the potency rate that is accommodated by major faults is given by the product of fault area and slip rate magnitude and micro-plate rotation rates. They then calculate a residual velocity field (which is the difference in magnitude of the observed velocity field) and a predicted velocity field (which is a function only of micro-plate rotations and earthquake cycle effects). The horizontal displacement rate gradient tensor, D, of the velocity, is assumed to be constant within each element.  Loveless and Meade used the method of Delauney triangulation (a unique triangulation, DT(S) for a set S of points in the Euclidean plane such that no point in S is inside the circumcircle of any triangle in the triangulation) of GPS stations within each crustal block. The results of their Delauney triangulation of residual velocity fields is given in Fig 13, left. The magnitude of the strain rate tensor is quite large in the lesser Himalayas and also near the Tehri dam.

Loveless and Meade also plotted the distance between modern earthquakes (with depths less than 33 km and magnitude M_W greater than 5) and historical (white outlined circles, M_W ³ 6.4) with respect to the surface trace of the nearest block geometry. They showed from this plot (Fig 13, right, click to expand) that ~ 65 % of the cumulative seismic energy release of earthquakes since 1976 has occurred in events within 25 km of a block boundary, and 90% within 95 km.  They therefore suggest that “earthquakes located within ~ 25 km of a block boundary can be considered to have occurred on a modeled fault segment.”

It is seen from Figs 12 and 13 that neither the residual velocity tensors nor the earthquake distances in the Tibetan plateau follow the fault and suture zone lines of Fig 12. Instead one could imagine, as I incline to, that the majority of the earthquakes occur at the perimeter of giant impact basins that I have indicated in Fig 13, right by black ovals in the context of Fig 9. Earthquake disasters at borders of Asteroid impact basins seem to have correlations with geodetic data in Fig 10.

I went through this effort of learning about the Tibetan Plateau, just to point out that before one learns from the orogenetic feature of the Himalayas in which Uttarakhand is located one has also to know about the influence of the Tibetan Plateau. The Chinese seems to be putting in a lot of effort to study the Tibetan plateau. I think one needs a serious contribution from the Indian side if one is interest in long-term stabilities.
 This exercise would be a waste of time if one is interested in financial gains. This seems to be the direction that the “authorities” of the Manmohan Singh, Montek Singh and Chidambaram kind would like to be taking even if they do so, so they say, in the name of “economic development” for the poor.
 Truth be said, the roads and dams are part of our infrastructure plans that require the inaccessibles to be brought to the immature unquenchables of all age. The cartoon below (click to expand) is from Ananth Shankar’s book of cartoons (The Crazy Desi Book Vol I,Travel; see https://www.facebook.com/TheCrazyDesiBook) put it

My value systems are developed from my childhood values a large portion of which come from Chennai, Tamil Nadu, where I grew up and where one often frequented the sea-side. Before going in to the sea, we were told the tamil equivalent of “look before you leap”: ஆழம் தெரியாமல் குதிக்க வேண்டாம் (which transliterates to “Depth not knowing, leaping not necessary”). I guess the hill-equivalent of this would be “Slopes not knowing, treading not necessary” or சாய்வு தெரியாமல் மலை ஏற வேண்டாம்.

Certainly not necessary for the road-building, dam-building, instant-pilgrim kind.