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# Temperature and solute-transport simulation in streamflow using a Lagrangian reference frame

January 1, 1980

A computer program for simulating one-dimensional, unsteady temperature and solute transport in a river has been developed and documented for general use. The solution approach to the convective-diffusion equation uses a moving reference frame (Lagrangian) which greatly simplifies the mathematics of the solution procedure and dramatically reduces errors caused by numerical dispersion. The solution procedure has the further advantages, relative to conventional Eulerian solution schemes, of being easy to understand in the physical sense, of being extremely stable numerically, and of providing an accounting system which is very useful for model calibration.

The model documentation is presented as a series of four programs of increasing complexity. The conservative transport model can be used to route a single conservative substance, such as dye, through a reach of a river. The simplified temperature model is used to predict water temperature in rivers, either with or without thermal loading, when few meteorological data are available. Only equilibrium temperature and windspeed are required. It is suggested that air temperature can be used to approximate equilibrium temperature. The complete temperature model is highly accurate but requires rather complete meteorological data. Finally, the 10-parameter model can be used to route as many as 10 interacting constituents through a river reach. The mathematical description of the interaction between the constituents, which does not need to be linear, is generally up to the user to supply. An example problem is solved for a three-parameter system involving temperature, dissolved oxygen, and biochemical oxygen demand.

For simplicity, all models are developed and presented assuming steady non-uniform flow. Generalization of the models to allow unsteady flow is extremely simple, involving the addition of no more than 18 cards to the program deck. The report is concluded by describing this generalization for any of the models. Before using the models with unsteady flow, a flow model must be used to calculate and store the necessary flow data at each cross section and time step. Such a flow model is available and documented.

Publication Year 1980 Temperature and solute-transport simulation in streamflow using a Lagrangian reference frame 10.3133/wri812 Harvey E. Jobson Report USGS Numbered Series Water-Resources Investigations Report 81-2 wri812 USGS Publications Warehouse