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Beginning from the shallow water equations (SWEs), a nonlinear self-similar analytic solution is derived for barotropic flow over varying topography. We study conditions relevant to the ocean slope where the flow is dominated by Earth's rotation and topography. The solution is found to extend the topographic β-plume solution of Kuehl (2014) in two ways. (1) The solution is valid for intensifying jets. (2) The influence of nonlinear advection is included. The SWEs are scaled to the case of a topographically controlled jet, and then solved by introducing a similarity variable, η = cxnxyny. The nonlinear solution, valid for topographies h = h0 − αxy3, takes the form of the Lambert W-function for pseudo velocity. The linear solution, valid for topographies h = h0 − αxy−γ, takes the form of the error function for transport. Kuehl's results considered the case −1 ≤ γ < 1 which admits expanding jets, while the new result considers the case γ < −1 which admits intensifying jets and a nonlinear case with γ = −3.
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Hölder‐Conditioned Hypsometry: A Refinement to a Classical Approach for the Characterization of Topography
Abstract The effective characterization of topographic surfaces is a central tenet of geomorphology. Differences in land surface properties reveal variations in structural controls and the nature and efficacy of Earth‐shaping processes. In this paper, we employ the Hölder exponents,α, characterizing the local scaling behavior of topography and commonly used in the study of the (multi)fractal properties of landscapes and show that the joint probability distribution of the area of the terrain with a given elevation andαcontains a wealth of information on topographic structure. The conditional distributions of the hypsometric integrals as a function ofα, that is,Ihyp|α, are shown to capture this structure. A multivariate analysis reveals three metrics that summarize these conditional distributions: Strahler's original hypsometric integral, the standard deviation of theIhyp|α, and the nature of any trend of theIhyp|αagainstα. An analysis of five digital elevation models (DEMs) from different regions of the United States shows that only one is truly described by the hypsometric integral (Mettman Ridge from central Oregon). In the other cases, the new metrics clearly discriminate between instances where topographic roughness is more clearly a function of elevation, as captured by the conditional variables. In a final example, we artificially sharpen the ridges and valleys of one DEM to show that while the hypsometric integral and standard deviation ofIhyp|αare invariant to the change, the trend ofIhyp|αagainstαcaptures the changes in topography.
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- PAR ID:
- 10452701
- Publisher / Repository:
- DOI PREFIX: 10.1029
- Date Published:
- Journal Name:
- Water Resources Research
- Volume:
- 56
- Issue:
- 5
- ISSN:
- 0043-1397
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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