Turbulent rivers

Abstract

Abstract. The existence of solutions describing the turbulent flow in rivers is proven. The existence of an associated invariant measure describing the statistical properties of this one-dimensional turbulence is established. The turbulent solutions are not smooth but Hölder continuous with exponent 3/4. The scaling of the solutions' second structure (or width) function gives rise to Hack's law (1957), stating that the length of the main river, in mature river basins, scales with the area of the basin l ∼ A h , h = 0.568 being Hack's exponent. Introduction.The flow of water in streams and rivers is a fascinating problem with many applications that has intrigued scientists and laymen for many centuries; see Levi [21]. Surprisingly it is still not completely understood even in one or two-dimensional approximations of the full three-dimensional flow. Erosion by water seems to determine the features of the surface of the earth, up to very large scales where the influence of earthquakes and tectonics is felt; see [33,34,32,6,4,36]. Thus water flow and the subsequent erosion give rise to the various scaling laws known for river networks and river basins; see [12,8,9,10,11].One of the best known scaling laws of river basins is Hack's law [16], which states that the area of the basin scales with the length of the main river to an exponent known as Hack's exponent. Careful studies of Hack's exponent, see [11], show that it actually has three ranges, depending on the age and size of the basin, apart from very small and very large scales where it is close to one. The first range corresponds to a spatial roughness coefficient of one half for small channelizing (very young) landsurfaces. This has been explained, see [4] and [13], as Brownian motion of water and sediment over the channelizing surface. The second range with a roughness coefficient of 2/3 corresponds to the evolution of a young surface forming a convex (geomorphically concave) surface, with young rivers that evolve by shock formation in the water flow. These shocks are called bores (in front) and hydraulic jumps (in rear); see Welsh, Birnir and Bertozzi [36]. Between them sediment is deposited. Finally there is a third range with a roughness

A review of volume‐area scaling of glaciers

“…The difficulty of measuring volume makes a test of the scaling theory more challenging, but similar obstacles have been overcome in other geophysical disciplines. For example, Hack's law relates river basin area to main channel length, and there is an extensive body of literature that theoretically derives, measures, and validates this important scaling relationship even though basin length might be interpreted (incorrectly) as appearing on both sides of the relationship [e.g., Hack, 1957;Birnir, 2008]. Observations support a particular mass balance closure (m = 2), a particular width closure (q = 3/5), and a particular equilibrium AAR closure (AAR = 0.577); see equation (126).…”

Mathematical Models for Erosion and the Optimal Transportation of Sediment

“…The second group has produced remarkable simulations of evolving channel networks; see [52,53], [18], [48] and [35]. The third group has lead to an increasing understanding of the physical mechanisms that underlie erosion and channel formation; see [40], [42], [30], [36], [28], [27], [29], [43], [22,23,24,25,21], [44,45,46], [39], [50], [9], [15], [7], [41].…”

An Approximation of the Invariant Measure for the Stochastic Navier-Stokes

“…Recently the existence of the invariant measure was established in the three-dimensional case [2]. It was established for uni-directional flow in [1] and for rivers in [3]. Below we will discuss how one can try to go about approximating the invariant measure in three dimensions.…”