Flow instabilities occur at all scales in planetary systems. In this thesis we examine three cases of such instabilities, on three very different length scales. In the first part, we test the idea that Archean granite-greenstone belts (GGBs) form by crustal diapirism, or Rayleigh-Taylor instabilities. GGBs are characterized by large granitic domes (50-100 km in diameter) embedded in narrow keel-shaped greenstones. They are ubiquitous in Archean (> 2.5 Ga) terrains, but rare thereafter. We performed finite element calculations for a visco-elastic, temperature-dependent, non-Newtonian crust under conditions appropriate for the Archean, which show that dense low-viscosity volcanics overlying a felsic basement will overturn diapirically in as little as 10 Ma, displacing as much as 60 % of the volcanics to the lower crust. This surprisingly fast overturn rate suggests that diapiric overturn dominated crustal tectonics in the hot conditions of the Early Earth, becoming less important as the Earth cooled. Moreover, the deposition of large volumes of wet basaltic volcanics to the lower crust may provide the source for the formation of the distinctly Archean granitic rocks which dominate Earth's oldest continents. The second part examines the origin of Venusian coronae, circular volcanic features unique to Venus. Coronae are thought to result from small instabilities (diapirs) from the core-mantle boundary, which are typical of stagnant-lid convection. However, most young coronae are located in a region surrounded by long-lived hotspots, typical of a more active style of mantle convection. Using analogue experiments in corn syrup heated from below, we show that the co-existence of diapirs and long-lived mantle plumes are a direct consequence of the catastrophic overturn of the cold Venusian lithosphere thought to have occurred ~ 700 Ma ago. In the last part we analyze the thermal effect of fluid flow through a full-scale experiment testing clay and concrete tunnel seals in a Deep Geological Repository for nuclear was finite element software, we were able to show that the formation of fissures in the heated chamber between the two seals effectively limited heat flow, and could explain the discrepancy between the predicted and measured temperatures.

Flow instabilities occur at all scales in planetary systems. In this thesis we examine three cases of such instabilities, on three very different length scales. In the first part, we test the idea that Archean granite-greenstone belts (GGBs) form by crustal diapirism, or Rayleigh-Taylor instabilities. GGBs are characterized by large granitic domes (50-100 km in diameter) embedded in narrow keel-shaped greenstones. They are ubiquitous in Archean (> 2.5 Ga) terrains, but rare thereafter. We performed finite element calculations for a visco-elastic, temperature-dependent, non-Newtonian crust under conditions appropriate for the Archean, which show that dense low-viscosity volcanics overlying a felsic basement will overturn diapirically in as little as 10 Ma, displacing as much as 60 % of the volcanics to the lower crust. This surprisingly fast overturn rate suggests that diapiric overturn dominated crustal tectonics in the hot conditions of the Early Earth, becoming less important as the Earth cooled. Moreover, the deposition of large volumes of wet basaltic volcanics to the lower crust may provide the source for the formation of the distinctly Archean granitic rocks which dominate Earth's oldest continents. The second part examines the origin of Venusian coronae, circular volcanic features unique to Venus. Coronae are thought to result from small instabilities (diapirs) from the core-mantle boundary, which are typical of stagnant-lid convection. However, most young coronae are located in a region surrounded by long-lived hotspots, typical of a more active style of mantle convection. Using analogue experiments in corn syrup heated from below, we show that the co-existence of diapirs and long-lived mantle plumes are a direct consequence of the catastrophic overturn of the cold Venusian lithosphere thought to have occurred ~ 700 Ma ago. In the last part we analyze the thermal effect of fluid flow through a full-scale experiment testing clay and concrete tunnel seals in a Deep Geological Repository for nuclear was finite element software, we were able to show that the formation of fissures in the heated chamber between the two seals effectively limited heat flow, and could explain the discrepancy between the predicted and measured temperatures.

In this dissertation, we consider two different strategies for improving the projection-based reduced order model (ROM) accuracy: (I) adding closure terms to the standard ROM; (II) using Lagrangian data to improve the ROM basis. Following strategy (I), we propose a new data-driven reduced order model (ROM) framework that centers around the hierarchical structure of the variational multiscale (VMS) methodology and utilizes data to increase the ROM accuracy at a modest computational cost. The VMS methodology is a natural fit for the hierarchical structure of the ROM basis: In the first step, we use the ROM projection to separate the scales into three categories: (i) resolved large scales, (ii) resolved small scales, and (iii) unresolved scales. In the second step, we explicitly identify the VMS-ROM closure terms, i.e., the terms representing the interactions among the three types of scales. In the third step, we use available data to model the VMS-ROM closure terms. Thus, instead of phenomenological models used in VMS for standard numerical discretizations (e.g., eddy viscosity models), we utilize available data to construct new structural VMS-ROM closure models. Specifically, we build ROM operators (vectors, matrices, and tensors) that are closest to the true ROM closure terms evaluated with the available data. We test the new data-driven VMS-ROM in the numerical simulation of four test cases: (i) the 1D Burgers equation with viscosity coefficient $nu = 10^{-3}$; (ii) a 2D flow past a circular cylinder at Reynolds numbers $Re=100$, $Re=500$, and $Re=1000$; (iii) the quasi-geostrophic equations at Reynolds number $Re=450$ and Rossby number $Ro=0.0036$; and (iv) a 2D flow over a backward facing step at Reynolds number $Re=1000$. The numerical results show that the data-driven VMS-ROM is significantly more accurate than standard ROMs. Furthermore, we propose a new hybrid ROM framework for the numerical simulation of fluid flows. This hybrid framework incorporates two closure modeling strategies: (i) A structural closure modeling component that involves the recently proposed data-driven variational multiscale ROM approach, and (ii) A functional closure modeling component that introduces an artificial viscosity term. We also utilize physical constraints for the structural ROM operators in order to add robustness to the hybrid ROM. We perform a numerical investigation of the hybrid ROM for the three-dimensional turbulent channel flow at a Reynolds number $Re = 13,750$. In addition, we focus on the mathematical foundations of ROM closures. First, we extend the verifiability concept from large eddy simulation to the ROM setting. Specifically, we call a ROM closure model verifiable if a small ROM closure model error (i.e., a small difference between the true ROM closure and the modeled ROM closure) implies a small ROM error. Second, we prove that a data-driven ROM closure (i.e., the data-driven variational multiscale ROM) is verifiable. For strategy (II), we propose new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct new Lagrangian ROMs. We show that the new Lagrangian ROMs are orders of magnitude more accurate than the standard Eulerian ROMs, i.e., ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs' accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis. / Doctor of Philosophy / Reduced order models (ROMs) are popular in physical and engineering applications: for example, ROMs are widely used in aircraft designing as it can greatly reduce computational cost for the aircraft's aeroelastic predictions while retaining good accuracy. However, for high Reynolds number turbulent flows, such as blood flows in arteries, oil transport in pipelines, and ocean currents, the standard ROMs may yield inaccurate results. In this dissertation, to improve ROM's accuracy for turbulent flows, we investigate three different types of ROMs. In this dissertation, both numerical and theoretical results show that the proposed new ROMs yield more accurate results than the standard ROM and thus can be more useful.