Stochastic Modeling and Analysis of Earth Surface Systems: Lakes, Rivers, and Oceans

Abstract

Observational records of natural processes are typically characterized by multiscale spatial and temporal variability. In this dissertation, variability inherent in three disparate Earth surface systems are studied; lakes, oceans, and rivers. For the systems under consideration, it is shown how random fast time scale variability in external driving forces are integrated or filtered by the system’s internal dynamics, which typically have a much slower response time compared to the forcing time scale, resulting in measurable output with multiscale temporal and or spatial variability. In chapter 2, the frequency (f) power spectrum, P(f), of lake water level fluctuations from a global analysis of lakes, are shown to exhibit power-law frequency dependence, P(f)~f^-b. The spectral exponent (b) is shown to be a function of lake surface area. A discrete surface growth model based on the equations governing aquifer-groundwater flow is developed and shown to reproduce the self-affine and surface-area-dependent fluctuations of lake water levels. This work shows that slow timescale groundwater dynamics within the lake system integrate or filter the faster timescale noisy mass inputs. In chapter 3, dynamic sea surface height topography over the range of O(10) km up to approximately 1000 km is characterized by piece-wise power-law wavenumber (v) power spectra, P(v), that are proportional to v^-2. The two scaling regions of the piecewise power-law are separated by a crossover wavenumber that is shown to depend on the baroclinic Rossby radius. Sea surface height is modeled as a stochastic field by considering the spatial response of the Ekman surface boundary layer to wind-stress-curl vorticity inputs at the ocean surface. This model captures the spatial and temporal scaling of SSH spectra as well as the observed latitude dependence in the crossover wavenumber. The model shows how spatial and temporal variability in SSH over the range of scales considered may be considered as the integrated response of the Ekman surface boundary layer to the spatially inhomogeneous fast timescale wind-stress-curl forcing inputs. In chapter 4, a case study is presented based on observed spatial and temporal multiscale variability for both hydrologic and geomorphic variables in the San Pedro river of Southeastern Arizona. The length (L) of discontinuous perennial reaches are observed to follow a power-law frequency-size distribution f(L)~L^-0.8. The power-law scaling is then shown to result from two scale separate controls on the underlying groundwater table. Within channel depth-to-bedrock exerts a “bottom-up” control on large scale hydrologic spatial variability, while the longitudinal channel profile gradient exerts a “top-down” control on the small scale hydrologic spatial variability. Two numerical models that are consistent with the geology and geomorphology of the San Pedro river are presented. It is then shown how stochastic spatial variability in bedrock topography and longitudinal channel morphology control the spatial scaling of surface hydrology in the San Pedro river. The overarching argument made in this dissertation is that multiscale variability in many Earth systems may be considered as the dynamical consequence and integrated response to high-dimensional and chaotic environmental driving forces.