Global sea level is rising at an increasing rate and communities and cities around the planet are in the way. While we know the historic and recent rates of sea level rise, projections for the future are difficult due to political, economic, and social unknowns, as well as uncertainties in how the vast ice sheets and glaciers of Antarctica and Greenland will respond to continued warming of the atmosphere and the oceans. It is clear, however, that sea level will continue to rise for centuries due to the greenhouse gases already in the atmosphere as well as those we continue to produce. A rising ocean leads to a retreating coastline, whether gradual inundation of low-lying shoreline areas or increased erosion of cliffs, bluffs, and dunes. Coastal armoring and beach nourishment have been the historical approaches to address coastal or shoreline erosion, but these are laden with economic and environmental costs, often short-lived, and have significant impacts on beaches; their approval by permitting agencies is also becoming more difficult, at least in California (Griggs and Patsch 2019) but also in a number of other states. Coastal communities and cities are already experiencing the impacts of rising seas and more will experience these impacts in the decades ahead. Many cities in California are beginning to discuss, consider, and plan for how they will adapt to higher sea levels, but not without controversy, especially concerning managed retreat. However, over the long run, they all will respond through relocation or retreat of some sort, whether managed or unmanaged. Sea level rise will not stop at 2050 or 2100. Effective adaptation will require a collaborative process involving many stakeholders, including coastal home and business owners, local governments, and state permitting agencies in order to develop and implement policies, plans and pathways for deliberate adaptation to the inevitable future. For many reasons, this is a complex problem with no easy or inexpensive solutions, but the sooner the science is understood and all parties are engaged, the sooner plans can be developed with clear trigger points for adaptive action, ultimately relocation or retreat.
Beaches form a significant component of the economy, history, and culture of southern California. Yet both the construction of dams and debris basins in coastal watersheds and the armoring of eroding coastal cliffs and bluffs have reduced sand supply. Ultimately, most of this beach sand is permanently lost to the submarine canyons that intercept littoral drift moving along this intensively used shoreline. Each decade the volume of lost sand is enough to build a beach 100 feet wide, 10 feet deep and 20 miles long, or a continuous beach extending from Newport Bay to San Clemente. Sea-level rise will negatively impact the beaches of southern California further, specifically those with back beach barriers such as seawalls, revetments, homes, businesses, highways, or railroads. Over 75% of the beaches in southern California are retained by structures, whether natural or artificial, and groin fields built decades ago have been important for local beach growth and stabilization efforts. While groins have been generally discouraged in recent decades in California, and there are important engineering and environmental considerations involved prior to any groin construction, the potential benefits are quite large for the intensively used beaches and growing population of southern California, particularly in light of predicted sea-level rise and public beach loss. All things considered, in many areas groins or groin fields may well meet the objectives of the California Coastal Act, which governs coastal land-use decisions. There are a number of shoreline areas in southern California where sand is in short supply, beaches are narrow, beach usage is high, and where sand retention structures could be used to widen or stabilize local beaches before sand is funneled offshore by submarine canyons intercepting littoral drift. Stabilizing and widening the beaches would add valuable recreational area, support beach ecology, provide a buffer for back beach infrastructure or development, and slow the impacts of a rising sea level.
We construct a broad class of solutions of the KP-I equation by using a reduced version of the Grammian form of the τ-function. The basic solution is a linear periodic chain of lumps propagating with distinct group and wave velocities. More generally, our solutions are evolving linear arrangements of lump chains, and can be viewed as the KP-I analogues of the family of line-soliton solutions of KP-II. However, the linear arrangements that we construct for KP-I are more general, and allow degenerate configurations such as parallel or superimposed lump chains. We also construct solutions describing interactions between lump chains and individual lumps, and discuss the relationship between the solutions obtained using the reduced and regular Grammian forms.
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