MIT12_009S11_lec10_11

MIT12_009S11_lec10_11 - 4 Scaling laws for rivers and...

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Unformatted text preview: 4 Scaling laws for rivers and runoff 4.1 Weathering and runoff Over the long term (10 5 10 9 yr), the slow processes of the rock cycle dominate the carbon cycle. We have already discussed volcanism as the long-term source of CO 2 , and burial as the long-term sink . Prior to burial, rocks are chemically weathered (e.g., leached). Dissolved carbon then becomes part of groundwater and river ows, the oceans, and ultimatelysedimentary rocks. Thus weathering may be thought of as the initiation of the burial sink. With respect to the carbon cycle, the most important of the chemical weath- ering reactions reads schematically as follows [1921]: CaSiO 3 + CO 2 CaCO 3 + SiO 2 . A similar reaction may be written by substituting Mg for Ca. Left-to-right, such reactions represent the uptake of CO 2 from the atmo- sphere, its transformation to HCO 3 during weathering of silicate minerals, and its eventual precipitation and burial in the oceans as carbonate minerals. Right-to-left, the reaction represents metamorphism and magmatism and the subsequent transfer of CO 2 back to the atmosphere and oceans by volcanism and related processes. In this lecture we consider aspects of transport implied in the forward reac- tion. From the viewpoint of the carbon cycle, transport is especially important if the reaction is transport-limited , i.e., if the rate of the forward reaction is determined more by transit times than by the time it takes for dissolution. 59 In this idealization, we imagine that a raindrop immediately dissolves miner- als, and we seek to understand the geometry of the path taken by the raindrop were it to continue to the ocean. We could proceed to associate time scales with travel paths, but we shall limit ourselves to purely geometric considerations. 4.2 Drainage basins and river networks Our idealized raindrop falls in a drainage basin , and eventually ows through a river network. River networks are among the most beautiful of Natures large-scale scale- invariant phenomena. Image courtesy of NASA. To see what we mean by scale-invariant, we briey return to random walks. 4.3 Scale invariance of random walks Define the rms excursion r = x 2 ( t ) 1 / 2 . We have previously shown that r t 1 / 2 . Now rescale time t bt and note that r ( t ) = b 1 / 2 r ( bt ) . 60 This simple manipulation yields a remarkable observation: the statistics of the random walk are unchanged by the simultaneous rescaling x b 1 / 2 x, t bt. This means that the random walk is statistically equivalent at all scales, e.g....
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MIT12_009S11_lec10_11 - 4 Scaling laws for rivers and...

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