Deep water waves having period T TildeTilde 12 s and signifi

Deep water waves, having period T TildeTilde 12 s and significant wave height H_0 TildeTilde 2 m, approach a long straight sandy coastline (with beach gradient S TildeTilde 0.02 and typical sediment fall velocity w_s TildeTilde 0.02 m/s) at an angle of alpha_0 TildeTilde 10 degree to the shore normal direction. Describe how the characteristics of these waves change as the waves propagate towards the coast, and indicate the form of the waves close to the shore and the characteristics and features of the nearshore region that you would expect to observe along this coast. If, further down the coast, the angle of wave approach in deep water increases to alpha_o TildeTilde 20degree how would you expect this stretch of coastline to change over time; would you expect it to erode, accrete or remain stable? Your answer should outline relevant physical processes responsible for the changes you highlight, incorporate relevant calculations to illustrate points that you make and clearly state any assumptions made. The assessment criteria that will be used when marking this question are provided on the last page of this examination paper. Shoaling/refraction equation: H=H_0 (1/2n c_0/c)^1/2 (cos alpha_0/cos alpha)^1/2 Snell\'s Law: sin alpha/C=constant Breaker Height (Komar and Gaughan, 1972): H_b=0.39g^1/5(TH^2_0)^2/5 Iribarren Number (Battjes, 1974): xi_b = s(h_b/L_0)^1/2[ 2.0 surging] Longshore Sediment Transport Rate: Q_1 = 89429H_b^5/2 sin alpha_bcos alpha_b Dimensionless Fall Velocity: Ohm=H_0/w_sT [ 5.5 dissipative beach] H = wave height, T = wave period, L= wavelength, C = phase speed, alpha = angle between wave crests and depth contours, n = ratio between group and phase speed of waves, S = beach gradient, w_s = sediment fall velocity, g = acceleration due to gravity (9.81 m/s^2), subscript_0 denotes deep water value, subscript_b denotes breakpoint value. Give a detailed account and explanation of the numerous ways in which tides are modified in shallow water. What are storm surges, why are they important and how are they caused? Given the following data for a sandy beach, recorded during a high spring tide on the north coast of Cornwall, compute the height of the combined surge and run-up relative to chart datum (CD). Show all of your calculations. Data: Wind speed (towards the coast) = 25 m/s Fetch = 150 km Wind drag coefficient = 3times 10^-3 Density of air = 1.225 kg/m^3 Density of water = 1025 kg/m^3 Average water depth = 60 m Acceleration due to gravity = 9.81 m/s^2 Change in atmospheric pressure relative to the mean value = 1000 N/m^2 The low-pressure system is moving slowly at negligible speed The significant offshore wave height = 4 m Height of high-tide = 8.2 m (relative to CD) Tidal Residual (R) Equation: W wind speed F fetch rho_A density of air rho density of water h mean water depth U_p velocity of atmospheric depression H_0 offshore significant wave height g acceleration due to gravity C_A coefficient of wind drag Delta P_A change in atmospheric pressure

Solution

1. It can be demonstrated that as a wave propagates from deep water in to the shore, the wave period will remain constant because the number of waves passing sequential points in a given interval of time must be constant. Other wave characteristics including the celerity, length, height, surface profile, particle velocity and acceleration, pressure field, and energy will all vary during passage from deep water to the nearshore area.

When a train of waves propagates toward the shore, at some point, depending on the wave characteristics and nearshore bottom slope, the waves will break. Landward of the point of wave breaking a surf zone will form where the waves dissipate their energy as they decay across the surf zone. As the waves approach the breaking point there will be a small progressive set down of the mean water level below the still water level. This setdown is caused by an increase in the radiation stress owing to the decreasing water depth as the waves propagate toward the shore. The setdown is maximum just seaward of the breaking point. In the surf zone, there is a decrease in radiation stress as wave energy is dissipated. This effect is stronger than the radiation stress increase owing to continued decrease in the water depth. The result is a progressive increase or setup of the mean water level above the still water level in the direction of the shore. This surf zone setup typically is significantly larger than the setdown that occurs seaward of the breaking point.

2a) The first and fundamental observation with respect to tides in shallow water is that the tide-generating force is of global scale. Only the largest water bodies such as the major oceans can therefore experience tidal forcing in the way described by Laplace. Smaller water bodies such as marginal seas or estuaries cannot produce a response to astronomical tidal forcing. If there is tidal movement in these regions, it is forced by the tidal currents of the deep ocean which enter and leave the region periodically at the connection to the ocean. Tides generated in this way are known as co-oscillation tides. Marginal seas have their own resonance frequencies, determined again by their dimensions. As a consequence, the amplitudes and phases of co-oscillation tides depend on the closeness of a resonance frequency to one of the tidal frequencies and on the amplitude of the tidal currents in the deep ocean at the connecting line with the marginal sea. This explains, for example, why mediterranean seas are virtually tide-free; their connection with the open ocean is so restricted that the oceanic tides cannot produce co-oscillation.

Shallow seas which are close to resonance with one of the tidal periods are of great importance for the world\'s fishing industry. The flow of strong tidal currents over a shallow ocean floor produces turbulence of sufficient intensity to keep the entire water column well mixed throughout most of the year. Nutrients which usually accumulate in the sediment and are no longer available to support marine life, are continuously kept in suspension under such conditions. These coastal seas are therefore among the most productive fishing regions of the world ocean, rivalling the great coastal upwelling regions and the fertile Southern Ocean. The North Sea or the Newfoundland Banks are two examples of regions where tidal mixing keeps nutrient concentrations in the water column at a high level.

Tides in shallow water are generally a mixture of propagating waves and standing waves. One major difference between these two types of waves is the phase relationship between elevation and tidal current. As could be seen from the example of the water bowl or tank, currents and water level are 90° (or a quarter period) out of phase: Currents are strongest when the water surface is flat and vanish when the water level is at its highest and lowest (high and low tide). In propagating waves, on the other hand, currents are strongest at high and low tide, i.e. they are in phase with the elevation. For a given coastal location the time of strongest tidal current relative to high tide therefore depends on the type of tidal wave in the region.

Sudden changes in water depth can lead to a change of the tide from a standing wave to a propagating wave. This occurs because the propagation speed of shallow water waves depends on the water depth. If such a wave encounters a sudden change of depth, its propagation speed is slower over the shallower region than over the deeper region; its propagation speed on either side of the sudden depth change is mismatched, and the wave cannot continue unchanged across the changing topography. This leads to partial reflection of the wave. If a wave approaches a steep rise of the sea floor, part of the wave continues as a propagating wave in the shallow water, while part of it is reflected back into the deeper water and combines with the incoming wave to form a partially standing wave. Tidal currents and elevation are thus in phase in the shallow part but out of phase, by a degree determined by the wave\'s reflection coefficient, in the deeper part. This explains the wide range of observed phase relationships between tidal currents and high or low tide in the world ocean\'s shelf seas.

2b) Storm surge is an abnormal rise of water generated by a storm, over and above the predicted astronomical tides. Storm surge should not be confused with storm tide, which is defined as the water level rise due to the combination of storm surge and the astronomical tide. This rise in water level can cause extreme flooding in coastal areas particularly when storm surge coincides with normal high tide, resulting in storm tides reaching up to 20 feet or more in some cases.

Storm surge is produced by water being pushed toward the shore by the force of the winds moving cyclonically around the storm. The impact on surge of the low pressure associated with intense storms is minimal in comparison to the water being forced toward the shore by the wind. The maximum potential storm surge for a particular location depends on a number of different factors. Storm surge is a very complex phenomenon because it is sensitive to the slightest changes in storm intensity, forward speed, size (radius of maximum winds-RMW), angle of approach to the coast, central pressure (minimal contribution in comparison to the wind), and the shape and characteristics of coastal features such as bays and estuaries. Other factors which can impact storm surge are the width and slope of the continental shelf. A shallow slope will potentially produce a greater storm surge than a steep shelf.