Date of Award

2009

Degree Type

Report

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

First Advisor

Jim Powell

Abstract

Mathematical models for the change in concentration of total dissolved nitrogen (TDN) in mountain lakes are developed based on the dynamics of coupled, well-mixed containers. Each includes a stratified lake structure without the complexity of a full fluid model. A lake is divided into a suite of compartments based on physical structure: warm upper layer (epilimnion), cold inflow and insertion layer (metalimnion), cold lower layer (hypolimnion), and a warm shallow shelf. With the compartments as the framework and literature values for uptake rates, death rates, half-saturation constants, and sinking rates, systems of equations are written for three models. The first is a system of differential equations including nutrient cycling within each compartment, including changes in TDN due to growth and death of seston as well as loss to and gain from lake sediments. With the literature values, flows, and TDN data taken at the inflow (Baker and Wurtsbaugh 2008), these equations are solved numerically to determine the concentration of TDN in each compartment. The second is a simplified version of the first model containing only fluxes of TDN between compartments and the flux in lake sediments. The third is a system of equations for the steady states of the first model found by making an assumption on the half-saturation constant. With TDN data taken at the outflow in 2002 (Baker and Wurtsbaugh 2008), terms of sedimentation fluxes are chosen to minimize the sum of the squared difference between the measured and predicted concentrations of TDN for each of the three systems. Each model is tested against data taken in 2003 (Baker and Wurtsbaugh 2008). The second model predicts observed TDN well in a stratified lake structure without the computational difficulty of a full fluid model. To determine the flows between compartments we solved differential equations for the transport of nutrients into lakes by cold plunging inflows (Hauenstein and Oracos 1984). The entrainment rates are treated as eigenvalues, and an eigenvalue problem is solved for the plunging inflow based on discharge data taken in 2003 (Arp 2006) and data taken from a field study on Bull Trout Lake (BTL) in central Idaho in June 2008. The results show that lake structure is a significant factor in relating input and output concentrations of TDN.

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