HEC 25 - Tidal Hydrology, Hydraulics, and Scour at Bridges

Chapter 2 - Tidal Hydrology and Boundary Conditions

2.1 Introduction

This chapter provides information and methods for estimating the driving forces, or boundary conditions, that will be used for tidal hydrodynamic modeling. At bridges located in tidal waterways, the peak flow and water surface elevation are a function of the combined effects of astronomical tides, storm surge, wind and upland runoff. These boundary conditions are imposed at open boundaries of the numerical models and, based on the geometric characteristics and flow resistance effects of the tidal waterway and bridge crossing, the model determines the design hydraulic condition for the bridge.

The number and complexity of the boundary conditions are related to the type of waterway. For example, a waterway consisting of a single entrance, such as a bridge crossing an inlet to a bay, may only require a single boundary condition that consists of only the astronomical tide. A simulation of a storm surge would also be included as a design condition. Multiple inlets connected to an embayment may require astronomical tides with different timing and ranges. In the most complex situations, boundary conditions for a design event could include astronomical tides with superimposed storm surge at several inlets, a wind field corresponding to the storm, and upland runoff from the drainage basin.

Understanding tide and storm surge conditions is important because coastal studies have demonstrated that unsteady flow analysis is crucial to the accurate representation of tidal hydraulics and scour (Pooled Fund Study by Ayres Associates 1994, 1997, 2002a, 2002b). Hydrologic and meteorological conditions are site specific and may involve multifaceted issues and considerations.

Developing and applying these conditions to unsteady flow models may require the consideration of coastal engineering concepts for some transportation projects. For these projects and locations, the complex conditions may require using the services of a qualified coastal engineer to ensure consideration of all relevant issues and information. For these reasons, detailing every hydrologic, hydraulic, sedimentation and meteorological issue, theory, and method is beyond the current scope of this manual. Instead, this chapter presents an overview of these boundary condition concepts for the highway hydraulic community.

This manual focuses on hurricanes as predominate storm events producing coastal boundary conditions. Along the east and gulf coasts, a rapid rise and fall of the hurricane storm surge is likely to produce the most intense flow conditions in a tidal waterway. However, this may not be the case in every region of the United States. Astronomical tides can also produce high flow velocities and should be analyzed for scour. Along the U.S. mid-Atlantic and New England coasts, storm events such as Nor'easters produce historically extreme coastal flooding and erosion. On the West Coast, winter storms and tsunamis may provide the critical combination of waves, velocities, and depth that would affect a bridge waterway. The Great Lakes have wind and wave forces resulting from storm events that affect bridge waterways. However, the approaches presented in this manual should be generally applicable to other tidal and storm conditions.

2.2 Astronomical Tides and Currents

The astronomical tide is the change in water level due primarily to the relative motions and the gravitational pull of the moon and sun. While the actual tide is due to the influence of the moon and sun, local atmospheric conditions and coastal geometry play a major role at a given location and time. The predicted tide for most locations is solely a function of the moon and sun. Appendix B, which is a reproduction of a NOAA document "Our Restless Tides," includes a detailed description of the nature and causes of tides.

2.2.1 Tide Types, Periods, and Levels

Figures 2.1 and 2.2 illustrate the terminology used to describe the relative positions of the tide. Tidal nomenclature categorizes these relative positions as mixed tide semidiurnal tide, and diurnal tide. The mixed tide has two highs and two lows each day. These are called higher high water, lower high water, higher low water and lower low water. The difference between a high and a low is called the tidal range and half the range is called the amplitude. The tidal period is the time between successive highs or successive lows. Coastal areas that exhibit semidiurnal tides (two highs and two lows per day) have two tidal periods per tidal day. The successive ranges of the semidiurnal tides are generally closer than in a mixed tide. Areas that have diurnal tides (one high and low per day) have a tidal period equal to the tidal day. In the United States, most west coast, gulf coast and east coast locations exhibit semidiurnal or mixed tidal behavior. However, changes in the earth-moon orientation and declination, as well as coastal geometry, may produce diurnal tides at some locations.

Tide station data include average tidal heights. These include mean tide level (MTL), mean low water (MLW), mean lower low water (MLLW), mean high water (MHW), and mean higher high water (MHHW). Mean tide level is the average of all high and low tides. Mean low water and mean lower low water are the average of all low tides and all lower low tides, respectively. Mean high water and mean higher high water are the average of all high tides and all higher high tides, respectively. The mean tide range is the difference between mean high and mean low water. Sections 2.10 and 2.11 include worked examples of calculating tide heights and tide ranges.

2.2.2 Tidal Days

As a result of the difference between the angular velocity of the moon's orbit and the angular velocity of the earth's rotation, a tidal day is approximately 24 hours and 50 minutes. As the earth makes one complete revolution, the moon has progressed in its orbit. Therefore, an additional 50 minutes of revolution is required for a lunar day. The fact that the tidal day is longer than 24 hours means that each day the high or low water will occur 50 minutes later than the previous day. In many engineering applications the tidal (or lunar) day is approximated as 25 hours.

2.2.3 Tidal Epochs

The principal tide producing forces have a period of approximately 18.6 years (called a tidal epoch ). Thus, a full cycle of tidal observations of about 19 years is needed at any one location in order to be able to minimize the error in the prediction of the tide. In most cases, the 19 year period is used to determine the mean of any one of the tidal parameters, i.e., mean high, mean higher high, mean low, mean spring, etc. Spring tides are larger than normal tides that occur approximately twice monthly during the new and full moons when the sun and moon forces are aligned. Neap tides are smaller than normal tides that occur at the first and third quarter moon phases.

Figure 2.1. The principal types of tides

Figure 2.2. Tide levels terminology.

National Tidal Datum Epoch (NTDE) is the specific 19-year period (epoch) adopted by the National Oceanographic and Atmospheric Administration (NOAA) and used to obtain these mean values for mean datums. The most recent NTDE included 1983 through 2001. When looking at older tidal records (such as the 1960-1978 NTDE) care should be taken to ensure that may have information that has changed during epochal revisions.

There are other factors that will affect both the timing and the magnitude of the tide. These include the land masses that the tide must move around, the friction between the water and the ocean bottom, other astronomical bodies, and the geometry of the oceans and sounds to name a few. In addition, the local meteorological conditions, including the wind and barometric pressure, will influence the rise and fall of the ocean (or lake). These latter parameters do not follow the same predictable periodic nature of the astronomical factors and therefore are not included in tidal predictions.

2.2.4 Tidal Predictions

NOAA provides tidal predictions at many U.S. and international locations. These predictions are available in published tables or via the internet. For many years NOAA printed these Tide Tables on an annual basis. More recently, NOAA and other (mostly private) entities have published these predictions electronically. NOAA predictions are available on the internet at http://tidesandcurrents.noaa.gov/ (Search terms: NOAA tide predictions).

NOAA tidal predictions are provided for both Reference Stations and Subordinate Stations (also referred to as Primary and Secondary Stations)along the U.S. coastline. There are approximately 250 Reference Stations and over 3000 total stations (Reference and Subordinate). Tide predictions are available at up to 6-minute intervals for Reference Stations. Figures 2.3 and 2.4 are examples of the detailed tide predictions available from NOAA. In these figures, the observed tides are also shown. An important note is that the predicted tide from NOAA and from other sources will generally give accurate estimates of the astronomical tides only. The actual tides, as recorded, will also have the effects of wind and barometric pressure affecting the reading.

Figure 2.3. NOAA predicted and observed tide for Annapolis, Maryland.

Figure 2.4. NOAA predicted and observed tides for Anchorage, Alaska.

Tide tables are available for all Reference Stations, such as shown in Table 2.1 for the Charleston Harbor Reference Station (only a few, representational days of data are shown). The tide tables include time and height for all high and low waters. Predicted tide level, times and other characteristics can be taken directly from these tides tables. For example on January 3, 2004, the high tide elevation of 5.3 feet (compared to mean lower low water) occurs at 4:45 am. The next low tide of 0.7 feet occurs at 11:14 am. This indicates a range of 4.6 feet. However, the range is less for the next tide cycle (high of 4.4 and low of 0.2 yields a range of 4.2 feet). The fact that there are two highs and two lows indicates either mixed or semidiurnal tide conditions.

Subordinate Station tide table values are determined by applying corrections to Reference Station values. These corrections are time and height differences, as illustrated in Table 2.2. Typically, the Subordinate Station lists the associated Reference Station (in the case of Table 2.2, this Reference Station is Charleston). The subsequent discussion describes the meaning of these various corrections.

Time Differences: The columns labeled "Time Differences" allow determination of the time of high and low tide at any station listed in this table. The values in these columns represent the hours and minutes to be added or subtracted from the time of high or low tide of the Reference Stations. A plus sign (+) indicates that the tide at the Subordinate Station occurs later than at the Reference Station and the difference should be added; a minus sign (-) indicates that it is earlier and should be subtracted.

To obtain the tide at a Subordinate Station on any date, apply the difference to the tide at the Reference Station for that same date. In some cases, however, to obtain an AM tide it may be necessary to use the preceding day's PM tide at the Reference Station or to obtain a PM tide it may be necessary to use the following day's AM tide.

The results obtained by application of the time differences will be in local time for the subordinate station. The necessary allowances for the change in date when crossing the international date line, or for different time zones have been included in the time differences listed.

Height Differences: The height of the tide, referred to the datum of nautical charts, is obtained by means of either height difference or ratios (found in the "Height Difference" columns). A plus sign (+) indicates that the difference should be added to the height at the reference station, and a minus sign (-) indicates that it should be subtracted. For most Subordinate Stations, use of a predicted height difference would give unsatisfactory predictions. In such cases they have been omitted and one or two ratios, indicated by an asterisk (*), are given. To obtain the height of tide at the Subordinate Station in these cases, multiply the height of tide at the Reference Station by the ratio listed. The result is normally rounded to the nearest 0.1 foot.

For some Subordinate Stations there is given, in parentheses, a ratio as well as a correction. In those instances, each predicted high and low water at the Reference Station should be first multiplied by the ratio and then the correction is added or subtracted from each product.

For example, the Limehouse Bridge location on the Stono River (shown in Table 2.2) has high and low tides 1hr43m and 1hr34m after the corresponding tides at the Charleston Reference Station. High and low tide levels are 1.08 and 1.32 times the corresponding tides at Charleston Reference Station (Table 2.1). Using this data the first two tides on January 3, 2004 are a high tide of 5.7 feet at 6:28 am (5.3 x 1.08 = 5.7 ft at 4:45 am + 1:43 = 6:28 am) and a low tide of 0.9 feet at 12:48 pm (0.7 x 1.32 = 0.9 ft at 11:14 am + 1:34 = 12:48 pm). Note that the computed heights are relative to the MLLW at the Subordinate Station and that conversion of these heights to a fixed datum would require additional information (the difference between MLLW and the fixed datum) at the Subordinate Station.

2.2.5 Tidal Range and Variability

As described earlier, the tide range is the elevation change from low water to high water for any given cycle. The specification of the tide range is one of the parameters needed to set the boundary conditions for modeling the hurricane surge. The NOAA Tide Tables present both the mean range and the spring range. The mean range is the average difference between high and low tide for the year. The spring range is the annual average of the spring tidal range. Spring tides occur when the sun and the moon combine to generate the greatest difference between high and low tide. There are typically two spring tides each month that coincide with new and full moons.

A hurricane that occurs during a spring high tide would potentially lead to the highest water levels when the surge is combined with the astronomical tide. Alternatively, if the storm surge peak coincided with the spring low tide, the net storm tide (surge plus astronomical tide) would be much lower.

There are significant differences in mean and spring tidal ranges along the United States coast. Table 2.3 illustrates some of these differences. These differences are due to the combination of the geometry and bathymetry of the coastline at each of these stations. It is interesting to note that there is not a uniform decrease in range as one moves from Maine to Florida. In fact, because of the extensive continental shelf, some of the largest tides on the U.S. eastern coast are in Georgia. Table 2.3 also illustrates the difference between the mean and the spring tide. In a conservative engineering analysis, the spring range is often used to account for maximum tidal flow conditions.

The annually published "NOAA Tide Table 2 (Tidal Differences and Other Constants)" presents the mean and spring ranges by location for each water body and major port. If a specific site is not listed, it may be possible to extrapolate from the nearest site for which these data are reported. Typically, Reference (or Primary) Stations listed in NOAA Tide Table 2 are located along the coast at major port entrances and the Subordinate (or Secondary) Stations are sites further inland and between the Reference Stations along the coastline. Once again, wind, pressure, and other factors effect the accuracy of the predictions.

2.2.6 Simulating Tide Constituents

Predicted tidal station high and low tide values generally do not contain sufficient stage versus time information to apply in an unsteady analysis. In such cases, mathematical approximations of astronomical tides are used. These equations can range from simple to more complex. Two such equations are described below.

Simple Tidal Constituent Equation: A location with a semidiurnal or diurnal tide may be approximated using the simple cosine relationship. For cosine functions using radians or degrees the relationship is:

A Simple Tidal Constituent Equation (2.1)

where:

H_t(t) = Astronomical tide at time t (ft, m)
a = Tidal amplitude (ft, m)
t = Time (hr)
T = Tidal period (hr)
Z = Vertical offset or datum adjustment (ft, m)

The tidal period, T will be approximately 12.5 or 25 hours depending upon whether the tide is semidiurnal or diurnal. The amplitude can be selected as half the range between mean high and mean low with or between mean higher high and mean lower low water. The values of the mean tide range and mean spring tide range are available in the NOAA tide tables. Because tide levels can vary so much from one tidal cycle to the next, it is often desirable to use this simple relationship based on mean tide amplitude (or, more conservatively, the spring tide amplitude) when developing a numerical model. Sections 2.10 and 2.11 include worked examples of determining the constants in the simple tidal constituent equation.

Complex Tidal Constituents Equation: A more complex equation allows better replication of all types (i.e., mixed, semidiurnal, and diurnal) of observed astronomical tides and to make predictions of tide levels. Since the various forces that produce astronomical tides cycle on regular periods, tides can be decomposed into a sum of harmonic terms of known amplitude and frequency. The decomposition requires tide measurements over a long period, at least one year, and depending on the length of time, should not include significant wind or hurricane events (Herbich 1990). NOAA publishes the harmonic constants at the Reference Stations for use in the following equation:

Complex Tidal Constituents (2.2)

where:

η = H_t(t) = Astronomical tide at time t (ft, m)
α₀ = Vertical offset or datum adjustment (ft, m)
α_I = Amplitude of the tidal constituent (ft, m)
σ_I = Angular frequency, or speed (degrees or radians/hr)
t = Time since start of epoch (hr)
δ_i = Phase angle, or epoch (degrees or radians)

Figure 2.5 is an example of harmonic constant data from NOAA. For their predictions, NOAA applies 36 constituent variables into equation 2.2. In general, very good astronomical tide predictions can be produced from the first 6 constituents. The constituent names relate to the astronomical force. For example M2 and S2 are the lunar and solar semidiurnal constituents. The S2 constituent has a speed of 30 degrees per hour (360 degrees in 12 hours), which indicates that the S2 constituent repeats twice per day.

The constants can be located at http://tidesandcurrents.noaa.gov/ (Search terms:NOAA harmonic constants). The maximum tide level can be estimated by adding all of the amplitudes. This is an unusually high tide that occurs when all of the forces coincide at a time such that (s_it + d_i) are all equal to a multiple of 360 degrees. As shown by the solid line in Figure 2.6 (as well as Figures 2.3 and 2.4), predictions that are made using Equation 2.2 can be very accurate in the absence of wind. Sections 2.10 and 2.11 include worked examples of calculating water levels using the complex tidal constituents equation.

Figure 2.5. Example harmonic constant data from NOAA.

Figure 2.6. Example of predictions from harmonic constant data.

2.2.7 Tidal Currents

Depending on the coastal location, normal U.S. tidal ranges can vary from less than 1 foot to over 20 feet. This range, over a typical semidiurnal half-period of about 6.25 hours (assuming a lunar day of 25 hours) can result in a significant head between the ocean and a bay. This head will in turn cause a flow, or a tidal current

Tidal nomenclature describes a flood tide as the tidal current that occurs during rising tide conditions (or when the tide enters a waterway from the ocean). An ebb tide refers to the flow released (or returning) from the waterway to the ocean during falling tide conditions. Since the astronomical tides are predictable, it follows that the tidal currents are also predictable. In fact, NOAA has prepared annual "Tidal Current Tables" similar to the "Tide Tables." One can also get tidal current predictions from NOAA (www.co-ops.nos.noaa.gov). These predictions are available for approximately 2000 U.S. coastal locations.

These predicted tidal currents, while very helpful for navigation, are not appropriate for design and should be used with extreme caution, even for model calibration purposes. Partially, the reason for these caveats is because waterway bathymetry can vary even over short distances, so even a waterway segment with essentially the same tidal elevation may have very different amounts of tidal current. If tidal currents are used for model calibration, judgment should be used in their interpretation. As described by NOAA on their tidal current website: "Currents are spatially variable, thus predictions should NOT be extrapolated even to near-by locations. Interpolation between two near-by locations should NOT be attempted. Use of such extrapolations can be hazardous."

Tide levels and tidal currents within an estuary, inlet or bay result from the astronomical tides at the ocean boundary. The waterway bathymetry, channel geometry, and any upstream runoff/riverine flow control how the wave translates up and through the tidal waterway and, therefore, how the tidal currents vary through the waterway. For example, Figure 2.7 illustrates semidiurnal tidally induced water levels within the nearly 200-mile length of the Chesapeake Bay. At the point of time represented in the illustration, a high tide is occurring at the Chesapeake Bay entrance (southeast on the map). Nearly 135 miles upstream, a low tide occurs near Solomons Island, Maryland. Between these two locations, the tidal prism is irregular, partially attributable to the numerous shallow tributaries with their associated storage. Normally the tidal "wave" takes approximately 12 hours to transverse the entire Chesapeake Bay. However, during the period represented in the figure, rainfall events in upstream rivers have resulted in large freshwater inflows that attenuated the tidal wave.

An other example of the complexity of tidal hydrodynamics is the James River in Virginia. Figure 2.8 shows tide level relative to a fixed datum. As the tide progresses upstream from Newport News, at the mouth, to the confluence with the Chickahominy River, the tide range and mean tide level decrease. From the Chickahominy River up to Richmond, Virginia, the tide range amplifies and mean tide level rises. Figure 2.9 shows tides at Newport News, the Chickahominy River and Richmond from tide gage data. The tide decreases in amplitude as it translates upstream to the Chickahominy River in approximately three hours. From the Chickahominy River the tide increases in amplitude as the wave propagates up to Richmond in approximately another four hours. The rate of propagation and the changes in tide level and amplitude depend on channel geometry, flow resistance, and upstream inflow. Therefore, computing flow velocities at any point along the James River requires information on the tides at the mouth, channel geometry up the estuary, and upstream flow conditions. Major tributaries must also be included in the numerical model.

Figure 2.7. Tidal levels and effects in the Chesapeake Bay.

Figure 2.8. Tide levels along the James River, Virginia.

Figure 2.9. Tide translation on the James River, Virginia.

The hydrodynamic models discussed in this manual (Chapter 4) are used to simulate tidal currents. When the boundary conditions for a particular site are defined as the local predicted tide, without the additional component of a storm surge, then the resulting model flow will be the tidal current. In terms of bridge scour analysis, most studies will be concerned with a combination of both the tidal currents as well as the additional current resulting from the additional head due to the storm.

2.3 Storms and Other Climatologic Conditions

In addition to astronomical tides, coastal hydrology is affected by storms and other local atmospheric or climatologic conditions. Whereas in a riverine analogy, the astronomical tides may be considered the "base flow", the coastal storm events are the equivalent of the longer recurrence interval flows used in design and analysis of large flows.

Often coastal storm events are associated with hurricanes and other tropical events. In reality, coastal storms may consist of many different types of systems including nor'easters, El Niño episodes, winter storms, and other local atmospheric and climatologic conditions. Figure 2.10 shows the recorded tracks of topical and extratropical storms along the North American coastline for a 20 year period from 1984 to 2003. This figure was obtained at http://maps.csc.noaa.gov/hurricanes/ (Search terms: historical hurricane tracks). Tropical storms reaching the east and gulf coasts are highly variable in magnitude and track. Tropical and extratropical storms rarely reach the U.S. west coast. In all cases, these storm events produce additional increases (or decreases) of coastal water surface. Additionally, these storm events also have an associated time period that must be considered in developing overall tidal hydrology. The sudden impact of tsunamis is another hydraulic event that may be very destructive.

Figure 2.10. Storm Tracks along the North American Coastline (1984-2003).

This section describes characteristics and manifestations of these storm events. Additional information on and specific details on each of these storms are available in the literature and Internet sites (FEMA 2003). In many cases, associated criteria and scales assist in "ranking" the severity of such storm events. However, none of these scales should be assumed to correspond to any design frequency - derivation of such storm frequencies are site specific and require statistical and engineering assessment and judgment.

2.3.1 Tropical Storms and Hurricanes

A hurricane is a type of tropical cyclone, the general term for all circulating weather systems (counterclockwise in the Northern Hemisphere) over tropical waters. Tropical cyclones are classified as follows: Tropical Depression - an organized system of clouds and thunderstorms with a defined circulation and maximum sustained winds of 38 mph; Tropical Storm - an organized system of strong thunderstorms with a defined circulation and maximum sustained winds of 39 to 73 mph; and Hurricane - an intense tropical weather system with a well defined circulation and maximum sustained winds of 74 mph or higher. The term "sustained wind" refers to surface wind speeds (10 m above the surface) that persist for durations of one minute.

Hurricanes originate in the tropical oceans, frequently in the eastern Atlantic Ocean and are then powered by the heat from the sea. The hurricanes typically are steered westward by easterly trade winds. The Coriolis effect provides the characteristic cyclonic spin of these storms. Around their core, winds grow with great velocity, generating violent seas. As the fierce winds accompanied by the low pressure move ashore, the storm surge grows and creates extensive flooding. In addition, the hurricane carries with it torrential rains and can produce tornadoes which cause significant structural damage.

Hurricanes are rated on the Saffir-Simpson Hurricane Scale (Table 2.4). The scale is a one through five rating (called categories) based on the hurricane's intensity. Although lower category hurricanes are more frequent, these ratings do not have any direct correlation the recurrence interval of surge and other flooding events. Rather, the Saffir-Simpson ratings are an indicator of the property damage and coastal flooding that results from a hurricane landfall. Table 2.4 also summarizes wind speed, storm surge, and surface pressures associated with all five of the hurricane categories.

Since storm surge values are highly dependent on the continental shelf slope in the landfall region, wind speed is the determining factor in the scale. Winds are evaluated using the maximum one-minute surface wind speeds. Appendix C shows the number of hurricanes of different categories that have crossed 50 nautical mile segments of the U.S. coastline between 1886 and 1999.

Damages from hurricanes extend well inland. Frequently, the most noted or newsworthy aspect of hurricane damage results from flooding along the coastal area. This is particularly important in low-lying areas such as the coastal barrier islands. Of course, the flooding continues upstream in every inlet open to the ocean. The damages for each level of hurricane increase with the intensity of the storm.

2.3.2 Extratropical and Nor'easter Events

Cyclonic events such as extratropical storms form whenever unstable air produces significant temperature and pressure differences. At times, such systems may stall off the coast and produce long (i.e., several days) periods of high winds and inland rainfall. These events rarely obtain hurricane level wind speeds; however, they can cause significant coastal hydrological effects and wave damages.

Many historically significant events on the upper Mid-Atlantic and New England coasts were a result of Nor'easter storm events. The "Ash Wednesday" storm of March 1962 was formed by the combination of several slow moving coastal low pressure systems along the Atlantic seaboard. This combined storm resulted in hurricane force winds and water levels 9 or more feet above mean water level in areas of Maryland and Delaware over a period of several days (to contrast, for this same region, the 1933 "hurricane of record" produced water elevations 7 feet above mean low water). Likewise, the popularized 1991 "All Hallow's Eve" or "Perfect Storm" produced 5 days of high wave action, coastal erosion, and washover (USGS 2003). These extratropical events are not limited to the Atlantic Coast; Florida's west coast experiences severe flooding from such events. During one March 1993 event, at a location north of Tampa Bay, the resulting inundation (and damages) was similar to those predicted to occur from a Category 1 hurricane (Citrus County 2000). Likewise, the Great Lakes coastal regions endure wave damages during winter extratropical events (USGS 2003).

Nor'easters should be considered relative to wave attack, erosion and design high water at transportation facilities. While these types of events are extremely destructive, from a bridge hydraulic and scour standpoint, the rate of rise and fall of these events is generally too gradual to produce high flow velocities in tidal waterways. In some locations, these types of events may control the hydraulic design and should be considered on a case by case basis.

2.3.3 Tsunamis, El Niño, and Winter Storm Events

Tsunamis ("harbor wave") normally result from an underwater disturbance (usually an earthquake) that triggers a series of waves that can travel many hundreds (or even thousands) of miles. In the open ocean, the waves may average 450 miles per hour. Reaching shallower waters, the waves decrease speed, but gain amplitude. Tsunamis appear on the coast as a series of successive waves where the period from wave crest to wave trough can range between 5 and 90 minutes (but normally between 10 and 45 minutes). Typically, the first of these waves is not the largest. A 1964 Alaskan earthquake sent tsunami waves between 10 and 20 feet high along the coasts of Washington, Oregon, and California. In regards to frequency, Hawaii and Alaska can expect a damaging tsunami on the average of once every seven years, while the West Coast experiences a damaging tsunami once every seventeen years (FEMA September 1993). The sudden impact of a tsunami can be extremely destructive, but from a bridge hydraulic and scour standpoint, they are outside the scope of this document.

El Niño refers to a periodic rise in equatorial Pacific Ocean surface temperatures that affect global weather patterns. Historical data reveals a relationship between El Niño and Southern California tropical cyclones and flood events (USGS 2003, FEMA 2004). Additionally, El Niño is responsible for increases of sea level (on the order of 10 inches) as eastward flowing water accumulates on the West Coast shore (USGS 2003). Some research also indicates that both El Niño and La Nina events have some relationship in affecting wind conditions and the California current (Schwing and Bograd 2003). Figures 2.11 and 2.12 illustrate differences in coastal water surface elevations at a Northern California bridge waterway during El Niño (October 1997) and after El Niño (April 1998) (USGS 1998) episodes. El Niño may need to be considered relative to wave attack and freeboard requirements. To account for the hydrodynamic effect of an El Niño event on bridge hydraulics and scour, the tides could be simulated with and without a 1 ft vertical offset to determine the worst case hydraulic condition.

During winter months along the West Coast and Great Lakes, changes in the jet stream and other metrological conditions may trigger storm events that impact coastal regions. For example, the infamous "Pineapple Express" event occurs when the jet stream dips into the vicinity of Hawaii (thus the "pineapple") and carries a fast, moisture laden storm system to Washington, Oregon, and California. Unlike tropical events, these winter storms do not behave as cyclonic systems. Instead they are characterized by high winds that drive waves onto coastal areas. Again, these conditions could be a possible design concern related to wave attack or freeboard requirements for transportation facilities, but are probably not a design case for bridge hydraulics and scour.

Figure 2.11. Coastal water levels during El Niño event.

Figure 2.12. Coastal water levels following El Niño event.

2.4 Storm Surges

During the infamous Hurricane Camille in 1969, a 25-foot storm surge inundated Pass Christian in Mississippi. The predicted surge level for Cedar Point, Florida (north of Tampa Bay) for a (simulated) Category 5 hurricane is 34 feet. In 1961, Hurricane Carla inundated Galveston, Texas with an 8-foot surge. Clearly, storm surge is extremely dangerous and needs to be considered in coastal hydrology.

For the purposes of this manual, a storm surge is the rise of water level above the astronomical tide as a result of a tropical and extratropical event. Normally associated with hurricanes, surges may be formed by a variety of storm events. A surge is characterized by both this increase in water elevation and the duration that the water inundates a location.

2.4.1 Factors in Surge Formation

Factors in surge formation are dependent on the type of storm event and location in the United States. Wind speed is the most important factor in surge formation during hurricanes, nor'easters, and lake storms. Important factors that may influence the level and duration of a storm surge include:

• Astronomical tides
• Rainfall
• Basin bathymetry/hydrography
• Maximum wind speed
• Forward speed of storm
• Radius of maximum winds
• Central pressure of storm
• Storm track
• Atmospheric pressure difference
• Surface wind waves and wave setup
• Earth's rotation
• Initial ocean state

In a cyclonic event such as a hurricane, the storm rotates about a low pressure center. The lower the central pressure, the stronger the wind. At sea the dominant effect is the low pressure that "pulls" the water in the center of the storm vertically upward. As the surge moves toward the coast and the water depth decreases, the importance of wind and the confining shoreline becomes paramount. The low pressure continues to pull water in the center of the storm vertically upward while the wind blowing over the water causes an onshore flow. Waves moving landward break at the shoreline and add to the surge height.

2.4.2 Surge Constituents

The factors listed above may influence development of different surge constituents. These constituents are commonly referred to as Stillwater, Wave Setup, and Fetch (there are others) water elevations.

Stillwater refers to the surge elevation from barometric pressure and the direct wind on the water surface. Conceptually, the pressure gradient introduces a pressure head to the water surface, resulting in an increase in the water surface elevation. The direct wind describes the increase in water surface elevation as a result of horizontal stresses at the wind/water interface. Direct wind is the primary element of estimated stillwater elevation.

Wave setup is the additional increase in water surface elevation as wind fields create wave fields and radiation stresses along the nearshore coastal region. Conceptually, waves breaking onto shore carry a significant volume of water. During a storm event this volume does not rapidly or easily return offshore. The differences in the rates of water coming ashore and returning offshore results in water "piling up" at the shore.

Similar to wave setup,Fetch typically refers to a length of water that allows formation of relatively unimpeded wind fields. The effect of fetches manifests itself as increased water surface elevations as the wind drives (piles up) water along the leeward shore. Fetch can be a primary surge constituent in lakes and other large, open, (unusually) inland bodies of water. In 2003, tropical storm Isabel created extreme surge conditions in the upper Chesapeake Bay, as a result of strong wind fields driving water 170 miles northward. The resulting surges exceeded stillwater elevations produced by the 1937 hurricane of record.

A storm surge event may not exhibit all (or even many) of these constituents. Typical coastal flood insurance studies (FIS) include tables of both estimated stillwater and wave setup water surface elevations for given frequencies. Methods of estimating a surge may not consider all of these constituents, so caution is warranted when reviewing a surge elevation. Selection of appropriate values of water surface for any specific site or project may require consultation with a qualified coastal engineer.

2.4.3 Surge Configuration

At any affected location, storm surge does not happen immediately, rather water elevations exhibit dynamic, time dependent characteristics, analogous to the formation of a riverine runoff hydrograph. Additionally, depending on many of the factors detailed above, the surge elevation and duration can vary by location.

Conceptually (and simplistically), imagine the storm surge as a "cone" of water. The highest water elevation is typically just right of the storm "eye" (or peak of the storm) and tapers down as distance increases from this "apex" of high water elevation. As this "cone" of water travels towards the coast and makes landfall, areas directly in the path (or storm track) are expected to see indications of surge first. Water levels increase and the peak surge occurs before the storm eye passes over. As the storm continues to track over land, water surface levels diminish. Coastal areas affected by the "cone", but not directly on the storm track (i.e., the path of the apex of the "cone") typically see surge later and not reach as high on elevation (although this may not be true as a result of the factors described above).

For example, once again consider the case of hurricane Carla that occurred in the Gulf of Mexico and traveled into Texas from the 7th to 12th of September 1961 (USACE 1984). Coastal areas just to the right of Carla's hurricane eye track experienced surge earlier and reported the highest surge elevations. Coastal areas close to, but not on the track, reported that surge began later and exhibited lower peak water elevations.

2.5 Predicting Storm Surge Hydrology

Prediction of storm surge is very important for determining the potential effects of the hurricane on coastal engineering works, evacuation, erosion and scour in waterways. A myriad of techniques, methods, and models exist for the formulation, analysis, and creation of storm surges in coastal hydrology. These methods have been applied to nearly every coastal or tidally influenced area in the United States. Some (but not all) methods and models include:

• Historical data probability analyses
• Synthetic storm surge hydrograph method (Pooled Fund Study)
• FEMA Surge model
• Sea, Lake, and Overland Surges from Hurricanes (SLOSH) model
• Advanced Circulation (ADCIRC) model
• Florida DOT Surge Hydrographs

These methods typically develop stillwater elevations and may not represent wave setup effects. The products from these techniques are generally those surge values contained within FEMA FIS documents and materials. Some approaches provide both surge level and duration information; others only provide the peak surge level.

These methods generally adopt one of two basic approaches as the basis of storm surge predictions: probabilistic - direct stochastic analysis of storm history; anddeterministic - surge derived from using equations or numerical model(s) that simulate physical characteristics and processes.

This manual cannot provide details on all background, factors and issues of these methods and approaches, including applicability. However, this section will provide information on overall concepts and provide details on simpler approaches, as appropriate.

2.5.1 Issues in Selection of Storm Surge Hydrology

Selection of the storm surge hydrologic approach requires a large degree of coastal engineering judgment including consideration of contributions of historic and local conditions. The judgment should include an assessment of the relative vulnerability and criticality of the transportation facility. As an analogy to this assessment, many States differentiate culvert design flow frequencies depending on the relative expected average daily traffic - an Interstate culvert may require sufficient size to pass a 50-year flow, whereas a local rural roadway may only need to allow a 10-year flow frequency. In a coastal setting, this judgment may entail considering the different risks associated with a single designated evacuation route (e.g. Florida Keys) versus a location that may provide alternative evacuation segments (e.g. Delaware beaches).

Similarly, other factors may influence the degree of detail in tidal and coastal hydrologic approaches. For example, the relatively shallow landward approaches of the Gulf Coast of Florida may result in different surge hydrology than those encountered from Alaska, where the continental shelf rapidly transitions to the deep Northern Pacific oceanic basin. Likewise, many historic surge events in New England are associated with Nor'easter storms, in contrast to winter storms or El Niño events that drive surge in California. Even the Great Lakes, with their negligible tidal affects, may experience hydrologic surge attributable to the length of wind driven fetch and corresponding wave setup.

2.5.2 Probabilistic Approaches for Surge Prediction

Probabilistic approaches apply stochastic approaches to historical water level or storm records to predict surge elevations (and potentially duration). These probabilistic techniques generally apply generally accepted techniques such as those described in WRC Bulletin 17B (WRC 1982). However, similar to riverine stream gage analyses, any probabilistic approach would be conducted using appropriate data selection and interpretation (WRC 1982).

A primary issue for applying this approach is typically the lack of historical data associated with many coastal areas. However, many coastal areas and seaports have been keeping tidal records for a significant length of time and this data source should not be discounted. While not as robust, review of historical water levels at or near a project site might yield information useful in validation of other approaches.

Finally, these records are useful in evaluating any changes in sea-level or land subsidence. Such an evaluation assists in making predictions of present and future conditions (the topic of sea-level change will be addressed later in this document).

2.5.3 Deterministic Approaches for Surge Prediction

Deterministic approaches are typically more numerically sophisticated and complex than probabilistic approaches, but allow derivation of surge estimates at a much more extensive coastal area. Additionally, while some information can be obtained from historic water level records, these techniques allow prediction for specific storm events. Reid and Bodine (1968) provided the seminal information on storm surge prediction including the effects of Coriolis force, surface slope, inverse barometer, astronomical tidal potential, wind stress, bottom stress and rainfall rate. More complete treatment is given by Westerink et al. (1994) who extends the work of Reid and Bodine and others to a global circulation model that incorporates the dynamics of the hurricane based upon the specified hurricane track, central pressure and maximum winds. The latter are available for historic storms from the National Hurricane Center of NOAA.

Modeling to predict storm surge in the sea and along the coastline is complicated and typically requires the use of super computers. However, for nearly all historic storms, these models have been run and the storm surge off the coastline has been calculated and. The results of these simulations may include only peak surge values or complete storm tide hydrographs that can be used as boundary conditions for bay or estuary models.

Synthetic Storm Surge Hydrograph Method

The Synthetic Storm Surge Hydrograph method is one (simple) deterministic approach. The method provides a means to generate time-dependent surge values needed in an unsteady flow analysis. Generally this method provides conservative estimates of coastal hydrology that might be appropriate for scour studies conducted for a bridge waterway project. To develop a synthetic time series, the storm surge can be idealized by (Cialone et al. 1993).

(2.6)

where:

D = R/f = half storm duration (hr)
R = Radius to maximum wind speed (n-mile, km)
f = Forward speed of storm (knots, kph)
S_p = Peak surge elevation for selected return period (ft, m)
t = Time (hr)
t₀ = Time of peak surge or land-fall (hr)

The critical variables in the method are the half storm duration (D) and the peak surge elevation (S_p). D is determined as the ratio of radius of maximum winds (R) to the forward speed of the storm (f). The appropriate values of R and f used to compute D can be determined from the data provided by the NOAA report NWS-38 (1987) and reproduced as Appendix D in this manual. The Pooled Fund Study recommends using the 50 percent values of R and f (Ayres Associates 2002a). These data give values of R and f as a function of distance measured along the shoreline from the U.S. - Mexico border. Using the 50 percent values of R and f generally results in values of D between 1.5 and 2.5 hours. The amount of time that water levels are above normal (surge duration) produced by Equation 2.6 is much longer than the value of D. This is true for Equation 2.6 and, of course, for the actual surge.

The value of S_pis assumed to be for either the desired design event for hydraulic design or the 100- or 500-year event for scour analyses. Appendix E provides a listing of the 100- and 500-year storm surge heights generated from ADCIRC processed output as part of the Pooled Fund Study (Ayres Associates 2002a). The heights, given in meters relative to local mean sea level, are for the nearest location to a given water body. Other sources, including FEMA Flood Insurance Studies, NOAA, USACE, and State studies can be used to determine values of S_P or S_t.

Equation 2.6 creates a symmetrical storm surge hydrograph as the shape is based on atmospheric pressure effects alone. As a result of the assumptions of storm duration in Equation 2.6, the shape tends to have rapidly increasing and decreasing limbs. This shape may not produce as accurate a boundary condition for a hydraulic study as other methods. However, in many cases, the shape might be ideal in a scour analysis where the rapid changes in depth (and velocity) would produce conservative results.

In some cases, the falling limb of the surge hydrograph tends to be steeper than the rising limb and the water surface elevation can drop below zero (Ayres Associates 2002a). Equation 2.6 well represents the rising hydrograph limb. An alternative synthetic surge hydrograph has been developed to better represent the falling limb. This alternative equation is:

for t > t₀

(2.7)

Equation 2.7 is the same as Equation 2.6 except for the additional term that should only be applied after the peak surge. At the peak (t = t₀) both equations are undefined, but at this point S_t = S_P. When applying these equations in a spreadsheet, a practical way of avoiding division by zero is to add a very small number to the time. Equation 2.7 can be used to represent the falling hydrograph limb because it more realistically represents the hydrograph shape. Figure 2.13 shows the two synthetic hydrograph shapes with a peak surge, S_P, equal to one. Also shown in Figure 2.13 are the average of 16 major storm surges along the East Coast as predicted by an ADCIRC model. This is the data used to fit the additional term of Equation 2.7. The data shown on Figure 2.13 indicate that Equation 2.6 works well for the surge rising limb and that Equation 2.7 better represents the surge falling limb.

Figure 2.13. Alternative synthetic storm surge hydrographs.

FDOT Storm Surge Hydrographs

Coastal engineering consultants to the Florida Department of Transportation (FDOT) recently completed a study that investigated storm surge techniques and developed a site-specific 50-, 100-, and 500-year storm surge hydrographs for locations along most of the Florida coast. The study contrasted and applied a variety of deterministic approaches and models. Details on this study can be found on the FDOT Hydraulic website: http://www.dot.state.fl.us/rddesign/dr/DHSH.htm (Search terms: Florida design hurricane surge hydrographs). Many other states have conducted studies identifying peak surge elevations along their coastlines for various recurrence intervals but do not have complete hydrographs.

2.6 Storm Tides

The Storm Tide is the combination of the astronomical tide and the storm surge. The resulting storm tide hydrograph can be applied as the boundary condition in hydraulic and scour studies. However, the temporal aspect of both astronomical tides and storm surges complicate this application. Considerations in developing the storm tide hydrograph may include:

• Hurricane forward speed
• Hurricane radius of maximum wind
• Maximum water surface elevation
• Timing of the hurricane relative to the astronomical tide
  
  

Each of these considerations will affect the shape, rate of rise and rate of fall of the storm tide hydrograph and, therefore, the bridge's design hydraulic condition. For example, Figure 2.14 shows the observed storm tide and predicted astronomical tides at two gages along the west coast of Florida during Hurricane Gordon (September 2000). An inspection of Figure 2.14 reveals several interesting observations:

• Gordon passed the Old Tampa tide gage at high tide and passed the Cedar Key gage approximately four hours later very nearly at low tide.
• The difference between the storm tide observations and astronomical tide predictions is the storm surge - the maximum storm surge at both locations appears to be approximately 4 feet.
• While peak storm tide elevations at both locations were similar (approximately +5.5 feet MLLW), if the timing of the peak storm surge and high astronomical tide at Cedar Key had coincided, rather than being out of phase, the peak storm tide may have reached +8 feet MLLW.
• The duration of the storm surge appears to be approximately 18 hours at Old Tampa Bay and 10 hours at Cedar Key, which should not be confused with the duration variable, D, in Equations 2.6 and 2.7.
  
  

Figure 2.15 shows the astronomical tide, actual storm tide, and synthetic storm tide from Equation 2.7 for the Old Tampa Bay gage. From the National Hurricane Center data, hurricane Gordon had a forward speed (f) of 10 miles/hr and a radius of maximum winds (R) of approximately 30 miles. The duration variable (D) in Equations 2.6 and 2.7 would be equal to 3 hours (30 miles / 10 mph = 3 hrs). The duration of the actual and computed surges are greater than 15 hours. The synthetic storm tide matches the measured storm tide very well for the rising limb and is too steep on the falling limb. Even with the overly steep falling limb, the rising limb usually produces the most extreme hydraulic condition. The synthetic hydrograph is not intended to replicate every storm, but is meant as a simple, three variable (surge height, forward speed, radius of maximum wind) equation to develop reasonable boundary condition storm tides for design conditions.

Figure 2.14. Observed storm tides for Hurricane Gordon.

Figure 2.15. Comparison of observed and synthetic storm tides for Hurricane Gordon.

2.6.1 Combining Storm Surge and Astronomical Tides

With the selection of S_p the storm surge plus tide (storm tide) hydrograph can be determined by combining Equation 2.1 or 2.2 with either Equation 2.6 or 2.7 to form the following equation:

Hydrograph Equation 2.8

(2.8)

However, in order to apply Equation 2.8 as a storm surge boundary condition, the astronomical tide and storm surge properties must be known. One could develop a large set of surge hydrographs using Equation 2.8 with a range of input parameters for the hurricane properties and timings, however, the number of simulations required will be excessive. Therefore, the recommended approach is to develop a set of design hydrographs that incorporate the expected hurricane properties.

Since the storm surge can occur at any point in the tide cycle, a conservative approach is to model the peak storm surge timed with four points on a tide cycle, high, low, mid-rising and mid-falling tide. These four conditions would be for the hydraulic analysis. It is important that each simulation have the same peak water surface, S_tot, for the desired recurrence interval. This is achieved by selecting the peak surge time, t₀, (high tide, mid-rising tide , etc.) and adjusting S_p in Equations 2.6 or 2.7 until the maximum value of S_tot in Equation 2.8 matches the desired storm tide peak. Section 2.10 and 2.11 include worked examples of developing the parameters for computing storm surge and astronomical tides.

2.6.2 Single Design Hydrograph Approach

Due to the complexity of many tidal hydrodynamic models, even these four simulations may require excessive effort, especially if multiple bridge alternatives are being investigated. If only one simulation is run, it is a reasonable approach to time the surge peak with the mid-rising tide. This method is called the Single Design Hydrograph (SDH) approach. Figure 2.16 shows the four design hydrographs for a base tide with a 4-foot (1.2 m) range, a datum adjustment of 1 foot (0.3 m) between mean tide level and a fixed datum, and an 11-foot (3.35 m) peak water surface corresponding to the design condition.

2.7 Wind Considerations

Wind is a significant component of the surge at a coastline. The U.S. Army Corps of Engineers Storm Surge Analysis manual (USACE 1986) indicates that wind is the greatest component of storm surge and that the peak surge occurs in the area of maximum winds. Hurricane winds rotate in a counterclockwise direction in the northern hemisphere and clockwise in the southern hemisphere due to the rotation of the earth. Many mathematical models have been proposed to simulate the wind field in a hurricane to assist in determining the location and level of the storm surge. One of the simplest is the Rankin vortex. As shown in Figure 2.17, the highest wind speeds occur on the right side of hurricanes because the wind and forward speed are additive.

Figure 2.16. Example surge design hydrographs.

Each of the sources of data for surge along the open coast (NOAA, FEMA, and ADCIRC) already include wind effects. Wind also affects flow conditions on inland waterways. If wind effects are included in a surge simulation within an estuary, bay or other tidal waterway, then additional considerations are required. These include (1) adjusting wind speeds to durations sustained long enough to realistically move water within the waterway, (2) adjusting wind speeds because the wind has been interrupted by land as it moves from ocean to an embayment, (3) using a realistic storm path that is consistent with the simulated surge, and (4) using realistic hurricane wind fields.

Adjusting wind speeds requires information on the individual numerical model being applied. A hurricane is defined by maximum 1-minute sustained wind speeds. The numerical model should use longer duration wind speeds. Figure 2.18 shows one method for the adjustments that could be made between different duration sustained winds. The other adjustment is required because wind speeds drop as the wind moves inland. Figure 2.19 shows one approach for adjusting wind speeds based on the type of land crossed by the wind as it moves from open ocean to inland embayment. Storm tracks for the study area should be evaluated to determine probable paths. The NOAA coastal service center website ( http://hurricane.csc.noaa.gov/hurricanes/ (Search terms: historic hurricane tracks) is a good resource for locating historic hurricane tracks. The model being used should have a cyclonic wind field generator to produce realistic storm wind patterns.

Figure 2.17. Schematic of hurricane windfield with a forward speed "f".

Figure 2.18. Ratio of wind speed for duration t to the 2-hour winds speed.

Figure 2.19. Wind speed adjustment for nearshore terrain.

2.8 Upland Runoff

Upland runoff can affect storm surge heights and flow conditions in tidal waterways if significant runoff discharges occur during the surge. Hurricanes can produce significant amounts of rainfall and extreme flooding in river systems much farther inland than the flooding caused by the surge. Upland flood discharges could reduce flood tide discharges and increase ebb tide discharges because the upland flood would be filling volume upstream of the bridge during the flood tide and replenishing storage during the ebb tide. Upland flood discharges should only be included in a surge analysis if (1) the flooding is caused by the same hurricane that is causing the surge and (2) it is likely that upland flood runoff can reach the bridge during the surge. If the flooding is caused by a prior hurricane or other weather system, then the surge and flood are statistically independent and should be analyzed separately. As the drainage basin size upstream of the bridge increases there is less chance that upland runoff can significantly affect the flow at a bridge during the storm surge. This is true for either the flood or ebb surge condition.

In general, it is recommended that constant upland runoff rates equal to the mean daily flow be used in storm surge analyses. A more conservative approach would be to include an upland flow hydrograph as an input to the storm surge analysis. This hydrograph should consist of a constant inflow equal to the average daily flow up to the point when the flood tide at the bridge peaks. After the peak, the hydrograph representing a flood that can be generated from a basin with a time of concentration equal to 6 to 10 hours could be included. This short time of concentration is recommended because this is the period when hurricane-produced intense rainfall can occur in near coastal areas (USACE 1984). For large tidal waterways, upland runoff is unlikely to have a significant affect on surge flows.

Although it is unlikely that the 100-year storm surge and the 100-year upland flood would coincide, it may be necessary to analyze each condition independently to determine scour conditions in a tidal waterway. The peak surge discharge will decrease for bridges located farther from the coast and at some point the upland flood condition will exceed the surge condition. The peak surge discharge should be compared to the upland flood discharge for the same return period and the larger discharge will generally be used as the design condition. It may be necessary to analyze each condition for scour and design using the worst case.

2.9 OTHER CONSIDERATIONS

2.9.1 Sea Level Changes

The tide is the diurnal (or semidiurnal) variation of the water level along the coast due to the influence of the moon and sun. There are much longer period fluctuations in coastal water levels that sometimes need to be considered in some engineering applications. These changes, over decades, may be important in the design and analysis of structures with potential useful lives of fifty to one hundred years. Ocean level rise and fall are taking place in the context of even longer periods (on the order of tens of thousands of years) that have seen coastal water levels as much as 500 ft lower than today's levels.

While it is always useful to be aware of the changes that have taken place in geologic time, engineers are more concerned with the potential for change in the next few decades. Figure 7.17 illustrates the changes determined from tide gage records for a few U.S. Atlantic coast sites. For the sites shown in this figure the trend has been for an increase in sea level. An often-used estimate for the trend along the mid-Atlantic coast is a rate of sea level rise of approximately 1 ft per 100 years. For other U.S. sites the trend has been for a smaller rate of rise, and for some sites along the Alaskan coast the trend has been a decrease in sea level. These changes in sea level are due to a complex interaction of geologic (subsidence, rebound, and uplift) and climatic factors. Since changes in sea level at any location are related to a number of factors, the terms "apparent" or "relative" ocean level rise are used.

NOAA monitors sea level changes through its network of tide gages. This information is available through their web site: http://tidesandcurrents.noaa.gov/sltrends/sltrends.html (Search terms: NOAA tide data).

The U.S. Environmental Protection Agency (EPA) has invested considerable resources in the prediction of future trends in sea level. The following is an excerpt from one of these EPA reports, "The Probability of Sea Level Rise," Titus and Narayanan (1995):

The Earth's average surface temperature has risen approximately 0.6°C (1°F) in the last century, and the nine warmest years have all occurred since 1980. Many climatologists believe that increasing atmospheric concentrations of carbon dioxide and other gases released by human activities are warming the Earth by a mechanism commonly known as the "greenhouse effect." Nevertheless, this warming effect appears to be partly offset by the cooling effect of sulfate aerosols, which reflect sunlight back into space.

Figure 2.20. Yearly sea level, Atlantic Coast (Shore Protection Manual, USACE).

Climate modeling studies generally estimate that global temperatures will rise a few degrees (C) in the next century. Such a warming is likely to raise sea level by expanding ocean water, and melting glaciers and portions of the Greenland Ice Sheet. Warmer polar ocean temperatures could also melt portions of the Ross and other Antarctic ice shelves, which might increase the rate at which Antarctic ice streams convey ice into the oceans. Along much of the United States coast, sea level is already rising 2.5-3.0 mm/yr (10 to 12 inches per century).

Based on these assumptions, which the EPA report explains in detail, the results can be summarized as follows:

1. Global warming is most likely to raise sea level 15 cm by the year 2050 and 34 cm by the year 2100. There is also a 10 percent chance that climate change will contribute 30 cm by 2050 and 65 cm by 2100. These estimates do not include sea level rise caused by factors other than greenhouse warming

   .

2. There is a 1 percent chance that global warming will raise sea level 1 meter in the next 100 years and 4 meters in the next 200 years. By the year 2200, there is also a 10 percent chance of a 2-meter contribution, and a 1-in-40 chance of a 3-meter contribution. Such a large rise in sea level could occur either if Antarctic ocean temperatures warm 5°C and Antarctic ice streams respond more rapidly than most glaciologists expect, or if Greenland temperatures warm by more than 10°C. Neither of these scenarios is likely.

3. By the year 2100, climate change is likely to increase the rate of sea level rise by 4.2 mm/yr. There is also a 1-in-10 chance that the contribution will be greater than 10 mm/yr, as well as a 1-in-10 chance that it will be less than 1 mm/yr.

4. Stabilizing global emissions in the year 2050 would be likely to reduce the rate of sea level rise by 28 percent by the year 2100, compared with what it would be otherwise. These calculations assume that we are uncertain about the future trajectory of greenhouse gas emissions.

5. Stabilizing emissions by the year 2025 could cut the rate of sea level rise in half. If a high global rate of emissions growth occurs in the next century, sea level is likely to rise 6.2 mm/yr by 2100; freezing emissions in 2025 would prevent the rate from exceeding 3.2 mm/yr. If less emissions growth were expected, freezing emissions in 2025 would cut the eventual rate of sea level rise by one-third.

Along most coasts, factors other than anthropogenic climate change will cause the sea to rise more than the rise resulting from climate change alone. These factors include compaction and subsidence of land, groundwater depletion, and natural climate variations. If these factors do not change, global sea level is likely to rise 45 cm by the year 2100, with a 1 percent chance of a 112 cm rise. Along the coast of New York, which typifies the United States, sea level is likely to rise 26 cm by 2050 and 55 cm by 2100. There is also a 1 percent chance of a 55 cm rise by 2050 and a 120 cm rise by 2100.

These predictions of future changes in sea level are useful as guides. For most engineering studies it is important to recognize the potential for an increase in the rate of sea level rise and plan accordingly. Such changes, if and when they occur, will of course have potentially significant impacts on tidal hydrodynamics and structure performance due to the net change in water depth.

2.9.2 High-Velocity Flows

Floodwaters moving at high velocities can lead to hydrodynamic forces on structural elements in the water column, including drag forces in the direction of flow and lift forces perpendicular to the direction of flow. Oscillations in lift forces correspond to the repeated shedding of vortices from alternate sides of the structural element (for example, these vortices can often be seen in the wakes behind bridge pilings in rapidly moving water). High-velocity flows can also move large quantities of sediment and debris. Current FEMA Flood Insurance Study (FIS) "V" zone mapping procedures cannot accurately predict locations where high-velocity flows and their impacts will be felt.

High velocity flows can be created or enhanced by the presence of manmade or natural obstructions along the shoreline and by "weak points" formed by bridges or shore-normal canals, channels, and drainage features. For example, anecdotal evidence after Hurricane Opal struck Navarre Beach, Florida, in 1995 suggests that large engineered buildings channeled flow between them, causing deep scour channels across this area and washing out roads and homes situated farther landward. Observations of damage caused by Hurricane Fran in 1996 at North Topsail Beach, North Carolina, show a correlation between storm cuts across the area and ditches and bridge locations along the frontage road (FEMA 1999).

2.10 Tide and Storm Surge Example Problems (SI)

The following examples illustrate conversion of tide levels and datums, the use of simple and complex tidal constituents equations, and the development of storm tide hydrographs. The data used for these examples were obtained primarily from NOAA websites referenced earlier in this chapter.

2.10.1 Example Problem 1 - Tide Heights (SI)

Using the data provided below for the Cooper River Entrance at Charleston, South Carolina, determine the mean tidal range, mean amplitude, and the datum adjustments to covert the tide heights from MLLW to MTL and from MLLW to elevations in NGVD29.

Data:

Station ID: 8665530         Publications Date: 04/29/03

Name: Charleston, Cooper River Entrance, South Carolina

NOAA Chart: 11524          Latitude: 32º 46.9'

Charleston Quad:         Longitude: 79º 55.5'W

Elevations of tidal datums referred to Mean Lower Low Water (MLLW), in METERS, FEET:

HIGHEST OBSERVED WATER LEVEL (09/21/1989) = 3.817, 12.52

Name: CHARLESTON, COOPER RIVER ENTRANCE

SOUTH CAROLINA

NOAA Chart: 11524 Latitude: 32° 46.9' N

USGS Quad: CHARLESTON Longitude:79° 55.5' W

Elevations of tidal datums referred to Mean Lower Low Water (MLLW),

in METERS, FEET:

HIGHEST OBSERVED WATER LEVEL (09/21/1989)= 3.817, 12.52

MEAN HIGHER HIGH WATER (MHHW)= 1.757, 5.76

MEAN HIGH WATER (MHW)= 1.648, 5.41

NORTH AMERICAN VERTICAL DATUM-1988 (NAVD)= 0.957, 3.14

MEAN SEA LEVEL (MSL) = 0.891, 2.92

MEAN TIDE LEVEL (MTL) = 0.853, 2.80

NATIONAL GEODETIC VERTICAL DATUM-1929 (NGVD) = 0.658, 2.16

MEAN LOW WATER (MLW) = 0.057, 0.19

MEAN LOWER LOW WATER (MLLW) = 0.000, 0.00

LOWEST OBSERVED WATER LEVEL (03/13/1993)= -1.245, -4.08

Solution:

Mean Tidal Range = MHW - MLW = 1.648 - 0.057 = 1.591 m

Mean Tidal Amplitude = Mean Tidal Range / 2.0 = 1.591/2.0 = 0.796 m

MLLW - MTL = 0.00 - 0.853 = -0.853 m (subtract 0.853 m from all tide heights relative to MLLW to obtain elevations in MTL).

MLLW - NGVD29 = 0.00 - 0.658 = -0.658 m (subtract 0.658 m from all tide heights relative to MLLW to obtain elevations in NGVD29)

2.10.2 Example Problem 2 - Simple Tidal Constituent Equation (SI)

Using the data and results of Example Problem 1, determine the parameters for the simple tidal equation for mean tide conditions (Equation 2.1) so the resulting astronomical tide function computes tide elevations in the NGVD29 datum. Assume a semidiurnal tide and use the engineering approximation of the length of a tidal day. Compute the tide height at 48.5 hours. What is the maximum tide elevation in NGVD29 that can be predicted by the resulting equation? Compare this result with MHW in NGVD29. Also, identify the timing of high, low, mid-rising and mid-falling tides.

Solution:

a = tidal amplitude = 0.796 m (from Example Problem 1)

Z = vertical datum offset = MTL - NGVD29 = 0.853 - 0.658 = 0.195 m

Note: A value of Z equal to zero in Equation 2.1 results in tide levels relative to mean tide level (MTL). Therefore, the adjustment in Equation 2.1 is the difference between MTL and NGVD29, not the difference between MLLW and NGVD29.

T = half the tidal day (engineering approximation) = 25/2 = 12.5 hrs.

H_t(max) = 0.796 + 0.195 = 0.991 m (NGVD29)

MHW = 1.648 - 0.658 = 0.990 m (NGVD29)

(Note that these results should be the same since the equation was developed for mean tide amplitude.)

In Equation 2.1, the cosine function is a multiple of 360° (2 radians) whenever t is a multiple of T. Therefore:

Cos(360t/T) = 1 at t=0,T,2T.... Therefore high tides occur at t = 0 hours and every 12.5 hours thereafter.

Cos(360t/T) = -1 at t=T/2,3T/2, 5T/2.... Therefore low tides occur at t = 6.25 hours and at 12.5 hour intervals thereafter.

The first mid-rising tide occurs one-quarter tide cycle after the first low tide. Therefore, mid-rising tides occur at t = (6.25 + 12.5/4) = 9.375 hours and at 12.5 hour intervals thereafter.

The first mid-falling tide occurs one-quarter tide cycle after the first high tide. Therefore, mid-falling tides occur at t = (0.0 + 12.5/4) = 3.125 hours and at 12.5 hour intervals thereafter.

2.10.3 Example Problem 3 - Complex Tidal Constituents Equation (SI)

Using the harmonic constants provided below for the Cooper River Entrance at Charleston, South Carolina and the results of Example Problems 1 and 2, compute the tide elevation in NGVD29 at t = 100 hours using Equation 2.2 and the first four constants. What is the maximum tide elevation in NGVD29 that could occur using the first eight constants and compare this value with MHW and MHHW in NGVD29.

Data:

Harmonic Constants (P2)

# -- Order in which NOS lists the constituents.

Name -- Common name used to refer to a particular constituent, subscript refers to the number of cycles per day. Click here for definitions.

Ampl -- One-half the range of a tidal constituent.

Epoch -- The phase lag of the observed tidal constituent relative to the theoretical equilibrium tide.

Speed -- The rate change in the phase of a constituent, expressed in degrees per hour. The speed is equal to 360 degrees divided by the constituent period expressed in hours.

Please refer to the Tide and Current Glossary for definitions of terms.

Amplitudes are in Meters

Phases are in degrees, referenced to UTC (GMT)

Latitude: 32° 46.9' N Longitude: 79° 55.5' W

8665530 CHARLESTON, COOPER RIVER ENTRANCE , SC--------------------------

Solution:

Since datum offset, a₀, of zero in Equation 2.2 provides tide levels relative to the mean tide level (MTL), the datum offset,a₀, for tide elevations in NGVD29 is 0.195 m (from Example Problem 2).

α₀=0.195m

α₁cos(σ₁t+δ)= 0.783cos(28.9841042*100+10.4) = 0.686 m

α₂cos(σ₂t+δ₂)= 0.119cos(30.0000000*100+36.1) = -0.109 m

α₃cos(σ₃t+δ₃)= 0.172cos(28.4397295*100+354.9) = 0.130 m

α₄cos(σ₄t+δ₄) = 0.105cos(15.0410686*100+199.7) = -0.011 m

η = 0.195 + 0.686 - 0.109 + 0.130 - 0.011 = 0.891 m (NGVD29)

η_max(first 8 constituents) = 0.195 + 0.783 + 0.119 + 0.172 + 0.105 + 0.033 + 0.079 + 0.006 + 0.008 = 1.50 m (NGVD29)

MHHW = 1.757 - 0.658 = 1.099 m (NGVD29) (from conversion in Example Problem 1)
MHW = 1.648 - 0.658 = 0.990 m (NGVD29) (from conversion in Example Problem 1).

The maximum tide from the first eight constituents is 0.40 and 0.50 meters higher than MHHW and MHW. This difference is expected because MHHW and MHW are averages of high tide conditions.

2.10.4 Example Problem 4 - Storm Surge Hydrograph (SI)

Using the simple tidal constituent equation developed in Example Problem 2 and the following data, develop the input parameters for a 100-year combined storm surge and astronomical tide (equation 2.8) for Charleston Harbor. Use the synthetic surge hydrograph (Equation 2.6) to represent the storm surge. Time the surge peak with the third mid-rising astronomical tide.

Data:

Peak Storm Tide Elevations:
Appendix E ADCIRC 100-year = 3.59 m (local Mean Sea Level)
MSL - NGVD29 = 0.891 - .658 = 0.233 m (0.23 m)
ADCIRC 100-year = 3.59 + .23 = 3.82 m (NGVD29)
FEMA Study (FIS) stillwater = 12.0 to 13.0 feet (NGVD29) = 3.66 to 3.96 m (NGVD29)
South Carolina Study = 13.1 feet (NGVD29) = 3.99 m (NGVD29)

The three estimates of 100-year surge elevation are reasonably consistent. The South Carolina Study value of 3.99 m (NGVD29) is selected as it is slightly conservative.

Hurricane Parameter:
Appendix D 50% levels:
Radius of Maximum Wind, R = 26 nmi.
Forward Speed, f = 11 knots.
Half Duration, D = R/f = 26/11 = 2.36 hours

FEMA FIS:
Average Radius of Maximum Wind, R_avg = 22.1 nmi.
Average Forward Speed, f_avg = 10.2 knots.
Average Half Duration, D_avg = R_avg/f_avg = 2.17 hours

The values are reasonably consistent. Use D = 2.36 from Appendix D. A slightly steeper surge rate of rise and fall would be produced by using the lower value of D.

Mid-rising tides occur at t = 9.375 hours and at 12.5 hour intervals thereafter. The third mid-rising tide in this data series will occur at 9.375 + 2x12.5 = 34.375 hours. Therefore, t₀ = 34.375 hours.

Solution:

The value of S_p must be adjusted using a trial and error process until the maximum value of S_tot is equal to one of the surge values (3.99 m in this case). This will occur at or near t₀. S_p was determined by trial and error using a spreadsheet and a value of S_p = 3.63 m was obtained. This yields the target value of S_tot equal to 3.99 at t = 35.0 hours. Had a different time for hurricane landfall been used, a different value of the surge peak would have been required to produce a 3.99 m storm tide elevation. The spreadsheet input and results are shown in Figure 2.21. The complex tidal constituents equation could also have been used as the astronomical tide component to provide a more realistic base tide condition. If the more complex astronomical tide condition is used, the selected time period should represent average tide range conditions.

Check the result at t = t₀ and t = 35 hours.

Figure 2.21. Solution to Example Problem 4 (SI).

2.11 TIDE AND STORM SURGE EXAMPLE PROBLEMS (U.S. Customary)

2.11.1 Example Problem 1 - Tide Heights (U.S. Customary)

Using the data provided below for the Cooper River Entrance at Charleston, South Carolina, determine the mean tidal range, mean amplitude, and the datum adjustments to covert the tide heights from MLLW to MTL and from MLLW to elevations in NGVD29.

Data:

Station ID: 8665530 PUBLICATION DATE: 04/29/2003

Name: CHARLESTON, COOPER RIVER ENTRANCE SOUTH CAROLINA

NOAA Chart: 11524 Latitude: 32° 46.9' N

USGS Quad: CHARLESTON Longitude: 79° 55.5' W

Elevations of tidal datums referred to Mean Lower Low Water (MLLW), in METERS, FEET:

HIGHEST OBSERVED WATER LEVEL (09/21/1989) = 3.817, 12.52
MEAN HIGHER HIGH WATER (MHHW) = 1.757, 5.76
MEAN HIGH WATER (MHW) = 1.648, 5.41
NORTH AMERICAN VERTICAL DATUM-1988 (NAVD) = 0.957, 3.14
MEAN SEA LEVEL (MSL) = 0.891, 2.92
MEAN TIDE LEVEL (MTL) = 0.853, 2.80
NATIONAL GEODETIC VERTICAL DATUM-1929 (NGVD) = 0.658, 2.16
MEAN LOW WATER (MLW) = 0.057, 0.19
MEAN LOWER LOW WATER (MLLW) = 0.000, 0.00
LOWEST OBSERVED WATER LEVEL (03/13/1993) = -1.245, -4.08

Solution:

Mean Tidal Range = MHW - MLW = 5.41 - 0.19 = 5.22 ft

Mean Tidal Amplitude = Mean Tidal Range / 2.0 = 5.22/2.0 = 2.61 ft

MLLW - MTL = 0.00 - 2.80 = -2.80 ft (subtract 2.8 ft from all tide heights relative to MLLW to obtain elevations in MTL).

MLLW - NGVD29 = 0.00 - 2.16 = -2.16 ft (subtract 2.16 ft from all tide heights relative to MLLW to obtain elevations in NGVD29)

2.11.2 Example Problem 2 - Simple Tidal Constituent Equation (U.S. Customary)

Using the data and results of Example Problem 1, determine the parameters for the simple tidal equation for mean tide conditions (Equation 2.1) so the resulting astronomical tide function computes tide elevations in the NGVD29 datum. Assume a semidiurnal tide and use the engineering approximation of the length of a tidal day. Compute the tide height at 48.5 hours. What is the maximum tide elevation in NGVD29 that can be predicted by the resulting equation? Compare this result with MHW in NGVD29. Also, identify the timing of high, low, mid-rising and mid-falling tides.

Solution:

a = tidal amplitude = 2.61 ft (from Example Problem 1)

Z = vertical datum offset = MTL - NGVD29 = 2.80 - 2.16 = 0.64 ft

Note: A value of Z equal to zero in Equation 2.1 results in tide levels relative to mean tide level (MTL). Therefore, the adjustment in Equation 2.1 is the difference between MTL and NGVD29, not the difference between MLLW and NGVD29.

T = half the tidal day (engineering approximation) = 25/2 = 12.5 hrs.

H_t(max) = 2.61 + 0.64 = 3.25 ft (NGVD29)
MHW = 5.41 - 2.16 = 3.25 ft (NGVD29)

(Note that these results should be the same since the equation was developed for mean tide amplitude.)

In Equation 2.1, the cosine function is a multiple of 360° (2 radians) whenever t is a multiple of T. Therefore:

Cos(360t/T) = 1 at t=0,T,2T.... Therefore high tides occur at t = 0 hours and every 12.5 hours thereafter.

Cos(360t/T) = -1 at t=T/2,3T/2, 5T/2.... Therefore low tides occur at t = 6.25 hours and at 12.5 hour intervals thereafter.

The first mid-rising tide occurs one-quarter tide cycle after the first low tide. Therefore, mid-rising tides occur at t = (6.25 + 12.5/4) = 9.375 hours and at 12.5 hour intervals thereafter.

The first mid-falling tide occurs one-quarter tide cycle after the first high tide. Therefore, mid-falling tides occur at t = (0.0 + 12.5/4) = 3.125 hours and at 12.5 hour intervals thereafter.

2.11.3 Example Problem 3 - Complex Tidal Constituents Equation (U.S. Customary)

Using the harmonic constants provided below for the Cooper River Entrance at Charleston, South Carolina and the results of Example Problems 1 and 2, compute the tide elevation in NGVD29 at t = 100 hours using Equation 2.2 and the first four constants. What is the maximum tide elevation in NGVD29 that could occur using the first eight constants and compare this value with MHW and MHHW in NGVD29.

Data:

Harmonic Constants (P2)

# -- Order in which NOS lists the constituents.

Name -- Common name used to refer to a particular constituent, subscript refers to the number of cycles per day.

Click here for definitions.

Ampl -- One-half the range of a tidal constituent.

Epoch -- The phase lag of the observed tidal constituent relative to the theoretical equilibrium tide.

Speed -- The rate change in the phase of a constituent, expressed in degrees per hour. The speed is equal to 360 degrees divided by the constituent period expressed in hours.

Please refer to the Tide and Current Glossary for definitions of terms.

Amplitudes are in Meters

Phases are in degrees, referenced to UTC (GMT)

Latitude: 32° 46.9' N Longitude: 79° 55.5' W

8665530 CHARLESTON, COOPER RIVER ENTRANCE , SC--------------------------

Solution:

Since datum offset,a₀, of zero in Equation 2.2 provides tide levels relative to the mean tide level (MTL), the datum offset, a₀, for tide elevations in NGVD29 is 0.64 ft (from Example Problem 2).

α₀= 0.64 ft

α₁cos(σ₁t+δ₁) = 2.569cos(28.9841042*100+10.4) = 2.251 ft

α₂cos(σ₂t+δ₂) = 0.390cos(30.0000000*100+36.1) = -0.356 ft

α₃cos(σ₃t+δ₃) = 0.564cos(28.4397295*100+354.9) = 0.425 ft

α₄cos(σ₄t+δ₄) = 0.345cos(15.0410686*100+199.7) = -0.037 ft

η = 0.64 + 2.251 - 0.356 + 0.425 - 0.037 = 2.92 ft (NGVD29)

η_max(first 8 constituents) = 0.64 + 2.569 + 0.390 + 0.564 + 0.345 + 0.108 + 0.260 + 0.020 + 0.025 = 4.92 ft (NGVD29)
MHHW = 5.76 - 2.16 = 3.60 ft (NGVD29) (from conversion in Example Problem 1)
MHW = 5.41 - 2.16 = 3.25 ft (NGVD29) (from conversion in Example Problem 1).

The maximum tide from the first eight constituents is 1.32 and 1.67 feet higher than MHHW and MHW. This difference is expected because MHHW and MHW are averages of high tide conditions.

2.11.4 Example Problem 4 - Storm Surge Hydrograph (U.S. Customary)

Using the simple tidal constituent equation developed in Example Problem 2 and the following data, develop the input parameters for a 100-year combined storm surge and astronomical tide (equation 2.8) for Charleston Harbor. Use the synthetic surge hydrograph (equation 2.6) to represent the storm surge. Time the surge peak with the third mid-rising astronomical tide.

Data:

Peak Storm Tide Elevations:
Appendix E ADCIRC 100-year = 3.59 m (local Mean Sea Level) = 11.8 ft (MSL)
MSL - NGVD29 = 2.92 - 2.16 = 0.76 ft (0.8 ft)
ADCIRC 100-year = 11.8 + 0.8 = 12.6 ft (NGVD29)
FEMA Study (FIS) stillwater = 12.0 to 13.0 feet (NGVD29)
South Carolina Study = 13.1 feet (NGVD29)

The three estimates of 100-year surge elevation are reasonably consistent. The South Carolina Study value of 13.1 ft (NGVD29) is selected as it is slightly conservative.

Hurricane Parameter:
Appendix D 50% levels:
Radius of Maximum Wind, R = 26 nmi.
Forward Speed, f = 11 knots.
Half Duration, D = R/f = 26/11 = 2.36 hours

FEMA FIS:
Average Radius of Maximum Wind, R_avg = 22.1 nmi.
Average Forward Speed, f_avg = 10.2 knots.
Average Half Duration, D_avg = R_avg/f_avg = 2.17 hours

The values are reasonably consistent. Use D = 2.36 from Appendix D. A slightly steeper surge rate of rise and fall would be produced by using the lower value of D.

Mid-rising tides occur at t = 9.375 hours and at 12.5 hour intervals thereafter. The third mid-rising tide in this data series will occur at 9.375 + 2x12.5 = 34.375 hours. Therefore, t₀ = 34.375 hours.

Solution:

The value of S_p must be adjusted using a trial and error process until maximum value of S_tot is equal to 13.1 ft. This will occur at or near t₀. S_p was determined by trial and error using a spreadsheet and a value of S_p = 11.93 ft was obtained. This yields the target value of S_tot equal to 13.1 at t = 35.0 hours. Had a different time for hurricane landfall been used, a different value of the surge peak would have been required to produce a 13.1 ft storm tide elevation. The spreadsheet input and results are shown in Figure 2.22. The complex tidal constituents equation could also have been used as the astronomical tide component to provide a more realistic base tide condition. If the more complex astronomical tide condition is used, the selected time period should represent average tide range conditions.

Check the results at t = t₀ and t = 35 hours.

Figure 2.22. Solution to Example Problem 4 (U.S. Customary).

<< Previous | Contents | Next >>