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Lake Erie is an enclosed, shallow sea with approximate mean dimensions of 60 feet in depth, 240 miles in length, and 40 miles in width. It is located in the region of confluence of the principal winter-time tracks associated with Alberta and Colorado lows, and therefore is exposed to wind action from severe cyclonic storms many of which reach their full intensity while well within range of influence upon the Lake. The associated wind tides are well known and in extreme cases have produced wind set-up in excess of 13 feet difference between Buffalo and Toledo at opposite ends of the longitudinal axis.
In this investigation numerical computations have been made for nine cases of record of extreme wind tide on Lake Erie. The computations are based upon an approximate, two-dimensional form of the Ekman boundary-layer equations, in which the viscous dimension is parameterized by an Ekman number. Effects of gravity, friction (with an eddy viscosity 40 cm2 sec−1) and the earth’s rotation are included.
The prediction equations are amenable to numerical integration by standard methods applicable to the momentum form of the dynamical equations; a pair of conjugate Richardson lattices is used for this purpose. Wind stress was obtained by an interpolation procedure based upon hourly surface-wind observations at six first-order stations located on the periphery of the Lake. A quadratic resistance formula with skin-friction coefficient 3.0 × 10−8 gave good results for computed wind set-up.
Although prediction of resurgences associated with the 14-hr free period was unsatisfactory, the average coefficient of correlation obtained between computed and observed set-up at various stations where hourly lake-level data are available is greater than 0.90. In general, the results of the investigation may be regarded as confirming that a sound basis exists for operational prediction of wind tides on Lake Erie by dynamical methods.

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... A different procedure has been employed by KIELMANN and KOWALIK [31]. The most elegant method has, however, been proposed by PLATZMAN 8 [43]. He treats the time derivative ∂(·)/∂t in (9.153) as if it were a real (or complex) number and writes (9.153) for constant ν as ...

... (9.174) Equation (9.173) is in the desired form. The four coefficient functions have been plotted by PLATZMAN [43] as functions of σ 0 . These graphs show that |D| is small in comparison to the moduli of the other coefficients for all values of σ 0 . ...

... The distinguished feature of this equation is that it has been obtained without taking side boundaries into account. In PLATZMAN's [43] words: 'An important step in delineating predictive characteristics of this equation is the determination of frequencies of free modes of vibration. In general, it is very difficult to do this if one requires that boundary conditions be met on the sides of a basin of even the simplest shape.' ...

In this chapter the intention is to describe the vertical and (eventually) also horizontal structure of the horizontal current
in lakes which are subjected to external wind forces. The water will be assumed to be homogeneous or stratified in two layers,
and the internal friction and the effects of the rotation of the Earth will play an important role in the establishment of
the current distribution.

... We are not interested in the temporal development of the flow field, which can be studied with time-varying solutions specifically designed for that purpose (e.g. Fjeldstad, 1930;Hidaka, 1933;Platzman, 1963;Madsen, 1977). The numerical results show that in many cases an equilibrium condition is achieved after a spin-up time that varies from some hours to some days of constant and uniform wind, a condition that many lakes experience from local breezes to large-scale synoptic winds. ...

... The path from Ekman's original solution to the description of steady currents in real enclosed basins was certainly long, but for the range of intermediate conditions several generalizations of the original Ekman theory have been proposed (see also Simons (1980), Hutter et al. (2011) and Defant (1962) for a review). Finite-depth and timedependent solutions followed the initial suggestion available in Ekman's work (Fjeldstad, 1930;Hidaka, 1933;Platzman, 1963;Madsen, 1977), together with the inclusion of horizontal pressure gradients associated to wind set-up (Welander, 1957;Birchfield, 1972), which were particularly relevant for the case of enclosed basins and were successfully applied to describe circulation in large lakes (Gedney and Lick, 1972). Additional complexity was added by taking into account density stratification (Lee and Ligget, 1970), with relevant contributions from estuarine studies (Kasai et al., 2000;Valle-Levinson et al., 2003), and non constant vertical turbulence (Thomas, 1975;Madsen, 1977;Svensson, 1979). ...

When investigating wind-induced steady circulation, the effect of the acceleration due to Earth's rotation is often neglected in narrow lakes, but the argument behind this assumption is blurred. Commonly, when the horizontal dimension is smaller than the Rossby radius, the Coriolis force is considered unimportant, but this is correct only for inertial currents and barotropic and baroclinic waves.
In this work, we revisit the classical Ekman transport solution for wind stress acting along the main axis of an elongated lake in steady-state conditions. We demonstrate that a secondary circulation develops and that the resulting crosswise volume transport, constrained in the closed domain, produces downwelling and upwelling that cannot be predicted by the standard Ekman formulas. We claim that the Rossby radius does not play any role in this process, which on the contrary is governed by the ratio between the actual depth and the thickness of the Ekman layer.
The theoretical analysis is supported by numerical experiments to show the dependence on latitude, width, depth and turbulence closure.

... We are not interested in the temporal development of the flow field, which can be studied with time-varying solutions specifically designed for that purpose (e.g. Fjeldstad, 1930;Hidaka, 1933;Platzman, 1963;Madsen, 1977). The numerical results show that in many cases an equilibrium condition is achieved after a spin-up time that varies from some hours to some days of constant and uniform wind, a condition that many lakes experience from local breezes to large-scale synoptic winds. ...

... The path from Ekman's original solution to the description of steady currents in real enclosed basins was certainly long, but for the range of intermediate conditions several generalizations of the original Ekman theory have been proposed (see also Simons (1980), Hutter et al. (2011) and Defant (1962) for a review). Finite-depth and timedependent solutions followed the initial suggestion available in Ekman's work (Fjeldstad, 1930;Hidaka, 1933;Platzman, 1963;Madsen, 1977), together with the inclusion of horizontal pressure gradients associated to wind set-up (Welander, 1957;Birchfield, 1972), which were particularly relevant for the case of enclosed basins and were successfully applied to describe circulation in large lakes (Gedney and Lick, 1972). Additional complexity was added by taking into account density stratification (Lee and Ligget, 1970), with relevant contributions from estuarine studies (Kasai et al., 2000;Valle-Levinson et al., 2003), and non constant vertical turbulence (Thomas, 1975;Madsen, 1977;Svensson, 1979). ...

When investigating wind-induced steady circulation, the effect of the acceleration due to Earth's rotation is often neglected in narrow lakes, but the argument behind this assumption is blurred. Commonly, when the horizontal dimension is smaller than the Rossby radius, the Coriolis force is considered unimportant, but this is correct only for inertial currents and barotropic and baroclinic waves. In this work, we revisit the classical Ek-man transport solution for wind stress acting along the main axis of an elongated lake in steady-state conditions. We demonstrate that a secondary circulation develops and that the resulting crosswise volume transport, constrained in the closed domain, produces down-welling and upwelling that cannot be predicted by the standard Ekman formulas. We claim that the Rossby radius does not play any role in this process, which on the contrary is governed by the ratio between the actual depth and the thickness of the Ekman layer. The theoretical analysis is supported by numerical experiments to show the dependence on latitude, width, depth and turbulence closure.

... This procedure appears most convenient for dealing with variable stratification and topography combined with baroclinicity. In the horizontal, the variables are staggered to form a lattice structure (Platzman, 1963). In a horizontal plane, the free surface, the vertical velocity, and the density are located at the center of squares, at the sides of which the velocity components are defined. ...

... As wind stress was used with a coefficient equal to 1.6 x l P 3 . A linear bottom stress, inversely proportional to the square of the depth with a coefficient of 50 cm2/s, was incorporated in the first run, approximating Platzman's (1963) solution of the Ekman equations. The results were averaged over 28 hours, eliminating inertial motions with periods of about 14 hours, as well as the major contributions of the free surface modes with periods a little over 1 day. ...

... Another approach to prescribing the bottom stress is to specify the vertical turbulent diffusion of momentum by a constant eddy viscosity . Platzman (1963) deduced a bottom friction coefficient as a function of the Ekman number ( d√f/2 ) in such a way that B → 0 for great depths and = 2.5 / 2 for shallow water. For Lake Erie, for instance, Platzman (1963) , and the components of the wind stress, τ s x and τ s y . ...

... Platzman (1963) deduced a bottom friction coefficient as a function of the Ekman number ( d√f/2 ) in such a way that B → 0 for great depths and = 2.5 / 2 for shallow water. For Lake Erie, for instance, Platzman (1963) , and the components of the wind stress, τ s x and τ s y . In principle, the atmospheric pressure gradients can be prescribed either from observations or from numerical weather prediction models. ...

In many countries, coastal dunes have proven environmentally and functionally efficient as important components of entire defence systems, which generally consist of both man-made and natural barriers against coastal flood and erosion. However, for dunes and natural barriers, such as sand pits and narrow islands, which may constitute an important part of the entire defence line, no systematic research has yet started to assess the safety under extreme storm conditions and the consequences of possible barrier breaching and overwash on the subsequent flooding and saltwater intrusion into the groundwater. Even the existing state-of-the-art prediction models for the erosion and breaching of dunes and coastal barriers (e.g. XBeach) still have serious limitations.
This report attempts to summarize the current knowledge on all relevant aspects related to the processes and models associated with (i) the storm surges, (ii) the initiation and development of breaches in dunes and similar coastal barriers induced by extreme water levels and waves, (iii) the flood wave propagation and inundation of the hinterland resulting from the barrier breaching, and (iv) the infiltration of saltwater into coastal aquifers and contamination of groundwater.
A particular focus is put on the coastal processes that might initiate a coastal barrier breaching either from the seaward, such as wave impact (wave collision), wave run-up, and run down, or from the landward, such as overtopping, overflow, combined flow, seepage, overwash, and washover processes. A brief overview on the state of the art breaching models, their capabilities and their limitation has been provided. Based on this overview, the XBeach model has been selected as the most appropriate model to study the breaching of coastal barriers under extreme storm surges. Very detailed overview of the XBeach model and its limitations has been presented in order to improve them in the future during this PhD study.
The coastal flooding of a hinterland has recognised as the induced result of the breaching. Therefore, the coastal flood risk and the flood propagation modelling has also discussed in order to predict the inundation extent and the resulting inundation depths. It is found that saltwater inundation, owing to coastal barrier breaching, results in aquifers salinity. Furthermore, natural remediation process of contaminated aquifers takes up to 20 years. Consequently, groundwater contamination as intangible and indirect damage that results from coastal flooding is suggested to be added to the whole flood damages in order to precisely estimate coastal flood risks.
In order to overcome the variety of processes (breaching, inundation, and infiltration and solute transport) and to simplify the flow and transport boundary conditions from process to another, the likelihood of coupling the modelling of these processes has been mentioned. It is found that coupling the modelling of these processes is very likely through XBeach.

... This procedure appears most convenient for dealing with variable stratification and topography combined with baroclinicity. In the horizontal, the variables are staggered to form a lattice structure (Platzman, 1963). In a horizontal plane, the free surface, the vertical velocity, and the density are located at the center of squares, at the sides of which the velocity components are defined. ...

... As wind stress was used with a coefficient equal to 1.6 x l P 3 . A linear bottom stress, inversely proportional to the square of the depth with a coefficient of 50 cm2/s, was incorporated in the first run, approximating Platzman's (1963) solution of the Ekman equations. The results were averaged over 28 hours, eliminating inertial motions with periods of about 14 hours, as well as the major contributions of the free surface modes with periods a little over 1 day. ...

Wind-driven circulations in the southwest Baltic are analyzed with reference to the combined effects of topography and stratification. Numerical results from a multi-layer nested model are compared with observations taken during the spring of 1975. It appears that the bathymetry and stratification of the southwest Baltic lead to substantial coupling of topographic and baroclinic effects.

... A series of studies have been carried out to quantify, understand and model wind tides in the 45 field. Among them, we can cite Kenney (1979) in Lake Winnipeg in Canada , Hellström (1941); Platzman (1963); Keulegan (1953); Gillies (1959) in Lake Erie in Canada and the United-States, Farrer (1957); Kivisild (1954); Saville (1952) in Lake Okeechobee in the United-States, Harris (1957); Platzman (1965); Hugues (1965) in Lake Michigan in the United-States, Nomitsu (1935); Hayami et al. (1996) in Lake Biwa in Japan, Nomitsu (1935) in Lake Kariba in Zambia and Zimbabwe, 50 De Lauro et al. (2018) in Venice lagoon in Italy or Metler et al. (1975) in Lake Balaton in Hungary. ...

The present paper is specifically focused on enclosed or semi-enclosed basins where the wind is the dominant driver of water surface tilting, leading to the so-called wind tide contributing to water levels rise. Wind-induced free surface tilting is studied using the 1-D steady form of the depth-averaged shallow water (Saint-Venant) momentum equation which reflects the depth-averaged local balance between surface slope and wind stress. Two contrasted field sites, the Berre and Vaccarès lagoons, have been monitored providing water level data along a reference axis. This study highlighted the occurrence of wind tides at the two field sites. The bimodal wind exposure ensured the robustness of the observations, with non-linear but symmetric behaviors patterns observed in winds from opposite directions. It is observed that the higher the wind speed, the steeper the slope of the free surface in accordance with the well known basic trend. In addition, a significant effect of depth is observed, with greater surface tilting in the shallower lagoon. The data analysis confirmed the robustness of such a simple approach in the present context. Using the additional assumption of constant, i.e. wind-independent, drag coefficients (CD) allowed a good match with the observations for moderate wind speeds for both sites. However, the depth effect required the CD to be increased in the shallower basin. Classical empirical wind-dependent CD parameterizations provide better wind-tide predictions than the constant-CD approach in very strong wind conditions but totally failed in predicting surface tilting in the shallower site, suggesting that physical parameters other than wind speed should be taken into account for the CD parameterization in very shallow lagoons.

... While hurricane-induced wind set-up events are more extreme than the types of events involved in the 2011 Richelieu River flooding, Chimney (2005) suggests that the use of calibrated relatively simple steady-state models can lead to satisfactory results, even in a similar context. A study by Platzman (1963) used more complex dynamic models to analyse the effects of the wind on wind set-up on Lake Erie. ...

The precision of Lake Champlain's water level estimation is a key component in the flood forecasting process for the Richelieu River. Hydrological models do not typically take into consideration the effects of the wind on the water level (also known as the wind set‐up). The objective of this study is to create an empirical wind set‐up forecast model for Lake Champlain during high wind events. The proposed model uses wind speed and direction across the Lake, as well as wind gusts as inputs. The model is calibrated to a subset of observations and evaluated on an independent sample, considering four wind speed bins. It is tested and compared to a variant of the Zuider Zee equation on twenty wind set‐up events that occurred between 2017 and 2019 using hindcast data from five different numerical weather prediction systems (GDPS, RDPS, HRDPS, NOAA and ECMWF). A quantile mapping‐based forecast calibration scheme is implemented for each of the forecast products to correct their biases. Results show that events are successfully predicted by the proposed model at least 72 hours in advance. These results are better than the other comparative models found in the literature and tested herein. Overall, significant improvements are obtained by including wind speed and wind gusts from different weather stations.
This article is protected by copyright. All rights reserved.

... The aim of the study is to understand the wind effect on mean water level variation in semienclosed shallow basins. The studied physical phenomenon is nearly steady water surface tilting due to wind stress, the so-called wind tide (Platzman (1963)). During strong wind conditions, wind tides can have significant consequences on low-lying areas such as submersion and flooding. ...

The aim of the study is to understand the wind effect on mean water level variation in semi-enclosed shallow basins. The studied physical phenomenon is nearly steady water surface tilting due to wind stress, the so-called wind tide (Platzman (1963)). During strong wind conditions, wind tides can have significant consequences on low-lying areas such as submersion and flooding. Two field sites are monitored in the S-E of France to characterize wind tides and more specifically to understand the relative effect of wind magnitude and depth on the mean water level dynamics.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/Q30I0taty9w

... The second group predicts storm surge numerically by using fluid dynamics and atmospheric driving forces [15]. Two-dimensional depth-integrated hydrodynamic models in the Cartesian structure grid system are commonly utilized to simulate storm surge, where governing equations are nonlinear shallow water equations with Boussinesq and hydrostatic approximations (e.g., [23]). The driving forces (i.e., pressure and wind fields) can be specified using parametric cyclone models [24,25] for hindcast/diagnostic purposes. ...

Storm surge induced by severe typhoons has caused many catastrophic tragedies to coastal communities over past decades. Accurate and efficient prediction/assessment of storm surge is still an important task in order to achieve coastal disaster mitigation especially under the influence of climate change. This study revisits storm surge predictions using artificial neural networks (ANN) and effective typhoon parameters. Recent progress of storm surge modeling and some remaining unresolved issues are reviewed. In this paper, we chose the northeastern region of Taiwan as the study area, where the largest storm surge record (over 1.8 m) has been observed. To develop the ANN-based storm surge model for various lead-times (from 1 to 12 h), typhoon parameters are carefully examined and selected by analogy with the physical modeling approach. A knowledge extraction method (KEM) with backward tracking and forward exploration procedures is also proposed to analyze the roles of hidden neurons and typhoon parameters in storm surge prediction, as well as to reveal the abundant, useful information covered in the fully-trained artificial brain. Finally, the capability of ANN model for long-lead-time predictions and influences in controlling parameters are investigated. Overall, excellent agreement with observations (i.e., the coefficient of efficiency CE > 0.95 for training and CE > 0.90 for validation) is achieved in one-hour-ahead prediction. When the typhoon affects coastal waters, contributions of wind speed, central pressure deficit, and relative angle are clarified via influential hidden neurons. A general pattern of maximum storm surge under various scenarios is also obtained. Moreover, satisfactory accuracy is successfully extended to a much longer lead time (i.e., CE > 0.85 for training and CE > 0.75 for validation in 12-h-ahead prediction). Possible reasons for further accuracy improvement compared to earlier works are addressed.

... Storm surge is the combined effect of (a) low atmospheric pressure allowing water to increase in elevation, this effect is significant in large storms (Feagin et al. 2010) and (b) wave-induced set up, as driven by radiation stress (Longuet-Higgins and Stewart 1962). Wind tides are changes in water level that occur within enclosed basins, such that the wind drives currents which push water into or out of the basins (Keulegan 1951;Platzman 1963). Surge is predominantly a regional-scale process -to model it requires knowledge of atmospheric pressure, waves, wind, and bathymetry. ...

Publicly available tidal predictions for coastlines are predominantly based on astronomical predictions. In shallow water basins, however, water levels can deviate from these predictions by a factor of two or more due to wind-induced fluctuations from non-regional storms. To model and correct these wind-induced tidal deviations, a two-stage empirical model was created: the Enhanced Tidal Model (ETM). For any given NOAA tide gauge location, this model first measured the wind-induced deviation based on a compiled dataset, and then adjusted the astronomical predictions into the future to create a 144-hour forecast. The ETM, when incorporating wind data, had only 76% of the error of NOAA astronomical tidal predictions (e.g. if NOAA had 1.0 ft. of error, ETM had only 0.76 ft. error from the observed water level). Certain ETM locations had approximately half (49%) as much prediction error as NOAA. With the improvement in tidal accuracy prediction, the ETM has the ability to significantly aid in navigation along with coastal flood prediction. We envision the ETM as a resource for industry and the public to make informed decisions that impact their livelihood.

... The SPLASH model has been partially documented in three publications, the mathematical technique in Jelesnianski (1967), and three operational techniques to run and interpret the results for forecasting purposes, Jelesnianski (1972Jelesnianski ( , 1976. The mathematical techniques are adapted from Platzman (1963). The tropical storm model used to generate surges is discussed in Jelesnianski and Taylor (1973). ...

A significant portion of the damage by hurricanes is the storm surges. The National Weather Service has developed a dynamical-numerical model to forecast hurricane storm surges. The model is used operationally for prediction, warning, and planning purposes. The model requires fixed oceanographic and real time meteorological input data. The oceanographic data were prepared for the Gulf and East coasts of the U.S. and are stored as an essential part of the program. Meteorological data for any tropical storm are supplied by the forecasters or planners using the model. The model was applied to hurricane Camille 1969. Comparison between the observed and computed surges for Camille was satisfactory for prediction purposes.

... The majority of the research on surface seiche damping has been based upon the measurements made by Endrös. Despite numerous reproductions and reference to these measurements [3,[14][15][16][17][18] no verification of the values he obtained, reproduced in "Appendix 1" (Table 4), has ever been carried out. This paper seeks to present a method through which the damping ratios and periods of surface seiches may be extracted using existing data without the need for labor intensive and subjective visual inspection of water elevation records. ...

Basin scale seiches in lakes are important elements of the total energy budget and are a driver of fluxes of important ecological parameters, such as oxygen, nutrients, and sediments. At present, the extraction of the damping ratios of surface seiches, which are directly related to the capacity of seiches to drive these fluxes through the increased mixing of the water column, is reliant on spectral analysis which may be heavily influenced by the transformation of water level records from the time domain to the frequency domain, and which are sensitive to the level of noise present within the data. Existing spectral-based methods struggle to extract the periods of surface seiches which are of similar magnitude due to the overlap between their spectral responses. In this study, the principles of operational modal analysis, through the random decrement technique (RDT), currently used primarily in the analysis of high rise structures and in the aeronautical industry and not previously applied within the fields of limnology or ecology, are applied to barotropic seiches through the analysis of water level data for Lake Geneva, Switzerland, and Lake Tahoe, USA. Using this technique, the autocorrelation of the measurements is estimated using the RDT and modal analysis can then be carried out on this time-domain signal to estimate periods of the dominant surface seiches and the corresponding damping ratios. The estimated periods show good agreement with experimental results obtained through conventional spectral techniques and consistent damping ratios are obtained for the dominant surface seiche of Lake Tahoe. The effect of input parameters is discussed, using data for the two lakes, alongside discussion of the application of RDT to the study of internal seiches and current barriers to its application. RDT has great potential for the analysis of both surface and internal seiches, offering a method through which accurate damping ratios of seiche oscillations may be obtained using readily available data without necessitating spectral analysis.

... For the first time, a viscous damping effect in the sea current dynamics was studied by Ekman (1905), then by Welander (1957), Platzman (1963), Jelesnianski (1970), and Jordan and Baker (1980). Mofjeld (1980) also investigated the influence of vertical viscosity on the propagation of barotropic Kelvin wave. ...

Effects of the viscosity, Earth rotation, and sphericity (beta-effect) on the long-wave dynamics are investigated based on the linear model. The basic equation for the complex amplitudes of gravitational long waves is obtained. It is shown that the viscosity plays a significant role in the long-wave dynamics. Stokes’ layer thickness is the criterion which separates two regimes of long-wave evolution: low viscosity and viscous flows. Two Stokes’ layers occur in the rotating fluid. The thickness of the first approaches to infinity as the frequency tends to inertial frequency. Considering the role of the Stokes’ layer as a criterion of viscosity influence, we can conclude that for the waves of the near-inertial frequency, viscosity always plays a significant role irrespective of ocean depths. The beta-effect leads to the planetary drift velocity occurrence in the equation. The planetary drift velocity can have either eastward or westward direction depending on the wave frequency. Thus, Earth sphericity causing the planetary drift plays a major role in the asymmetry of the eastward and westward directions in wave dynamics. Friction is another reason for the asymmetry of the eastward and westward directions in wave dynamics. Damping decrements of the westward and eastward waves differ with the biggest difference for waves with the near-inertial frequencies. Group velocities of eastward and westward waves are nonsymmetrical too. Moreover, in a certain range of the near-inertial frequencies, group velocities of both westward and eastward waves are directed exceptionally eastward. Thus, the beta-effect and fluid viscosity can be the reasons for the asymmetry of western and eastern bays in the tidal wave dynamics.

... The horizontal transport equations are solved through the application of the Navier-Stokes momentum equations for incompressible and turbulent flow and solved for the pressure, Coriolis and frictional forces every time step. The governing equations are integrated over the entire depth of the water column as derived by Platzman 1963, in which the dissipation is determined solely by an eddy viscosity coefficient, and modified with a bottom slip coefficient proposed by Jelesniansky (1967). The bottom stress is not determined by the depth-averaged velocity. ...

In this study, the first ever Sea, Lake, Overland Surges from Hurricanes (SLOSH) grid was built for the Yucatan Peninsula. The SLOSH model was used to simulate storm surges in the coastal area of the states of Yucatan and Campeche (Mexico). Based on climatology, more than 39,900 hypothetical hurricanes covering all possible directions of motion were synthesized. The storm intensity (category), forward speed, radius of maximum winds and the tide anomaly were varied for each hypothetical track. According to these scenarios, the potential storm surge and associated inundation threat were computed. Subsequently, the Maximum Envelope of Water (MEOW) and the Maximum of the MEOWs (MOMs) were calculated to assess the flood hazard induced by tropical cyclones under varying conditions. In addition, for each MOM, the socioeconomic vulnerability aspects were taken into account in order to assess the hurricane flood risk for the states of Yucatan and Campeche. Results show that the most vulnerable areas are the surroundings of Terminos lagoon, Campeche City and its neighboring areas in the state of Campeche. For Yucatan, the towns located in the Northwest (Celestun, Hunucma and Progreso) and the eastern part of the state presented the highest risk values. The methodology used in this study can be applied to other coastal zones of Mexico as well as places with similar attributes. Furthermore, the MEOW and MOM are very useful as a decision-making tool for prevention, preparedness, evacuation plans, mitigation of the flood hazard and its associated risk, and also for insurance companies.

... For Lake Ontario, Simons (1973a, b) gave a value of 2 9 10 4 -4 9 10 4 cm 2 s -1 in shallow as well as deep water, and this leads to B = 50/D 2 to 100/D 2 in C.G.S. units. Another approach for prescribing the bottom stress is to specify the vertical turbulent diffusion of momentum by a constant eddy viscosity v. Platzman (1963) deduced a bottom friction coefficient as a function of the Ekman number, D ffiffiffiffiffiffiffiffiffi f =2v p , in such a way that B ? 0 for great depths and B = 2.5v/D 2 for shallow water. For Lake Erie, Platzman took v = 40 cm 2 s -1 , which gives B = 100/D 2 in C.G.S. units. ...

Cyclone tracks over the Great Lakes of North America shift, both East–West as well as North–South. The reasons for the shifts are various small-scale as well as large-scale processes associated with the general circulation of the atmosphere. The East–West shift has an approximate periodicity of 10 years, while the North–South shift occurs roughly with a periodicity of 20 years. The East–West shift is more important than the North–South shift. The amount of shift could be as much as a few hundred kilometers. The implication of these shifts for storm surges in the Great Lakes is considered.

... Taylor 1919 [9]. Ramming and Kowalik 1980 [7]; however, recall that (23.5) 2 is only one possible form of possible functional relations between R and u, see in particular Platzman's parameterization [6], treated in detail in Vol. I, Chap. ...

In this chapter the intention is to describe the horizontal velocity distribution in a homogeneous lake by the spatially three-dimensional dynamical equations, based on the hydrostatic pressure assumption on the one hand, and their spatially two dimensional depth integrated reduction on the other hand. Comparison of the two sets of solutions for wind forcing, uniform in space and Heaviside in time, from various directions discloses the conditions when the depth averaged equations likely yield valid approximations of the three dimensional situation. Lake Zurich is used as an example. The extensive computations reveal that the problem of approximate determination of the barotropic velocity distribution in a homogeneous lake needs careful scrutiny. We shall analyze this problem by applying layered versions of the equations of motion in the hydrostatic pressure assumption to Lake Zurich and comparing the wind-induced current distribution obtained for a number of wind scenarios of a one layer and an eight-layer model. It may be deduced that depth integrated models deliver horizontal currents in homogeneous lakes of extremely shall depths (ca 5 m) only.

... The equations of motion in the Cartesian coordinate system used in SLOSH were first developed by Platzman (1963) and later modified by Jelesnianski (1967) to include a bottom slip coefficient: ...

Major coastal cities, which have large populations and economies, easily suffer from the losses due to hurricane wind and storm surge hazards. Although current design codes consider the joint occurrence of high wind and surge, information on site specific joint distributions of hurricane wind and storm surge along the U.S. Eastern Coast and Gulf of Mexico is still sparse and limited. In this paper, joint probability distributions of combined hurricane wind and storm surge for the City of Charleston, SC is developed. A stochastic hurricane model was used to simulate 5,000 years of synthetic hurricanes. The simulated hurricanes were inputted into the ADCIRC (Advanced Circulation) surge prediction model to compute the surge heights at selected locations. The calculated peak wind speeds and surge heights were employed to generate the joint probability distributions at each location. These joint distributions developed can be used in a multi-hazard design or risk assessment framework to consider the combined effects of hurricane wind and storm surge hazards.

... The equations of motion in the Cartesian coordinate system used in SLOSH were first developed by Platzman (1963) and later modified by Jelesnianski (1967) to include a bottom slip coefficient: ...

Combined effects of hurricane wind and surge can pose significant threats to coastal cities. Although current design codes consider the joint occurrence of wind and surge, information on site-specific joint distributions of hurricane wind and surge along the US Coast is still sparse and limited. In this study, joint hazard maps for combined hurricane wind and surge for Charleston, South Carolina (SC), were developed. A stochastic Markov chain hurricane simulation program was utilized to generate 50,000 years of full-track hurricane events. The surface wind speeds and surge heights from individual hurricanes were computed using the Georgiou’s wind field model and the Sea, Lake and Overland Surges from Hurricanes (SLOSH) model, respectively. To validate the accuracy of the SLOSH model, the simulated surge levels were compared to the surge levels calculated by another state-of-the-art storm surge model, ADCIRC (Advanced Circulation), and the actual observed water elevations from historical hurricane events. Good agreements were found between the simulated and observed water elevations. The model surface wind speeds were also compared with the design wind speeds in ASCE 7-10 and were found to agree well with the design values. Using the peak wind speeds and maximum surge heights, the joint hazard surfaces and the joint hazard maps for Charleston, SC, were developed. As part of this study, an interactive computer program, which can be used to obtain the joint wind speed and surge height distributions for any location in terms of latitude and longitude in Charleston area, was created. These joint hazard surfaces and hazard maps can be used in a multi-hazard design or risk assessment framework to consider the combined effects of hurricane wind and surge.

... By an analysis similar to the one discussed by Platzman [6], the stability condition for system ' ...

A system of finite difference equations for storm surge prediction has been constructed, using forward time differences. The scheme was tested for special simple geometrical configurations, and it was found to be stable without intro- ducing smoothing operators. The variation with time of the total energy was, in each case, the test of stability. The small-scale oscillation of the energy with time (characteristic of forward difference schemes) was studied in detail. A method of reducing this effect is suggested. A completely implicit finite difference scheme is discussed from the point of view of stability and convergence. It is shown how the requirement of a convergent iterative process actually introduces a severe restriction on the ratio ALIAS, thus canceling the advantages of the otherwise unconditionally stable implicit schemes.

... Most lakes are, however, not sufficiently large that the rotation of the Earth would in any even marginal form, play a role in the oscillation characteristics of homogeneous lakes. 2 The Great Lakes, the Baltic Sea, the (semi)-bounded ocean basins (say, Adriatic Sea in the Mediterranean Sea, Black Sea, Lake Baikal, the Caspian Sea) are safe candidates where the rotation of the Earth plays a significant role. References on these are by Mortimer [35,37,38], in particular with emphasis on Lakes Michigan and Superior by Mortimer and Fee [42] and Mortimer [41], on Lake Erie by Platzman [44] and Platzman and Rao [50], on Lakes Ontario and Superior by Rao and Schwab [52] and on Lake Huron by Schwab and Rao [57]. Platzman [48] reports on a barotropic seiche analysis of the Atlantic and Indian Oceans and [49] and on gravitational seiches of the entire World Ocean. ...

We have already given a detailed description of rotation affected external and internal waves in idealized containers of constant
depth: straight channels, gulfs, rectangles and circular and elliptical cylinders. Pure Kelvin and Poincaré waves were shown
to describe the oscillating motion in straight channels and their combination yielded the solution of the reflection of the
rotation affected waves at the end wall of a rectangular gulf. The typical characterizations of Kelvin and Poincaré waves
were seen to prevail (with some modification) in the fluid motion of rotating circular and elliptical cylinders with constant
depth. The behaviour was termed Kelvin-type if for basin-scale dynamics the amplitudes of the surface and isopycnal displacements
and velocities are shore-bound (i.e. large close to the boundaries and smaller in the interior of the basin), the motion cyclonic
(that is counter-clockwise on the N.H.) and frequencies sub- or (less often) superinertial. Alternatively, for Poincaré-type
behaviour, the surface and isopycnal displacements and velocities have large amplitudes in off-shore regions, their motion
is anti-cyclonic and frequencies are strictly superinertial.

... Such investigations have been carried out by Hansen (1956) and Miyazaki (1965). Instead of treating bottom stress as an extrapolation of present forces, Platzman (1963) considered the time history of present and past forces using Ekman's theory and derived a diflerentd operator for bottom stress in series form. Jelesnianski (1967) used a modification of this scheme in numerical computations of hurricanegenerated storm surges along coastal areas. ...

A transient Ekman's transport equation, in which bottom stress is formed as a convoluted integral in terms of surface stress and surface slope, and a continuity equation are used as predictors to compute storm surges in a model basin. Driving forces in the basin are analytically computed, using a model storm to represent actual meteorological conditions. A coastal boundary condition that relates surface slope to surface stress is developed by balancing slope and drift transports normal to a vertical wall. At interior grid points of the basin, sea-surface heights are computed by numerical means, using the prediction equations. These sea-surface heights are then extrapolated to the coast to agree with the coastal surface slope given by the boundary condition. Coastal storm surges computed in this manner are compared with observed su-%ges to test the model developed in this study.

... The above scheme is centrally differenced in time and uses the Jacobian difference operators devised by Arakawa (1966) to ensure that various integral constraints on the convective terms of the differential equations are preserved in the finite-difference formulation. Finally, the diffusion terms are evaluated at the preceding time level, as indicated by the subscript n-1, to prevent computational instability arising from these terms (Platzman 1963). The numerical procedure consists of successively solving (8)-( 10) until the desired state of flow development, usually the steady state, has been reached. ...

Earlier models of fire plumes based on simple entrainment laws and neglecting dynamic pressure have failed to produce the relatively shallow inflow over the fire perimeter known as fire wind. This inflow is of prime importance in fire modelling as it normally provides much of the air required for combustion; for this reason we have carried out a very simple numerical experiment on two-dimensional natural convection above a strip heat source with the intention of simulating those aspects of fire behaviour involved in the generation of fire wind without attempting the formidably difficult task of detailed fire modelling. Our results show clearly that fire wind is driven by the dynamic pressure field which is generated by and intimately related to the region of strong buoyant acceleration close above the ground boundary. Throughout our parametric range there is a concentrated region of large horizontal pressure gradient in a neighbourhood above the perimeter of the fire, and elsewhere the pressure gradients play a lesser role.

... If these constraints arenot observed computational instability will arise owing to the uncontrolled aliasing of long and short wavelengths (Phillips 1959). The diffusion terms also need careful treatment and Platzman (1963) has shown that the diffusion terms of (12) and (13) must be evaluated a t times (n-1) At to ensure (linear) computational stability of the difference equations. In the notation of Lilly (1964) this is indicated by the subscript 'lag'. ...

Amongst the more important laboratory experiments which have produced concentrated vortices in rotating tanks are the sink experiments of Long and the bubble convection experiments of Turner & Lilly. This paper describes a numerical experiment which draws from the laboratory experiments those features which are believed to be most relevant to atmospheric vortices such as tornadoes and waterspouts.
In the numerical model the mechanism driving the vortices is represented by an externally specified vertical body force field defined in a narrow neighbourhood of the axis of rotation. The body force field is applied to a tank of fluid initially in a state of rigid rotation and the subsequent flow development is obtained by solving the Navier–Stokes equations as an initial-value problem.
Earlier investigations have revealed that concentrated vortices will form only for a restricted range of flow parameters, and for the numerical experiment this range was selected using an order-of-magnitude analysis of the steady Navier–Stokes equations for sink vortices performed by Morton. With values of the flow parameters obtained in this way, concentrated vortices with angular velocities up to 30 times that of the tank are generated, whereas only much weaker vortices are formed at other parametric states. The numerical solutions are also used to investigate the comparative effect of a free upper surface and a no-slip lid.
The concentrated vortices produced in the numerical experiment grow downwards from near the top of the tank until they reach the bottom plate whereupon they strengthen rapidly before reaching a quasi-steady state. In the quasi-steady state the flow in the tank typically consists of the vortex at the axis of rotation, strong inflow and outflow boundary layers at the bottom and top plates respectively, and a region of slowly-rotating descending flow over the remainder of the tank. The flow is cyclonic (i.e. in the same sense as the tank) in the vortex core and over most of the bottom half of the tank and is anticyclonic over the upper half of the tank away from the axis of rotation.

Hurricanes and other extreme precipitation events can have devastating effects on population and infrastructure that can create problems for emergency responses and evacuation. Projected climate change and associated global warming may lead to an increase in extreme weather events that results in greater inundation from storm surges or massive precipitation. For example, record flooding during Hurricane Katrina or, more recently, during Hurricane Harvey in 2017, led to many people being cut off from aid and unable to evacuate. This study focuses on the impact of severe weather under climate change for areas of Harris County, TX that are susceptible to flooding either by storm surge or extreme rainfall and evaluates the transit demand and availability in those areas. Future risk of flooding in Harris County was assessed by GIS mapping of the 100-year and 500-year FEMA floodplains and most extreme category 5 storm tide and global sea level rise. The flood maps have been overlaid with population demographics and transit accessibility to determine vulnerable populations in need of transit during a disaster. It was calculated that 70% of densely populated census block groups are located within the floodplains, including a disproportional amount of low-income block groups. The results also show a lack of transit availability in many areas susceptible to extreme storm surge exaggerated with sea level rise. Further study of these areas to improve transit infrastructure and evacuation strategies will improve the outcomes of extreme weather events in the future.

This chapter begins with studying steady state layer flows through cylindrical conduits (ellipse, triangle, rectangle) and use of the Prandtl membrane analogy. This study of the Navier-Stokes fluids is important in geophysical fluid dynamics and is manifest in Ekman’s theory and its extensions, where non-inertial effects chiefly influence the details of the fluid flow, evidenced in the Ekman spiral in atmospheric and oceanic boundary flows and in free geostrophic flows as their outer solutions. Extensions of the behavior exhibited by the assumption of constant (turbulent) viscosity are based on influences of depth dependence of the viscosity which influences the circulation pattern of such steady flows. Unsteady flows are analyzed for viscous flows along an oscillating wall and the growth of a viscous boundary layer as a response of an initial tangential velocity jump with time. The chapter closes with the study of an axial laminar jet and viscous flows in a converging two-dimensional channel.

This article attempts to present the state of the art of the finite-element modeling techniques in surface and subsurface hydraulic problems. Fundamentals of finite-element techniques are introduced and presented based on the convective-dispersion equation. Before discussing any applications, the author shows the Rayleigh-Ritz and the Galerkin finite-element formulations, their equivalence, and their essential difference. He examines in detail the finite-element approximations, and outlines the finite-element algorithm for preparing a computer code.

We review experimental and theoretical studies of linear and nonlinear internal fluid waves and argue that their discovery is based on a systematic development of thermometry from the early reversing thermometers to the moored thermistor chains. The latter (paired with electric conductivity measurements) allowed development of isotherm (isopycnal) time series and made the observation of large amplitude internal waves possible. Such measurements (particularly in the laboratory) made identification of solitary waves possible and gave rise to the emergence of very active studies of the mathematical description of the motion of internal waves in terms of propagating time-dependent interface motions of density interfaces or isopycnal surfaces. As long as the waves remain stable, i.e., do not break, they can mathematically be described for two-layer fluids by the Korteweg-de Vries equation and its generalization. When the waves break, the turbulent analogs of the Navier–Stokes equations must be used with appropriate closure conditions to adequately capture their transformation and flux of matter to depth, which is commonly known as meromixis.

Storm surges are the world’s foremost natural hazard. The global storm surge problems are reviewed, starting with global climatology, the tracks of tropical and extra-tropical cyclones, and the region where major surges occur. The storm surge prediction problems discussed from a mathematical point of view and the input data, boundary conditions and the meteorological forcing terms are explained. The particular uses of various types of grids is elucidated; special problems such as inclusion of tidal flats and ice cover are considered. The similarities and differences between tropical cyclone-generated surges and extra-tropical cyclone-generated surges are discussed.

The fundamental physical principles governing the motion of lake waters are the conservation laws of mass, momentum and energy. When diffusion processes of active or passive tracer substances are also considered, these laws must be complemented by transport equations of tracer mass. All these statements have the form of balance laws and in each of them flux terms arise, for which, in order to arrive at field equations, phenomenological postulates must be established. Hydrodynamics of lakes can be described by a Navier-Stokes-Fourier-Fick fluid or its simplifications. Its field equations are partial differential equations for the velocity vector v, the pressure p, the temperature T and, possibly, the mass concentrations cα (α =1, ..., N) of N different tracers (i.e. a suspended sediment, phosphate, nitrate, salinity, etc.). Boundary conditions for v, p, T and cα must also be established; in view of the fact that surfaces may deform and that evaporation may occur, these are not alltogether trivial. In fact the equations of motion of the free or of internal surtaces of density discontinuity — these are the so-called kinematic surface equations — serve as further field equations with the surface displacements as unknown boundary variables. Additional boundary conditions have to be formulated at the lake bottom and along the shore. The latter play a more significant role in physical limnology than in oceanography because for many phenomana the boundedness of the lake domain will affect the details of the processes while oceans may for the same processes be regarded as infinite or semi-infinite. This, for instance, implies that by and large wave spectra in the ocean are continuous, while they are often quantized in lakes.

In this section the hydrodynamic equations are formulated, mainly in order to state basic principles and introduce a notation. Simple solutions are then developed for water movements in a narrow rectangular basin subjected to steady wind directed along its length. Vertical structures of current are derived for both one- and two-layered systems representing, respectively, a lake during conditions of winter homogeneity and summer stratification. In spite of their simplicity, for the most part achieved by linearization, the use of constant coefficients of eddy viscosity and the neglect of the Coriolis force, the solutions illustrate some important facts about the dynamics of wind-driven flows in a long narrow lake. Perhaps the main interest of the analysis lies in the actual construction of closed solutions, satisfying appropriate boundary conditions, for lake circulation.

Insurance rate filings involving hurricane perils are generally based on complex, numerical models. Evaluation of such rate filings are further complicated if the model is proprietary so that state regulators are shielded from the inner workings of the models. To circumvent this difficulty while adhering to proprietary restrictions, Watson and Johnson (2003) developed an ensemble of 324 public domain models that bracket the published results of proprietary models while having the advantage of full disclosure of methodology. Moreover, the collection of models can provide regulators an appreciation of the state of the art of hurricane risk modeling to assist them in evaluating future rate filings. This methodology was applied for the North Carolina Department of Insurance but similar studies can be rapidly completed for other states as well. The results provide regulators with an independent, public domain spectrum of values to assess specific rate filings.

A Gulf of Mexico performance evaluation and comparison of coastal circulation and wave models was executed through harmonic analyses of tidal simulations, hindcasts of Hurricane Ike (2008) and Rita (2005), and a benchmarking study. Three unstructured coastal circulation models (ADCIRC, FVCOM, and SELFE) validated with similar skill on a new common Gulf scale mesh (ULLR) with identical frictional parameterization and forcing for the tidal validation and hurricane hindcasts. Coupled circulation and wave models, SWAN+ADCIRC and WWMII+SELFE, along with FVCOM loosely coupled with SWAN, also validated with similar skill. NOAA's official operational forecast storm surge model (SLOSH) was implemented on local and Gulf scale meshes with the same wind stress and pressure forcing used by the unstructured models for hindcasts of Ike and Rita. SLOSH's local meshes failed to capture regional processes such as Ike's forerunner and the results from the Gulf scale mesh further suggest shortcomings may be due to a combination of poor mesh resolution, missing internal physics such as tides and nonlinear advection, and SLOSH's internal frictional parameterization. In addition, these models were benchmarked to assess and compare execution speed and scalability for a prototypical operational simulation. It was apparent that a higher number of computational cores are needed for the unstructured models to meet similar operational implementation requirements to SLOSH, and that some of them could benefit from improved parallelization and faster execution speed.

The present study deals with the application of a physical mathematical model of the Adriatic Sea to the problem of predicting the sea level variations at Venice. The equations of motion are considered. Taking into account the peculiar topography of the region, a unidimensional model of the Adriatic Sea is formulated. The conditions for the computation stability are determined; a first test of the model is carried out by computing the variation in time of some fundamental quantities like the mass and energy of the basin. Thereafter, the main effects due to the geometry of the basin are studied by using ideal ‘wind quanta’. Agreement is found between the results of the model and those obtained by means of classical hydrodynamics. Applications of the model to actual cases of sea level rise at Venice follow. Finally, on the basis of a critical analysis of the results, some considerations on the predictability of the phenomenon and on the requirements of the predicted meteorological fields are offered.

The mean water circulations of Lake Superior during June-September, 1973, is obtained by hydrodynamical modeling. The time interval covers most of the stratification period during which the temperature data have been adequately collected and analysed. The computed results show reasonable agreement with observed current meter readings. In particular, the generally counterclockwise circulation and some features of the Keweenaw current are obtained in the results. A computed map showing the frequent upwelling and downwelling zones is also given. These zones are often referred to in the description of the physical, chemical and biological regimes of the lake.The computed currents provide adequate description of the advective transports during the period. By parameterizing the turbulent diffusion in an advection-diffusion model, it is shown that the formulation proposed in previous studies appears to be also applicable for simulating the chloride transport in Lake Superior. Based on these studies, the mixing times of conservative materials have been estimated to be about 2 to 3 years for Lake Superior, depending on the location of the source.

A descriptive analysis of the response of Lake Erie to the passage of
the blizzard cyclone of January 26, 1978, is presented. This intense
extratropical cyclone, the worst ever to cross the Ohio Valley and Great
Lakes regiion of the United States, set numerous record low sea level
pressure readings at nearly every recording station surrounding Lake
Erie and subjected the lake region to sharp temperature drops and high
winds. The lake surface was significantly ice covered during the storm
event; it remained virtually intact on the entire Western Basin, and
partial ice cover breakup occurred over the Central Basin. The
investigation of the water level fluctuations induced by the cyclone are
based on data acquired from normal meterological and water level
monitoring stations surrounding the lake. The most unusual aspects of
the water surface fluctuations include the observance of a pressure
suction induced rise in water level in the Western Basin before the
storm passed north of the lake; a maximum storm surge setup occurring
between Marblehed, Ohio, and Port Colborne, Canada, and not between the
ends of the lake; and a separate oscillatory surge occurring at Port
Stanley, Canada. The probable causes and reasons for these fluctuations
are thoroughly analyzed in the context of existing theories that deal
with how a lake surface responds to external atmospheric forcing
functions such as wind stress, sea level pressure changes, and
resonance. The effect of the ice cover on the water level fluctuations
is also presented.

A linearized stratified lake of rectangular shape with dimensions and
other physical parameters comparable to those of the Great Lakes is
presented to examine the currents induced by prescribed wind and thermal
conditions at the boundary. The boundary layer of thickness
E½ is identified at the surface and the solid
boundary, where the wind and wall effects are most important. The
influence of thermal input is restricted in the interior. The upwelling
and downwelling phenomena and a relatively strong coastal current are
significant in the linear stratified lake. Their relations to the input
functions established in the analysis substantiate observations in the
Great Lakes.

Remote sensing and long-term monitoring of closed and climatically
sensitive open lakes can provide useful information for the study of
climatic change. Satellite radar altimetry offers the advantages of
day/night and all-weather capability in the production of relative lake
level changes on a global scale. A simple technique which derives
relative lake level changes is described with specific relevance to the
TOPEX/POSEIDON geophysical data record data set. An assessment of the
coverage and global tracking performance of both the NASA radar
altimeter and the solid state altimeter over these lakes is discussed,
and results are presented for the first 1.75 years of the mission. Lake
level time series were acquired for 12 closed lakes, six open lakes, and
three reservoirs, providing information in many cases where ground gauge
data are unobtainable or the lake is inaccessible. The results, accurate
to ˜4 cm rms, mark the beginning of a very accurate lake level
data set, showing that TOPEX/POSEIDON can successfully contribute to the
long-term global program.

The purpose of this study is to examine variations in the response of an island system (the Hawaiian Islands, in this case) to an incoming tsunamilike wave pulse approaching the system along various azimuths. Simulations were carried out numerically by using an explicit finite-difference analog for the linearized equations of motion and continuity for long waves in a variable depth ocean. The model topography is based on the submarine topography of the Hawaiian Island region. Island coastlines are fully reflecting, so no attempt to simulate runup was made. Qualitative comparisons between model results and historical data from tsunamis approaching along similar azimuths show that the model produces realistic simulations. Azimuths were chosen for waves approaching from four general geographic areas: South America, Alaska, Aleutians-Kuriles-Japan-Philippines, and Southwest Pacific. Nearly all distant tsunamis striking Hawaii have come from one of these areas. Our conclusions are: (1) Tsunami response in the overall system does not vary greatly over small (10°–15°) changes in azimuth but does vary significantly over large changes (>60°). (2) Local response may vary greatly with azimuth, but certain areas seem to respond strongly to tsunamis approaching from almost any direction. (3) Topographic focusing seems to play the dominant role in determining localized response.

The steady-state wind-driven currents in Lake Erie are investigated. A
numerical solution for the mass-transport stream function and the
three-dimensional velocities as a function of depth and horizontal
position is obtained and compared with measurements. The agreement is
good. This report shows that the currents depend strongly on bottom
topography and boundary geometry.

The hydrodynamic equations governing the water-level response of a lake to wind stress are inverted to determine wind stress from water-level fluctuations. In order to obtain a unique solution, the wind-stress field is represented in terms of a finite number of spatially dependent basis functions with time-dependent coefficients. The discretized version of the inverse equation is solved by a least-squares procedure to obtain the coefficients, and thereby the stress. The method is tested for several ideal cases with Lake Erie topography. Real water-level data is then used to determine hourly values of vector wind stress over Lake Erie for the period 5 May–31 October, 1979. Results are compared with measurements of wind speed and direction from buoys deployed in the lake. Calculated stress direction agrees with observed wind direction for wind speeds > 7.5 m s−1. Under neutral conditions, calculated drag coefficients increase with the wind speed from 1.53 × 10−3 for 7.5−10 m s−1 winds to 2.04 × 10−3 for 15−17.5 m s−1 winds. Drag coefficients are lower for stable conditions and higher for unstable conditions.

Some early (pre-Second World War) storm surges on the Great Lakes produced storms that were extensive enough to influence the whole Great Lakes system were reconstructed making use of scant meteorological and oceanographic data and descriptions from popular literature. Port Colborne, on the northeastern shore of Lake Erie exhibited consistently great storm surges. Negative surges were observed in narrow connecting waters such as the Detroit River. The simple analytical techniques used here showed that the water level changes could be accounted for mostly by the large-scale pressure system.

A previously developed Great Lakes ice forecasting model for winter navigation aid has been calibrated using observed data from specific Lake Erie ice transport events. The model is based on the macroscopic continuum hypothesis for the fragmented ice field and the internal ice stress is represented by a viscous-plastic type constitutive law. The external driving force includes the time-dependent wind and water current fields as well as the thermodynamic source/sink terms for the ice mass conservation. By adjusting the model coefficients employed in the constitutive equations and the thermodynamic factors, the computed results are in reasonable agreement with the short-term observations for the 1978-79 ice season in Lake Erie. Major findings include the profound effects of the wind-driven water currents and the ice melt at the icewater interface on the ice regime of Lake Erie during the late stage of the ice season.

The 19th typhoon of 1991 (hereafter referred to as Typhoon 9119) caused remarkable storm surges more than 3 m in height with great inundation resulting along the coast in the western part of the Seto Inland Sea of Japan. This paper discussed the generation mechanism of storm surges due to Typhoon 9119 using numerical simulations with the conventional two-dimensional model. Some discrepancies between calculated storm surges and those observed were found, the reason for which was assumed to be lack of accuracy in the typhoon model. A method was developed to estimate wind field in a typhoon using sea level data and adopting the D. J. Schwab method (1982) Dynamics of Atmosphere and Oceans, 6, 251–278. Comparison of the estimated wind field with the observed wind field showed good agreement. Using the estimated wind field and newly determined normal modes of the analyzed region, the detailed structure and the generation mechanism of the storm surge resulting from Typhoon 9119 are identified.

Numerical models were used to compute water circulations throughout the 1970 shipping season for Lake Erie and for the 1972 International Field Year on Lake Ontario. Simultaneous computations of surface elevations were compared with observed water levels to adjust the model results after the fact. As a by‐product of these simultations, effective stress coefficients over water can be estimated. The results support earlier evidence that the effective wind stress over water is larger than indicated by atmospheric boundary layer measurements.

A new method for solving the linearized equations of motion is presented in this paper, which is the implementation of an outstanding idea suggested by Welander: a transport approach to the convolution method. The present work focuses on the case of constant eddy viscosity and constant density but can be easily extended to the case of arbitrary but time-invariant eddy viscosity or density structure. As two of the three equations of motion are solved analytically and the main numerical ‘do-loop’ only updates the sea level and the transport, the method features succinctness and fast convergence.
The method is tested in Heaps' basin and the results are compared with Heaps' results for the transient state and with analytical solutions for the steady state. The comparison yields satisfactory agreement. The computational advantage of the method compared with Heaps' spectral method and Jelesnianski's bottom stress method is analysed and illustrated with examples.
Attention is also paid to the recent efforts made in the spectral method to accelerate the convergence of the velocity profile. This study suggests that an efficient way to accelerate the convergence is to extract both the windinduced surface Ekman spiral and the pressure-induced bottom Ekman spiral as a prespecified part of the profile.
The present work also provides a direct way to find the eigenfunctions for arbitrary eddy viscosity profile. In addition, mode-trucated errors are analysed and tabulated as functions of mode number and the ratio of the Ekman depth to the water depth, which allows a determination of a proper mode number given an error tolerance.

A non-linear two-dimensional vertically stratified cross-sectional model of a constant depth basin without rotation is used
to investigate the influence of vertical and horizontal diffusion upon the wind-driven circulation in the basin and the associated
temperature field. The influence of horizontal grid resolution, in particular the application of an irregular grid with high
resolution in the coastal boundary layer is examined. The calculations show that the initial response to a wind impulse is
downwelling at the downwind end of the basin with upwelling and convective mixing at the opposite end. Results from a two-layer
analytical model show that the initial response is the excitation of an infinite number of internal seiche modes in order
to represent the initial response which is confined to a narrow near coastal region. As time progresses, at the downwind end
of the basin a density front propagates away from the boundary, with the intensity of its horizontal gradient and associated
vertical velocity determined by both horizontal and vertical viscosity values. Calculations demonstrate the importance of
high horizontal grid resolution in resolving this density gradient together with an accurate density advection scheme. The
application of an irregular grid in the horizontal with high grid resolution in the nearshore region enables the initial response
to be accurately reproduced although physically unrealistic short waves appear as the frontal region propagates onto the coarser
grid. Parameterization of horizontal viscosity using a Smagorinsky-type formulation acts as a selective grid size-dependent
filter, and removes the short-wave problem although enhanced smoothing can occur if the scaling coefficient in the formulation
is too large. Calculations clearly show the advantages of using an irregular grid but also the importance of using a grid
size-dependent filter to avoid numerical problems.

A time-independent dynamical model of storm surge along island coasts using orthogonal curvilinear coordinates is presented. The curved annulus between an island coast and an arbitrary deep-water boundary is mapped conformally onto a rectangular image.
Two configurations of island coasts are investigated; circular and elliptic coasts. The corresponding coordinates are circular polar and elliptic respectively. The linearized vertically-integrated equations of motion are used to model storm surges with two assumptions: (i) bottom stress is proportional to horizontal transport, and (ii) storm forces are shear stresses on water surface.
Analytical solutions are presented for three dynamical cases: (i) a constant-depth basin acted upon by a uniform storm stress, (ii) variable-depth basin acted upon by a uniform-direction variable-magnitude stress, and. (iii) a basin with closed depth contours acted upon by vortex-shaped storm stress.
The obtained solutions clarify the relative importance of the various parameters and variables that affect surge height distribution along island coasts. These solutions may be used to test a time-dependent, numerical dynamical storm model.

Oscillations of an enclosed basin due to a suddenly blowing wind which is periodic for several periods and constant otherwise, are studied. If quadratic bottom friction is taken into consideration a forced oscillation and seiches and, in addition, non-linear oscillations given by the higher harmonics and the sum and difference frequencies, are obtained. The results are compared with observations made in the Baltic Sea where the forced oscillation shows a period of 120 hours. The main energy of the non-linear components concentrates on 58.7–60 hours and on about 35 hours.Es werden die Schwingungen eines abgeschlossenen Meeresgebietes als Folge eines Windes betrachtet, der anfangs periodisch, spter konstant ist. Bercksichtigt man quadratische Bodenreibung, so entstehen neben der erzwungenen Schwingung und den Seiches nicht-lineare Schwingungen. Ihre Frequenzen sind durch ganzzahlige Vielfache der Frequenzen der linearen Schwingungen sowie durch Summen- und Differenzfrequenzen gegeben. Die Ergebnisse werden zur Interpretation der in der Ostsee beobachteten Energiekonzentrationen bei 120h, 60h und 35h herangezogen.On tudie les oscillations d'un bassin ferm, dues un vent soufflant brusquement, de faon priodique pendant plusieurs priodes, et, puis d'une manire constante. Si on considre le frottement quadratique sur le fond, on obtient des seiches, une oscillation force et, en outre, des oscillations non linaires provoques par les harmoniques les plus leves ainsi que par la somme et la diffrence des frquences. On compare les rsultats avec les observations faites en mer Baltique o l'oscillation force a une priode de 120 heures. La plus grande partie de l'nergie des composantes non linaires se concentre sur 58.7–70 heures et sur environ 35 heures.

A numerical model has been designed to study the storm surge induced by typhoon along the coast of Taiwan. The governing equations
have been expressed in spherical coordinate system, and a finite difference method has been used to solve them. In the system
of hydrodynamical equations, the nonlinear advection and lateral eddy viscosity terms are prominent in shallow coastal waters.
Air pressure gradient and wind stresses are the driving forces in the model of typhoon surge. The model has been verified
with storm surges induced by Typhoons Herb in 1996, and by typhoons Kai-Tak and Bilis in 2000.

The limited amount of information contained in a set of meteorological predictors precludes any precise statement concerning which one of a number of possible future events will occur. For purposes of operational decision making the probability distribution over the possible events for given values of the predictors is required. The mathematical exposition of a technique for obtaining this distribution is presented. An objective procedure is proposed for excluding from the analysis any redundant or nonsignificant information.
Two numerical examples are provided which illustrate the application of the technique where the predictors are selected using the proposed procedure.

The linearized hydrodynamic equations for storm surges are solved in analytic form for a very simple model basin and an arbitrary field of wind and pressure to show that a solution can be obtained as an integral of the product of the atmospheric forcing function and an influence function whose value tends to zero with increasing time lags. In practical cases t,his solution can be computed as a weighted sum of the meteorological observations during a short period before the storm surge observation. A finite difference scheme for a slightly more goncral basin is then developed and the solution given formally in terms of a polynomial involving bot'h vectors and matrices. It is shown that this solution is equivalent to the analytic solution and that both are equivalent to a li~lear function of the meteorological measurements of wind and pressure which must be used to obtain a descript,ion of any actual forcing function for storm surges. The technique can be generalized to provide the solution for basins of almost any shape. The difficulties and uncertainties involved in the hydrodynamic solution are discussed, and the advantages of using a statistical method to dc4erminc the solution of the problem when sufficient data are available are shom-n.

The dates of incidence of extreme wind tide on Lake Erie have been determined for the 20-year period 1940 through 1959, for all cases in which the difference in lake level between Buffalo and Toledo exceeded 6 feet. A frequency-intensity analysis shows that a set-up in excess of 10 feet may be expected once every 2 years. Extreme wind tides occur mainly in the 6-month period October through March; more than 70 percent of the cases fall in the three months November, December, and January. November is the month of most frequent incidence, having more than one-third of the total number of cases in the period studied. The observed seasonal variation of extreme set-up frequency is interpreted as a reflection of the seasonal variation of storm frequency and of storm-track location. Secondary, but important factors are: seasonal variation of storm intensity and seasonal variation of thermal stability of the atmospheric boundary layer. The tendency for marked temperature stratification to be present in t...

In connection with recent work of predicting hurricane storm tides in New York Bay, it became desirable to adopt the most suitable value of surface wind stress over water at high wind velocities that current knowledge will sustain. To this end a review of available literature of field and laboratory experiments on the subject was undertaken and the collated results are presented. To some extent it has been found possible to resolve some of the wide disparities that have existed up to now between field and laboratory results by the quite simple expedient of adapting the laboratory data for wind speeds at 10-cm height (usually) to prototype conditions of wind speeds at 10-m height through use of the Karman-Prandtl equation for verticle velocity distribution. Some 47 authorities are quoted in arriving at Cd values of 0.0024±.0005 for strong winds and 0.0015±.0008 for light winds, in a wind stress relationship of the form in which ρa is the mass density of the air and U the wind velocity at 10-m height. Although there is fairly satisfactory unanimity now regarding the Cd value for high winds, the situation is far less satisfactory for the case of light winds.

This chapter discusses the problem of predicting the storm surge once the meteorological conditions are given. The existing crude methods can predict the storm-surge amplitudes from given meteorological conditions with accuracy of the order of 10–20%. With the more general numerical and empirical methods to be discussed, it seems possible to predict the storm-surge amplitude from given meteorological conditions with an error of only a few percent. For the prediction of a storm surge in a given meteorological case, one has to introduce initial values, but these play a small role because of the presence of strong forcing functions. If these forcing functions are sufficiently familiar over a past time, one can start from a sea at rest. The forcing functions can continually control the development and prevent the growth of the initial errors. Finally, the basic model equations are in themselves both simpler and more accurate than in the meteorological case. The thermodynamic processes in the sea can be safely neglected and the equations can, with good approximation, be linearized.

The idea of forecasting the weather by calculation was first dreamt of by Lewis Fry Richardson. The first edition of this book, published in 1922, set out a detailed algorithm for systematic numerical weather prediction. The method of computing atmospheric changes, which he mapped out in great detail in this book, is essentially the method used today. He was greatly ahead of his time because, before his ideas could bear fruit, advances in four critical areas were needed: better understanding of the dynamics of the atmosphere; stable computational algorithms to integrate the equations; regular observations of the free atmosphere; and powerful automatic computer equipment. Over the ensuing years, progress in numerical weather prediction has been dramatic. Weather prediction and climate modelling have now reached a high level of sophistication, and are witness to the influence of Richardson's ideas. This edition contains a new foreword by Peter Lynch that sets the original book in context. © Stephen A. Richardson and Elaine Traylen and Peter Lynch 2007.

In 1923 Ekman developed a theory for the sea-level changes produced in a deep sea by the action of a steady wind. In the present paper the theory has been extended to a shallow sea, for which the Ekman “depth of frictional influence” d is comparable to the actual depth h. It is pointed out that the wind-stress divergence may be of the same importance as the wind-stress curl in this case.The theory is furthermore extended to the transient case. It is demonstrated how the velocity profile and the flow can be expressed in terms of the local time-histories of the wind-stress and the surface slope, and a single integro-differential equation is derived for the sea-level elevation. This equation could possibly be used for prediction of meteorologic tides and storm surges.

Caption title. Arkiv för matematik, astronomi och fysik, Bd. 2, no. 11. Xerographic reproduction of the JHU copy, on double leaves folded once in Japanese style, and bound with leaves inverted. Bibliographical foot-notes.

Storms of the Great Lakes

- E B Garriott

Beobachtungen über die Dämpfung der Seiches in Seen

- A Endrös

Winds and water levels on Lake Erie

- D K A Gillies

Détermination des dénivellations et des courants de marée. Proceedings, Seventh Congress on Coastal Engineering, The Hague

- F Gohin

Wind effects on lakes and rivers. Ingeniörsvetenskaps-akademien, Handlingar 158

- B M O Hellström

A mathematical investigation on the development of wind currents in heterogeneous waters

- Koji Hidaka

1956: A procedure for numerical integration of the primitive equations of the two-parameter model of the atmosphere

- A Eliassen

1902: Wind velocity and fluctuations of water level on Lake Erie

- Alfred J Henry

1959: Winds, wind set-ups, and seiches on Lake Erie, part 2. U. S. Corps of Engineers, Lake Survey

- Ira A Hunt

Non-stationary ocean currents. Memoirs, Imperial Marine Observatory

- Koji Hidaka

Coast effect upon the ocean current and the sea level. Memoirs, College of Science, Kyoto Imperial University (Series A). I. (with T. Takegami) Steady state

- Takaharu Nomitsu

1957: Modification of the quadratic bottom-stress law for turbulent channel flow in the presence of surface wind-stress

- R O Reid

Long and short period oscillations in Lake Erie. State of Ohio, Department of Natural Resources, Division of Shore Erosion

- James L Verber

1922: Effects of winds and of barometric pressures on the Great Lakes

- John F Hayford

Winds, wind set-ups, and seiches on Lake Erie, part 1. U. S. Corps of Engineers, Lake Survey

- Ira A Hunt
- Jr

Theoretical investigations of typhoon surges along the Japanese coast

- M Miyazaki
- T Ueno
- S Unoki

On the development of the slope current and the barometric current in the ocean

- Takaharu Nomitsu

The currents of western Lake Erie

- Franklyn C W Olson

A treatise on limnology

- G Hutchinson
- Evelyn

A theory of the rising stage of drift current in the ocean

- Takaharu Nomitsu

Theory of ocean tides. International Union of Geodesy and Geophysics, General Assembly at Helsinki

- C L Pekeris
- M Dishon

Turbulent transfer in the lower atmosphere

- C H B Priestley