title: Hydrodynamic and water quality modelling of the lower kaituna river and maketu estuary

authors: Goodhue, Nigel David

subjects: numerical

abstract: The Maketu Estuary is a shallow intertidal estuary (2.3 km2) located in the Bay of Plenty, North Island, New Zealand. The Kaituna River contributes the largest freshwater flow into the estuary through control gates. Lake Rotoiti and indirectly Lake Rotorua supply the base flow to the Kaituna River, with tributaries along the 50 km reach also significantly contributing to the flow. Water quality within the river is affected by elevated nutrients, faecal coliforms, high oxygen demand and algae concentrations derived from the lakes as well as contributions from tributaries and industrial and urban discharge. Through the use of a coupled hydrodynamic-biogeochemical numerical model ELCOM-CAEDYM, this study aims to examine the nutrient, phytoplankton and hydrodynamics of the Maketu Estuary and lower Kaituna River. 

Water quality and hydrodynamic measurements were sourced from Environment Bay of Plenty's data archives as well as a number of instrument deployments to collect water velocity, tidal elevation and salinity and temperature measurements during the course of this study. Included in the field work was a survey of the lower river and estuary bathymetry.

Model simulations predicted that the maximum residence time in the Maketu Estuary is 1.5 days, occurring in the inner western region.  Residence time in the lower river (mouth to 8.5 km upstream) is in the order of hours although some variations were predicted near the river mouth. Growth rates of four phytoplankton groups where assessed over a 15 day period in January 2004. In the Kaituna River ELCOM-CAEDYM predicted that the community growth rates were small with the exception of a slight increase in biomass of the two freshwater groups in a semi-detached river bend. The increase in the loop was correlated with an increase of residence time. In the estuary, marine diatoms showed the highest growth rates in the western region which is expected to relate to retention time and available nutrients. Dinoflagellates showed the smallest variation in predicted growth rates, most likely due to their broad salinity tolerance. The two freshwater species showed a reduction in abundance when mixed with marine water. A principle limiting factor to phytoplankton growth in both the river and estuary is the low residence time.

A number of scenarios were simulated in the river and estuary by altering the forcing conditions in the model. A simulation of the increased nutrient load associated with the Rotoiti diversion wall revealed that phytoplankton growth in the river and estuary will not be significantly affected. Because of the close proximity of the control gates to the river mouth, a proportion of water drawn through the structure can be marine. By opening the old river channel, model simulations predicted that a reduction in salinity would be possible, however the outcome of complete freshwater is probably not achievable. Increasing the discharge volume from the river into the estuary was also simulated. The results indicated that increasing the freshwater inflow at Fords Cut would reduce the salinity in the estuary while increasing the net (residual) flow towards the estuary mouth. Increasing the flow would also result in a greater range of salinity in regions of the estuary. Changing the inflow location to the historic Papahikahawai Channel also affected the salinity in the estuary. The most significant effect of an inflow at this location was a reduction of the residual currents in the western region of the estuary.

text: Hydrodynamic and water quality modelling of the lower kaituna river and maketu estuary The Maketu Estuary is a shallow intertidal estuary (2.3 km2) located in the Bay of Plenty, North Island, New Zealand. The Kaituna River contributes the largest freshwater flow into the estuary through control gates. Lake Rotoiti and indirectly Lake Rotorua supply the base flow to the Kaituna River, with tributaries along the 50 km reach also significantly contributing to the flow. Water quality within the river is affected by elevated nutrients, faecal coliforms, high oxygen demand and algae concentrations derived from the lakes as well as contributions from tributaries and industrial and urban discharge. Through the use of a coupled hydrodynamic-biogeochemical numerical model ELCOM-CAEDYM, this study aims to examine the nutrient, phytoplankton and hydrodynamics of the Maketu Estuary and lower Kaituna River. 

Water quality and hydrodynamic measurements were sourced from Environment Bay of Plenty's data archives as well as a number of instrument deployments to collect water velocity, tidal elevation and salinity and temperature measurements during the course of this study. Included in the field work was a survey of the lower river and estuary bathymetry.

Model simulations predicted that the maximum residence time in the Maketu Estuary is 1.5 days, occurring in the inner western region.  Residence time in the lower river (mouth to 8.5 km upstream) is in the order of hours although some variations were predicted near the river mouth. Growth rates of four phytoplankton groups where assessed over a 15 day period in January 2004. In the Kaituna River ELCOM-CAEDYM predicted that the community growth rates were small with the exception of a slight increase in biomass of the two freshwater groups in a semi-detached river bend. The increase in the loop was correlated with an increase of residence time. In the estuary, marine diatoms showed the highest growth rates in the western region which is expected to relate to retention time and available nutrients. Dinoflagellates showed the smallest variation in predicted growth rates, most likely due to their broad salinity tolerance. The two freshwater species showed a reduction in abundance when mixed with marine water. A principle limiting factor to phytoplankton growth in both the river and estuary is the low residence time.

A number of scenarios were simulated in the river and estuary by altering the forcing conditions in the model. A simulation of the increased nutrient load associated with the Rotoiti diversion wall revealed that phytoplankton growth in the river and estuary will not be significantly affected. Because of the close proximity of the control gates to the river mouth, a proportion of water drawn through the structure can be marine. By opening the old river channel, model simulations predicted that a reduction in salinity would be possible, however the outcome of complete freshwater is probably not achievable. Increasing the discharge volume from the river into the estuary was also simulated. The results indicated that increasing the freshwater inflow at Fords Cut would reduce the salinity in the estuary while increasing the net (residual) flow towards the estuary mouth. Increasing the flow would also result in a greater range of salinity in regions of the estuary. Changing the inflow location to the historic Papahikahawai Channel also affected the salinity in the estuary. The most significant effect of an inflow at this location was a reduction of the residual currents in the western region of the estuary.

year: 2007

title: The structure and average discrepancies of lattice rules for numerical integration

authors: Reddy, Muni Vasudewan

abstract: Lattice rules are equal-weight quadrature rules which are used in the approximation of multidimensional integrands over the s-dimensional unit cube [0,1]ˢ.  One of the problems encountered in the study of such rules is the unavailability of a unique representation. It is known that any lattice rule may be expressed in a canonical D - Z form in which D is a diagonal matrix whose diagonal entries are known as the invariants and Z is an integer matrix. Although D is unique in this canonical form, Z may be chosen in many different ways. Except for the case of so-called projection-regular and prime-power rules, no such unique Z is available. In the latter case of prime-power rules, the unique D - Z form developed is known as an ultratriangular form. Associated with each ultratriangular form is a set of column indices. Any lattice rule may be decomposed into prime-power components. In this thesis, a unique D - Z form is defined for a special class of lattice rules for which the component prime-power rules have a consistent set of column indices. This new unique form includes the known unique forms for projection-regular and prime-power rules as special cases. We also use the ultratriangular form for prime-power lattice rules to derive a formula to calculate the number of prime-power rules having a given set of invariants and column indices. 

The existing theory of lattice rules that is based on the generator matrix of the dual lattice has made the assumption that its representation in the so-called Hermite normal form is upper triangular. However, since projection-regular rules have a unique Z-matrix which is unit upper triangular, the corresponding generator matrix for the dual lattice is lower triangular. This suggests that a lower triangular Hermite normal form might be appropriate for study. We consider this situation and give the conditions on the lower triangular Hermite normal form which allow a projection-regular rule to be easily recognized. 

Number-theoretic rules are a class of lattice rules which are known to be particularly suitable for the approximation of multidimensional integrals in which the integrands are periodic. In the case of non-periodic integrands there is numerical evidence that the average L₂ discrepancy for these rules is smaller than the expected value for Monte-Carlo rules when the dimension s is less than 18. For non-periodic integrands, a vertex-modified version of the number-theoretic rule has been previously proposed. In s-dimensions these vertex-modified rules contain 2ˢ weights which may be chosen optimally so that the discrepancy is minimized. We shall compare the average discrepancy for these optimal vertex-modified number-theoretic rules with that for normal number-theoretic and Monte-Carlo rules. A similar comparison is also carried out between the averages for number-theoretic rules and for 2ˢ copy rules with approximately the same number of points. 

In the case of periodic integrands it has been shown that the average of Pα and the values of R for 2ˢ copy rules are smaller than those for number-theoretic rules. For this periodic case, we use an analogue of the L₂ discrepancy to carry out a similar comparison.

text: The structure and average discrepancies of lattice rules for numerical integration Lattice rules are equal-weight quadrature rules which are used in the approximation of multidimensional integrands over the s-dimensional unit cube [0,1]ˢ.  One of the problems encountered in the study of such rules is the unavailability of a unique representation. It is known that any lattice rule may be expressed in a canonical D - Z form in which D is a diagonal matrix whose diagonal entries are known as the invariants and Z is an integer matrix. Although D is unique in this canonical form, Z may be chosen in many different ways. Except for the case of so-called projection-regular and prime-power rules, no such unique Z is available. In the latter case of prime-power rules, the unique D - Z form developed is known as an ultratriangular form. Associated with each ultratriangular form is a set of column indices. Any lattice rule may be decomposed into prime-power components. In this thesis, a unique D - Z form is defined for a special class of lattice rules for which the component prime-power rules have a consistent set of column indices. This new unique form includes the known unique forms for projection-regular and prime-power rules as special cases. We also use the ultratriangular form for prime-power lattice rules to derive a formula to calculate the number of prime-power rules having a given set of invariants and column indices. 

The existing theory of lattice rules that is based on the generator matrix of the dual lattice has made the assumption that its representation in the so-called Hermite normal form is upper triangular. However, since projection-regular rules have a unique Z-matrix which is unit upper triangular, the corresponding generator matrix for the dual lattice is lower triangular. This suggests that a lower triangular Hermite normal form might be appropriate for study. We consider this situation and give the conditions on the lower triangular Hermite normal form which allow a projection-regular rule to be easily recognized. 

Number-theoretic rules are a class of lattice rules which are known to be particularly suitable for the approximation of multidimensional integrals in which the integrands are periodic. In the case of non-periodic integrands there is numerical evidence that the average L₂ discrepancy for these rules is smaller than the expected value for Monte-Carlo rules when the dimension s is less than 18. For non-periodic integrands, a vertex-modified version of the number-theoretic rule has been previously proposed. In s-dimensions these vertex-modified rules contain 2ˢ weights which may be chosen optimally so that the discrepancy is minimized. We shall compare the average discrepancy for these optimal vertex-modified number-theoretic rules with that for normal number-theoretic and Monte-Carlo rules. A similar comparison is also carried out between the averages for number-theoretic rules and for 2ˢ copy rules with approximately the same number of points. 

In the case of periodic integrands it has been shown that the average of Pα and the values of R for 2ˢ copy rules are smaller than those for number-theoretic rules. For this periodic case, we use an analogue of the L₂ discrepancy to carry out a similar comparison.

year: 2000

title: The "false generosity" of treaty settlements: innovation and contortion

authors: Te Aho, Linda

abstract: In Aotearoa New Zealand the ways in which indigenous claims to lands and waters are addressed are often looked to as good, or even best, practice by indigenous peoples around the world. While things are far from perfect, in recent decades Maori have succeeded in changing perceptions about the Treaty of Waitangi as the foundation of the nation, and work steadily continues on settling outstanding Treaty claims. The resulting Treaty of Waitangi settlements are negotiated arrangements which aim to remove a sense of historical grievance and achieve significant rebuilding of the Maori economy. They are seen by some as dynamic and powerful steps towards economic independence, as a means of recognising special relationships to lands and waters, and a necessary prerequisite to improved relationships between the state and the indigenous Maori in the future. ¹ Critics see the settlements and the processes followed to reach them as too heavily weighted in the government's favour. ² They argue that the settlements do not sufficiently compensate for actual losses. They are said to pit Maori against Maori. Diverse claimant groups are effectively forced to negotiate within standardised and fixed parameters. For these and other reasons, the settlement agreements, policies and processes have been labelled as divisive and compromising self-determination. ³

text: The "false generosity" of treaty settlements: innovation and contortion In Aotearoa New Zealand the ways in which indigenous claims to lands and waters are addressed are often looked to as good, or even best, practice by indigenous peoples around the world. While things are far from perfect, in recent decades Maori have succeeded in changing perceptions about the Treaty of Waitangi as the foundation of the nation, and work steadily continues on settling outstanding Treaty claims. The resulting Treaty of Waitangi settlements are negotiated arrangements which aim to remove a sense of historical grievance and achieve significant rebuilding of the Maori economy. They are seen by some as dynamic and powerful steps towards economic independence, as a means of recognising special relationships to lands and waters, and a necessary prerequisite to improved relationships between the state and the indigenous Maori in the future. ¹ Critics see the settlements and the processes followed to reach them as too heavily weighted in the government's favour. ² They argue that the settlements do not sufficiently compensate for actual losses. They are said to pit Maori against Maori. Diverse claimant groups are effectively forced to negotiate within standardised and fixed parameters. For these and other reasons, the settlement agreements, policies and processes have been labelled as divisive and compromising self-determination. ³

year: 2017