title: The lore of lakes

authors: Green, John D.

abstract: Studies of lakes and their many layers of silt and sediment can tell us a great deal, from centuries-old deforestation practices to the climate changes over thousands of years. The study of the history of lake ecosystems is a fast developing field known as palaeolimnology.

text: The lore of lakes Studies of lakes and their many layers of silt and sediment can tell us a great deal, from centuries-old deforestation practices to the climate changes over thousands of years. The study of the history of lake ecosystems is a fast developing field known as palaeolimnology.

year: 1992

title: Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere

authors: Kalnins, Ernie G.

subjects: superintegrability

abstract: We show that the symmetry operators for the quantum superintegrable system on the 3-sphere with generic 4-parameter potential form a closed quadratic algebra with 6 linearly independent generators that closes at order 6 (as differential operators). Further there is an algebraic relation at order 8 expressing the fact that there are only 5 algebraically independent generators. We work out the details of modeling physically relevant irreducible representations of the quadratic algebra in terms of divided difference operators in two variables. We determine several ON bases for this model including spherical and cylindrical bases. These bases are expressed in terms of two variable Wilson and Racah polynomials with arbitrary parameters, as defined by Tratnik. The generators for the quadratic algebra are expressed in terms of recurrence operators for the one-variable Wilson polynomials. The quadratic algebra structure breaks the degeneracy of the space of these polynomials. In an earlier paper the authors found a similar characterization of one variable Wilson and Racah polynomials in terms of irreducible representations of the quadratic algebra for the quantum superintegrable system on the 2-sphere with generic 3-parameter potential. This indicates a general relationship between 2nd order superintegrable systems and discrete orthogonal polynomials.

text: Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere We show that the symmetry operators for the quantum superintegrable system on the 3-sphere with generic 4-parameter potential form a closed quadratic algebra with 6 linearly independent generators that closes at order 6 (as differential operators). Further there is an algebraic relation at order 8 expressing the fact that there are only 5 algebraically independent generators. We work out the details of modeling physically relevant irreducible representations of the quadratic algebra in terms of divided difference operators in two variables. We determine several ON bases for this model including spherical and cylindrical bases. These bases are expressed in terms of two variable Wilson and Racah polynomials with arbitrary parameters, as defined by Tratnik. The generators for the quadratic algebra are expressed in terms of recurrence operators for the one-variable Wilson polynomials. The quadratic algebra structure breaks the degeneracy of the space of these polynomials. In an earlier paper the authors found a similar characterization of one variable Wilson and Racah polynomials in terms of irreducible representations of the quadratic algebra for the quantum superintegrable system on the 2-sphere with generic 3-parameter potential. This indicates a general relationship between 2nd order superintegrable systems and discrete orthogonal polynomials.

year: 2011

title: Ecological modelling of water quality management options in Lake Waahi to support Hauanga Kai species: Technical Report

authors: Allan, Mathew Grant

abstract: This report describes a numerical modelling approach for Lake Waahi to assess the effects of various future scenarios on water quality attributes known to affect the appearance of the lake’s multiple hauanga kai species across differing spatial scales. Hauanga kai refers to customary gathering and use of naturally occurring and cultivated foods. Numerical models were used to support shared learning and development of future scenarios in response to different management options and environmental pressures. The shallow lakes of the Waikato region have been particularly impacted by anthropogenic eutrophication. This is due to the fact that many of these lakes have already undergone a degree of natural eutrophication due to their geomorphic properties, such as their low elevation and shallow depth. The main contributing factors to water quality decline in shallow Waikato lakes have been identified as: high external nutrient and sediment loads, internal loading exacerbated by high sediment resuspension and bioturbation from pest fish (carp), reduced water depth due to installation of water control structures, reduction in wetland filtering potential due to draining, and stock access to wetlands. A 1-D hydrodynamics model, the General Ocean Turbulence Model was coupled to a complex ecological model (PClake) via the Framework for Aquatic Biogeochemical Models (FABM). FABM is an open source FORTAN-based code that facilitates the coupling of hydrodynamic and ecological models. FABM-PCLake is a redesigned PCLake. The consensus from the model simulations carried out in the study suggests that, for Lake Waahi to undergo restoration whereby the lake is restored to clear-water stable state, both external load needs to be reduced significantly by greater than 50%, along with changes in lake ecological structure, resulting from re-establishment of macrophytes aided by the control of carp biomass. These finding supports the general view within the literature that for lake restoration success, external load reduction (catchment load) is critical. Finally, if the ultimate goal of lake restoration in Lake Waahi is to address the decline in tuna (shortfin eel - Anguilla australis) biomass, a return to a clear-water stable state should facilitate improvement in tuna habitat and provide the conditions for development of a healthy eel population. Coupled modelling approaches will provide a more complete picture of potential responses of Hauanga kai species to land and lake management scenarios.

text: Ecological modelling of water quality management options in Lake Waahi to support Hauanga Kai species: Technical Report This report describes a numerical modelling approach for Lake Waahi to assess the effects of various future scenarios on water quality attributes known to affect the appearance of the lake’s multiple hauanga kai species across differing spatial scales. Hauanga kai refers to customary gathering and use of naturally occurring and cultivated foods. Numerical models were used to support shared learning and development of future scenarios in response to different management options and environmental pressures. The shallow lakes of the Waikato region have been particularly impacted by anthropogenic eutrophication. This is due to the fact that many of these lakes have already undergone a degree of natural eutrophication due to their geomorphic properties, such as their low elevation and shallow depth. The main contributing factors to water quality decline in shallow Waikato lakes have been identified as: high external nutrient and sediment loads, internal loading exacerbated by high sediment resuspension and bioturbation from pest fish (carp), reduced water depth due to installation of water control structures, reduction in wetland filtering potential due to draining, and stock access to wetlands. A 1-D hydrodynamics model, the General Ocean Turbulence Model was coupled to a complex ecological model (PClake) via the Framework for Aquatic Biogeochemical Models (FABM). FABM is an open source FORTAN-based code that facilitates the coupling of hydrodynamic and ecological models. FABM-PCLake is a redesigned PCLake. The consensus from the model simulations carried out in the study suggests that, for Lake Waahi to undergo restoration whereby the lake is restored to clear-water stable state, both external load needs to be reduced significantly by greater than 50%, along with changes in lake ecological structure, resulting from re-establishment of macrophytes aided by the control of carp biomass. These finding supports the general view within the literature that for lake restoration success, external load reduction (catchment load) is critical. Finally, if the ultimate goal of lake restoration in Lake Waahi is to address the decline in tuna (shortfin eel - Anguilla australis) biomass, a return to a clear-water stable state should facilitate improvement in tuna habitat and provide the conditions for development of a healthy eel population. Coupled modelling approaches will provide a more complete picture of potential responses of Hauanga kai species to land and lake management scenarios.

year: 2018