Air temperatures are sometimes used as substitutes for stream temperatures. To examine the errors associated with this procedure, linear relationships between stream temperatures, T, and air temperatures, Ta, recorded for 11 streams in the central U.S. (Mississippi River basin) were analyzed. Weather stations were an average 42 miles (range 0 to 144 miles) from the rivers. The general equations, Tw= 5.0 + 0.75 Ta and Tw= 2.9 + 0.86 Ta with temperatures in °C, were derived for daily and weekly water temperatures, respectively, for the 11 streams studied. The simulations had a standard deviation between measurements and predictions of 2.7°C (daily) and 2.1°C (weekly). Equations derived for each specific stream individually gave lower standard deviations, i.e., 2.1°C and 1.4°C, respectively. Small, shallow streams had smaller deviations than large, deep rivers. The measured water temperatures follow the air temperatures closely with some time lag. time lags ranged from hours to days, increasing with stream depth. Taking into account these time lags improved the daily temperature predictions slightly. Periods of ice cover were excluded from the analysis.

Stream temperature estimation from air temperature

Influence of uncertain boundary conditions and model structure on flood inundation predictions.

Experimental results are presented concerning the discharge characteristics, the boundary shear stress and boundary shear force distributions in a compound section comprising of one rectangular main channel and two symmetrically disposed flood plains. Equations are presented giving the shear force on the flood plains as a percentage of the total shear force in terms of two dimensionless parameters. The experimental shear force results are used to derive ancillary equations for the lateral and vertical transfer of momentum within the cross section. The apparent shear force acting on the vertical interface between one flood plain and the main channel is shown to increase rapidly for low relative depths and high flood plain widths. Equations are also presented giving the proportion of the total flow which occurs in the various sub areas. The division of flow based on linear proportion of the areas is shown to be inadequate on account of the interaction between the flood plain and main channel flows.

Flood Plain and Main Channel Flow Interaction

Since the development of the entropy theory by Shannon in the late 1940s and of the principle of maximum entropy (POME) by Jaynes in the late 1950s there has been a proliferation of applications of entropy in a wide spectrum of areas, including hydrological and environmental sciences. The real impetus to entropy-based hydrological modelling was provided by Amorocho and Espildora in 1972. A great variety of entropy applications in hydrology and water resources have since been reported, and new applications continue to unfold. This paper reviews the recent contributions on entropy applications in hydrology and water resources, discusses the usefulness and versatility of the entropy concept, and reflects on the strengths and limitations of this concept. The paper concludes with comments on its implications in developing countries. #1997 by John Wiley & Sons, Ltd.

The use of entropy in hydrology and water resources

Time-series for river gauging stations are core blue-skies and applied
research resources for understanding impacts of climate and anthropogenic
change on basin hydrology. River flow archives hold vital information
for evidence-based assessment of past hydrological variability,
and support hydrological modelling of future changes. River discharge
is an integration of basin input, storage and transfer processes to the
gauging point. It is important to set basin outlet data in regional to
global and long-term contexts: to better understand nested scales of
variability; to pinpoint locations and time periods most sensitive to climate
and human impacts; to make predictions for ungauged basins;
and to inform decision makers on water security issues, and where and
when to take measures to mitigate water hazards and stress, including
floods and droughts (Dai et al., 2009; Bonnell et al., 2006; Feyen &
Dankers, 2009; Haddeland et al., 2006; Hannah et al., 2005). Thus, there
is clear rationale for supporting large-scale (i.e. regional to continental to
global) river flow archives. Notable examples of such databases include
that held by the WMO Global Runoff Data Centre (GRDC) and the
UNESCO Flow Regimes from International Experimental and Network
Data (FRIEND) European Water Archive (EWA). For large-scale river
flow archives to be valuable research resources, they must be fit for
purpose. However, these databases are at risk due to a possible decline
in network coverage, associated time-series truncation, growing human
impact on (near-) natural flows, and increasingly restricted access to
national-scale data. This commentary aims: (1) to demonstrate largescale
river flow datasets are crucial to advance hydrological science and
solve operational issues; (2) to assess the current status of large-scale
river flow datasets; and (3) to propose ways forward to consolidate historical
data and secure future river flow data.

Large-scale river flow archives: importance, current status and future needs.

The concept of entropy based on probability has been applied in modeling the vertical distributions of the velocity, shear stress, and (suspended) sediment concentration in open‐channel flows. A velocity distribution equation derived by the entropy‐maximization principle has advantages over the Prandtl‐von Karman universal velocity distribution equation. The (maximized) entropy functions derived for the velocity distribution and sediment transport are effective in reflecting the effects of the size of suspended sediment, coarseness of bed material, and sediment concentration. They can, therefore, be used as variables for characterizing and comparing various open‐channel flows. Through their sensitivities to the various parameters needed to describe the flow and sediment transport characteristics, the entropy functions can also be used to obtain much‐needed, new equations for the estimation of parameters difficult to determine owing to insufficient numbers of equations available. The entropy concept provid...

Entropy and Probability Concepts in Hydraulics

Velocity distribution equations for open channel flows are derived and compared. These equations are derived by a combined application of: (1) A probabilistic formulation of the velocity distribution problem; (2) the entropy concept in the selection of the probability distribution function of velocity; (3) a geometrical technique in modeling a curvilinear coordinate and the coordinate transformation between this and the Cartesian coordinates; and (4) the basic hydrodynamics concerning the rates of transport of mass, momentum, and kinetic energy by the flow through an open channel cross section. A technique to estimate the parameters of these velocity distribution equations is also developed. The equations are capable of modeling and simulating the velocity distribution from the channel bed to the water surface, which may have the maximum velocity occurring on or below the water surface.

Velocity Distribution in Open Channel Flow

Equations based on the entropy concept have been derived for describing the two‐dimensional velocity distribution in an openchannel cross section. The velocity equation derived is capable of describing the variation of velocity in both the vertical and transverse directions, with the maximum velocity occurring on or below the water surface. Equations for determining the location of mean velocity have also been derived along with those, such as the entropy function, that can be used as measures of the homogeneity of velocity distribution in a channel cross section. A dimensionless parameter of the entropy function named the M number has been found useful as an index for characterizing and comparing various patterns of velocity distribution and states of open‐channel flow systems. The definition and demonstrated usefulness of this parameter indicate the importance and value of the information given by the location and magnitude of maximum velocity in a cross section, and suggest the need for future experime...

Entropy and 2-D velocity distribution in open channels

The maximum velocity in a channel cross section is highlighted with respect to its information content and relation to the mean velocity and entropy parameter. Under a wide range of discharge and water depth, a channel section seems to have propensity to establish and maintain an equilibrium state that corresponds to a value of entropy parameter. The entropy parameter of a channel section can be determined from the relation between the mean and maximum velocities. A technique has been developed to determine the discharge from a velocity profile on a single vertical passing through the point of maximum velocity in a channel cross section. The technique is an efficient way to estimate the discharge in streams and rivers, that can be used to continuously update the flow resistance during an unsteady flow, to enhance a filtering scheme designed to reduce uncertainties in flow forecasting. The use of the constant value of entropy parameter predetermined for a channel section can simplify the discharge estimation.

Maximum and Mean Velocities and Entropy in Open-Channel Flow

Maximum velocity in a channel section often occurs below the water surface. Its location is linked to the ratio of the mean and maximum velocities, velocity distribution parameter, location of mean velocity, energy and momentum coefficients, and probability density function underpinning a velocity distribution equation derived by applying the probability and entropy concepts. The mean value of the ratio of the mean and maximum velocities at a given channel section is stable and constant, and invariant with time and discharge. Its relationship with the others in turn leads to formation of a network of related constants that represent regularities in open-channel flows and can be used to ease discharge measurements and other tasks in hydraulic engineering. Under the probability concept, the ratio of mean and maximum velocities being constant means that the probability distribution underpinning the velocity distribution and other related variables is resilient, and that the same probability distribution is governing various phenomena observable at a channel section and explains the regularities in open-channel flows.

Chao-Lin Chiu

Papers

Entropy and Probability Concepts in Hydraulics

Velocity Distribution in Open Channel Flow

Entropy and 2-D velocity distribution in open channels

Maximum and Mean Velocities and Entropy in Open-Channel Flow

Maximum Velocity and Regularities in Open-Channel Flow