## Y JBcurlTp

at ox where p is pressure; t is time; x is the distance eastward, y northward, and z upward; ris wind stress;/is the Coriolis parameter; p0 is a constant representative water density; and yn is the long Rossby wave speed for the nth vertical mode. The term B is an amplitude factor determined by the vertical structure function. Details are given in Sturges et al. (1998). The wave speed, -y, can be computed directly from the (known) density distribution in the vertical direction. One very great advantage of our model formulation is that everything in the model is either physics or observations; there are no "free parameters" to tune to make the results look right. The wind stress, r, is obtained from an oceanic wind compilation, the Comprehensive Ocean-Atmosphere Data Set, or "COADS" archive (Slutz et al., 1985).

The physical meaning of the terms in (1) is that the first term is the time rate of change of pressure at any point, which could be at the sea surface or deeper. The second term, the product of the wave speed, y, times the local slope of the pressure surface, represents the rate at which the free waves travel, given no external forcing. The last term represents the forcing of the large-scale winds. The curl is the familiar mathematical operator; this term is often call the "Ekman pumping" because it represents the local convergence of any surface "Ekman transport" by the local winds. That is, if more surface transport is being driven into the region than is going out the other side, there must be a convergence; the only place it can go is down, forcing near-surface water deeper into the ocean. Such vertical velocities are extremely small (order of 10~5 cm/s) but of major dynamical importance in the ocean. Of course, this term can be positive or negative.

An important thing to notice here is that the model contains no frictional terms that can be conveniently "tuned" to get an answer we like. The wave speeds, y, are computed a priori from the density field, as are the vertical structure functions. The wind stress is determined from historical ships' observations (available from the National Climatic Data Center at www.ncdc.noaa. gov). The model produces an output that mimics the variability of the sea surface of the central North Atlantic Ocean since the end of WWII; a simple movie of this output is presented on the CD-ROM that accompanies this volume.

Figure 7.4 has the mean trend, approximately 18 cm/century (the "rise of sea level" signal), removed, since there is a long-term rise of sea level even in mid-ocean. The important point, however, is that we know the signals at Bermuda are propagating to the west. We can be certain, therefore, that they will strike the western boundary of the ocean. Thus we are led to a very important conclusion: the decadal variability of sea level on the U.S. east coast is most likely the result of the impact of such waves.

### 7.2.3 Observed Signals in the Pacific Ocean

The fluctuations in the sea level record at Hawaii are not terribly different, qualitatively, from those at Bermuda. And the waves that travel past Hawaii are going to the west, just like those at Bermuda. They will never reach the U.S. mainland. The only waves sufficiently energetic to reach the U.S. west coast, by contrast, are those that are trapped at the equator and which (because of quite different physics) travel to the east. Large, low-frequency signals at near-equatorial latitudes, such as the famous ENSO (EI Nino, Southern Oscillation) signals, are rather fast and travel along the equator from west to east. When they strike the continent, they then propagate poleward along the coast, in both hemispheres. See, for example, Clarke (1992) for details of the reflection at the boundary and Chelton and Davis (1982) for an insightful discussion of the processes by which waves propagate up the U.S. west coast. White (1977) made an interesting discovery about annual period waves that travel across the Pacific, but of course, the frequencies of interest for the sea level rise problem are at periods of many years. Other studies of wave propagation in the Pacific include those of Qui (1997) and Cummins et al. (1986).

### 7.3 THE UNDERLYING PHYSICS OF THE PROBLEM

We have only recently learned that the cause of the decadal signals on the east coast of continents is from wind forcing over the open ocean. This work is described by Hong et al. (2000). Other candidate forcing mechanisms such as variations in local winds along the coast or salinity variations caused by changes in river runoff effects have been studied and found to be unimportant for these sea level signals (Hong et al. 2000). Although the propagation of long Rossby waves across the ocean is a relatively simple process to model, the question of how to understand the passage of these waves across or through the Gulf Stream was the major problem attacked by Hong et al. (2000).

In general, we would expect that as a long Rossby wave passes through the Gulf Stream, the strong horizontal shear in the stream would destroy most of the identifiable wave characteristics. Yet it has long been known that at periods as long as a year the annual cycle of stored heat causes the ocean to go up and down at the coast just as out in the open ocean. In the case of the decadal signals, however, the waves clearly must propagate through the Gulf Stream to reach the southern U.S. east coast, and it was not understood how this could happen.

Figure 7.5, taken from Hong et al. (2000), shows a comparison between the observed sea level variability at a coastal station on the coast of Delaware and the computed sea level variability from the Rossby wave model. The mean slope has been removed. The importance of this result is that the low-frequency (interannual and longer) variability at the coast has been computed entirely from a model that uses only winds over the full width of the Atlantic Ocean as its forcing mechanism. Therefore the addition of variations in salinity, of river inputs, of atmospheric pressure variability, or of local winds along the coast can only serve to make small, minor improvements in the results shown in Fig. 7.5.

Figure 7.5 Comparison between coastal sea level and a Rossby wave model calculation. Sea level at Lewes, Delaware, on the U.S. east coast, is shown by the dashed curve; the model output is shown by the solid curve. The model is forced only by open ocean winds. Adapted from Hong et al. (2000).

It is worth emphasizing that the sea level signals of Fig. 7.5 are produced by the interactions between the winds and the ocean within the model of Hong et al. (2000). That is, they are an oceanic phenomenon and not in the wind field itself. To put it another way, the model output of Fig. 7.5 is in no way a simple linear modification of the input wind forcing signals. It represents the complex response of the ocean to a nearly random wind forcing. Although we understand this process enough to model it adequately, the underlying nature of this response is not thoroughly understood.

7.3.1 Correcting Observed Sea Level Rise for Wind-Induced Decadal Signals

At this point the next logical step should be obvious: using the ocean winds as inputs, we compute the model sea level signals and subtract the wind-forced part to determine the true "rise of sea level" at the coasts. Unfortunately, doing the calculation requires knowing the wind field. These long term winds are not available. During World Wars I and II, taking wind data over the oceans took a back seat to hiding convoys of ships from submarines. As a result, there are large gaps in the wind data set during the war years, making it much more difficult to calculate the wind-induced sea level signals. There are techniques for attempting to interpolate across gaps in a record, but one has little assurance that simple methods produce reliable results. No one knows, obviously, what the winds were, but certain statistical properties of the wind field can be assumed to remain unchanged from one time period to the next. This is clearly a research-level project, and the authors are attempting to complete the work even as this is written.

APPENDIX: SOME DETAILS OF THE SIGNAL PROCESSING OF TIDE GAUGE DATA

Figure 7.A.1 (upper panel) shows an example at Charleston, South Carolina, of what the signal looks like when we do not remove the tides or other high-frequency motions. Figure 7.A.1 (lower panel) shows these variations after the tides have been removed but the month-to-month variability has not been removed. The most noticeable variability is an irregular annual cycle. This mid-latitude annual variability has been shown to be caused by the stored heat in the upper hundred meters of the ocean, which is in turn associated with the annual solar cycle.

The variability in Fig. 7.A.1 (lower panel) however, is obviously very irregular. The reason for this, perhaps surprisingly, is that the strongest effect at periods of a few days is associated with longshore currents, associated in turn with local wind forcing. Figure 7.A.2 shows the signal in this so-called "wind band," or synoptic band. The variations shown in Fig. 7.A.2 have been filtered

Time (year)

Figure 7.A.1 Examples of sea level at Charleston, in various forms. Upper panel shows the actual hourly values, in which the main feature that emerges is the tidal cycle. Lower panel shows monthly mean values. In this presentation the annual cycle is dominant, but irregular (mostly wind-driven) features are still apparent.

Time (year)

Figure 7.A.1 Examples of sea level at Charleston, in various forms. Upper panel shows the actual hourly values, in which the main feature that emerges is the tidal cycle. Lower panel shows monthly mean values. In this presentation the annual cycle is dominant, but irregular (mostly wind-driven) features are still apparent.

to remove the daily tides out to periods of about 30 h, but to retain power at periods longer than 72 h. The literature on the subject of wind-driven coastal currents is enormous; see, for example, Csanady (1982), Mitchum and Clarke (1986), or Clarke and Van Gorder (1986). We wish to emphasize that the decadal signal of roughly 10 cm/century, while an overwhelmingly significant feature over many years, is relatively small when viewed from a high-frequency perspective. The filtering operations must be done carefully. It is remarkable that coastal tide gauge measurements are sufficiently accurate to allow these small signals to emerge from the background of noise and other effects. Careful analysis of the way the tidal system is kept in calibration, however, shows that this level of accuracy is built into the methodology (Mitchum, 1994).

In examining the lowest frequency signals it is usually adequate to ignore the effects of atmospheric pressure fluctuations because these are small at long periods. At the time scale of a few days to a few weeks, as weather systems pass over a region, the effects of atmospheric pressure signals cause substantial sea level variability, and so must be taken into account. Yet for

Time (hour)

Figure 7.A.2 Sea level at Charleston, as in Fig. 7.A.I. The main feature here is the wind-forced response associated with local oscillatory longshore flows. The signal here shows the energy only at periods longer than 30 hours, suppressing the tidal variations.

Time (hour)

Figure 7.A.2 Sea level at Charleston, as in Fig. 7.A.I. The main feature here is the wind-forced response associated with local oscillatory longshore flows. The signal here shows the energy only at periods longer than 30 hours, suppressing the tidal variations.

periods as long as a year these signals tend to be small at low latitudes and grow larger at higher latitudes. Obviously, a great deal of signal processing takes place before the long-term rise of sea level of Fig. 7.1 becomes evident. It must be done carefully. Small nonlinear terms cannot be safely neglected.

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