Friday, June 17, 2011

Earth

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Planet Ocean


The single most striking fact about the Earth is that its awash with water. Dominating our planets surface and affecting the lives of everyone, even those who live far inland, the Earths ocean -- the vast expanse of water circling the globe and comprising the Atlantic and Pacific Oceans and numerous smaller seas -- has long been a source of wonder and awe. From the earliest recorded times, men and women have sought to understand the behavior of the ocean and of the life within it. Our knowledge of the ocean is far from complete, but is steadily advancing -- thanks in great part to new developments in mathematics.


Millennia of trial-and-error experience led to practical and sometimes elegant solutions to problems in ship-building, navigation, fishing strategy, and the anticipation of oceanic activity ranging from rough seas to the rhythm of tides. During the last few centuries, our understanding of the ocean has become increasingly scientific. The observations and accumulated wisdom of mariners throughout the ages have been augmented by detailed measurements of water temperature and salinity and by greater physical understanding of the watery forces that cause waves and currents.


The scientific approach brought with it the need for mathematical analysis. Oceanography today uses mathematical equations to describe fundamental ocean processes and requires mathematical theories to understand their implications. Researchers use statistics and signal processing to weave together the many separate strands of data from sonar buoys, shipboard instruments, and satellites. Partial differential equations describe the mechanics of fluid motion, from the surface waves that rock sea-going ships to the deep currents that sweep around the globe. Numerical analysis has made it possible to obtain increasingly accurate solutions to these equations; dynamical systems theory and statistics have provided additional insights. Todays oceanographers are really mathematicians, in the best tradition of Galileo and Newton. Mathematics, you might say, is the salty language of modern oceanography.


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Whats math got to do with it?


At its most elemental, any ocean process is all about change. Measurable quantities may change as time passes (for example, tidelines on a beach move from low to high twice a day) or may change from location to location (for example, pressure on a submarine increases as it dives deeper into the sea), but most quantities such as temperature and salinity change based on both position and time. The areas of mathematics which are critical to the description of changing processes are calculus and differential equations. In particular, partial differential equations (PDEs, for short) are used to describe quantities that change continuously in time and space. All areas of oceanography rely heavily on these subjects.


Marine geology and marine geophysics, for example, study the structure of the Earth as a whole and the changes it has undergone through time. Seismic studies for oil exploration, predictions of tsunamis (devastating waves created by deep sea earthquakes), and investigations into the formation of the largest mountain ranges on our planet (the oceanic ridges) are among the interests of marine geologists and geophysicists. Chemical oceanography studies the chemistry of aquatic environments, with special attention to interactions between the Earths crust, the so-called biota (micro-organisms, plants, and animals), and the atmosphere. Marine chemists are particularly interested in understanding both the natural phenomena and the human-generated changes affecting the chemistry of the worlds oceans, rivers, and lakes.


Similarly, biological oceanography studies how marine lifeforms interact with each other and with their ocean environment. Marine biologists chart the populations of biota in estuaries, in coastal zones, and in the open sea, with the ultimate goal of mathematically modeling and predicting their growth and migration patterns. Both biological and chemical oceanographers are concerned with ecosystem modeling mathematical representations of interactions between the oceans biological and chemical constituents, such as plants and animals and the nutrients they feed on. Can you imagine learning about whales, for example, without asking about what they eat? Ecosystem modeling, especially in coastal contexts, is concerned with immediate practical issues such as how to predict the amount of biological productivity, the fate of pollutants, and the appearance of harmful algal blooms.


Marine geology, chemistry, and biology occur within the context of the dynamical behavior of the ocean, which is the province of physical oceanography. Physical oceanographers study the full spectrum of circulation patterns of the ocean, from breaking waves on stormy beaches to the great currents and eddies that transport mass and energy (mostly in the form of heat) around the globe and that interact with atmospheric dynamics to drive the weekly weather and the Earths long-term climate. Physical oceanographers rely on geophysical fluid dynamics to characterize the behavior of fluids (such as ocean waters) on a rotating globe (the Earth). The Earths rotation pushes on large-scale fluid flows in much the same manner as the rotation of a merry-go-round pushes on a person walking a radial line from its center to its rim. This phenomenon, called the Coriolis effect, must be included in any description of large-scale ocean phenomena. (The Coriolis force is not strong enough to affect small-scale fluid behavior such as water draining from an ordinary household bathtub!)


The notion that partial differential equations may be used to describe the motion of physical fluids goes back at least to the Swiss mathematician Leonhard Euler. In 1755, he gave the first physically and mathematically successful description of the behavior of an idealized fluid. The Euler equations, as theyre called today, are a set of nonlinear PDEs which express Newtons law of force equals mass times acceleration for a non-viscous fluid -- the watery equivalent of a frictionless mechanical system.


In 181, Claude Navier improved on Eulers equations by including the effects of viscosity. Oddly enough, the equations he obtained are correct, even though the physical assumptions on which he based his derivation were wrong! In 1845, George Gabriel Stokes rederived the same set of equations, but on a more sound theoretical basis. The result, known as the Navier-Stokes equations, forms the starting point for all modern fluid dynamics studies. Together with the laws of thermodynamics, which were developed in the latter half of the nineteenth century, they are the basis for modern physical oceanography.


The study of nonlinear PDEs is a huge field that underlies much of applied mathematics. With certain notable exceptions, the presence of nonlinearity makes it virtually impossible to obtain exact solutions to these equations. This is certainly true of the Navier-Stokes equations. Consequently, much work is being carried out in computational fluid dynamics, with the goal of using computers to approximate numerically the solution of the Navier-Stokes (and Euler) equations. Researchers also attempt to simplify the equations in order to emphasize key physical features and to reduce the computational problem to a manageable size. An ongoing challenge for oceanographers and mathematicians is to understand enough about the physical meaning of the Navier-Stokes equations to make sensible simplifications. The goal is to work with simplified versions that still provide useful approximate descriptions and predictions.


What could be so difficult about simplifying the equations of fluid dynamics? There are two major obstacles which any study of ocean behavior must overcome the vast range of temporal and spatial scales present in the ocean and the tendency of fluid flows to be unstable. Physical oceanography must contend with turbulent eddies that span mere centimeters and last mere seconds; traveling surface gravity waves with wavelengths of kilometers and periods of minutes to hours; ocean tides with wavelengths of thousands of kilometers and periods of half a day; and ocean currents with spatial extents of thousands of kilometers and lifetimes measured in centuries. The computation of ocean circulation on these scales, from a millimeter up to the size of the Earth, is an enormous problem. Current theory and technology cannot approximate behavior over such a wide scope.


Similarly, the tendency toward instability complicates the prediction of fluid behavior. Even in a stable flow, the trajectory of an idealized fluid particle can be unpredictable. The eventual path of a fluid particle, or some object carried by the flow, can be highly sensitive to its initial position. Put two floating objects -- say Tom Hanks and a volleyball -- side by side in the ocean, wait a few days, and the chance of finding them still together is a Hollywood coincidence. Instability makes matters that much worse.


The basic problem is that small disturbances to a flow may, if they have the right structure, draw energy from the flow and grow rapidly until they are so large as to alter the flow in fundamental ways. This kind of instability can lead to turbulence; one atmospheric example is gusts of wind on a breezy day. The mathematical and physical elements of oceanic instabilities are similar to those that operate in the atmosphere and make the prediction of storms so very difficult for meteorologists. In some ways the surprising fact is that large-scale patterns, such as the Gulf Stream, are so long-lived despite the oceans tendency toward instability.


Aspects of physical oceanography


Physical oceanography has many subdisciplines, including planetary-scale circulation and climate, coastal oceanography, equatorial oceanography, internal waves and turbulence, and surface waves and air-sea interaction. While the phenomena studied by these subdisciplines certainly interact in complicated ways, most oceanographers specialize in one. A comprehensive account of all these areas would fill many, many volumes of an oceanic encyclopedia, but here are a few examples to suggest the tang of modern physical oceanography.


Planetary-scale circulation and climate


During 18-8, an environmental condition called El Niño was blamed for a multitude of natural disasters severe damage to the Pacific Oceans coral populations; droughts in Indonesia and the Amazon rain forests that led to destructive wildfires; and the loss of over 000 lives in the United States due to great storms that caused floods in the Gulf states and torrential rains and high tides in California. In 18, the return of El Niño led to the death by starvation of thousands of seals and sea lions in the California channel islands because the fish on which they normally feed were driven away by atmospheric and oceanic conditions.


What is El Niño? Basically, its a warming of the upper layers of the tropical Pacific Ocean, caused by interaction with the atmosphere. Normally the winds over the Pacific form a circular pattern above the equator near the sea surface, the trade winds blow west across the Pacific, from South America to Indonesia, where they cycle up through the atmosphere to form the Upper Westerlies, blowing east back across the Pacific. These strong winds drive the ocean to create an upwelling of cooler, nutrient-rich waters along the tropical coast of South America and along the equator. During El Niño years, the trade winds weaken and upwelling is reduced. This causes surface temperature to rise over a vast area of the ocean, and these temperature changes greatly affect the local climate.


Normally, high rainfall occurs north of the equator and in the tropical southwest Pacific area. In El Niño years, the areas of high rainfall are over the ocean, rather than over Indonesia and Australia. The weakened trade winds and reduced upwelling reduce the nutrients available to the phyto- and zoo-plankton that form the foundation of the marine food chain. This has proved disastrous for Peruvian fisheries, and has necessitated a ban on fishing off the coast of Peru during these years. In normal years, about 0 percent (by weight) of the entire worlds fish harvest has been caught there! El Niño effects can lead to more hurricanes in the Pacific, fewer hurricanes in the Gulf of Mexico, and droughts and floods throughout the world.


A related effect of El Niño is a dramatic increase of surface atmospheric pressure over Indonesia and Australia. This atmospheric portion of the El Niño effects is called the Southern Oscillation. The El Niño Southern Oscillation (ENSO) pattern can occur two or three times a decade.


Modelling the ENSO phenomenon has been a great challenge for oceanographers, requiring the use of sophisticated mathematical techniques. While researchers have a pretty good understanding of the physical dynamics that cause El Niño Southern Oscillation, accurate predictions are very hard to make. Oceanographers and meteorologists find it difficult predicting even when an El Niño year will occur -- let alone predicting the number and intensity of hurricanes that may form during that year!


A central enigma to physical oceanographers is the structure of the thermocline, the distribution of water temperatures throughout the ocean. Due to variations in solar and other incoming thermal energy, the ocean is not heated uniformly at the surface. This variable heating contributes to the existence of ocean currents, which in turn lead to variations in water temperatures throughout the full depth of the ocean. The temperature variations in the surface waters can have an enormous and immediate impact on all life in and out of the ocean, especially through their influence on climate, as observed in the studies of El Niño. During the last twenty years, several breakthroughs in physical and mathematical understanding of the thermocline have been achieved, through the work of a group of geophysical fluid dynamicists including Joseph Pedlosky, at the Woods Hole Oceanographic Institution, Peter Rhines, now at the University of Washington, and William Young, now at the Scripps Institution of Oceanography.


Internal waves and turbulence


In c. 600 B.C., the despot Periander sent off, by ship, the sons of certain noble families with orders that the boys be castrated. Though under full sail, the ship suddenly halted dead in the water. According to the historian Pliny, the cause was a kind of mollusk which attached itself to the ships hull, preventing its progress and thus rescuing the boys. Pliny provides other accounts of ships under full power being suddenly held fast in the water, often blaming not a mollusk but a small clinging fish called a Remora. Even one Remora could, it was supposed, halt an entire ship!


Becalmed ships continued to trouble navigators of coastal and polar waters through the centuries. In fact, Norwegian sailors encountered it so frequently in their fjords that their word now defines the effect dödvand, in English dead water. Eventually, mariners recognized that dead water appears where there is a great influx of fresh, cold water forming a layer over the salty sea.


In old mariners lore, ships were held by fresh water sticking to the hull. Sailors tried many ways to get out of dead water pouring oil on the waters in front of the ship; running the entire crew up and down the ship; working the rudder; drawing a heavy rope under the ship, stem to stern; banishing monks from the ship; and even firing guns into the water or using oars and handspikes to beat the water.


The phenomenon of dead water was finally explained scientifically when the theory of internal waves was developed. These are waves that can occur at the boundary between two fluids of different densities. For example, in 176, Benjamin Franklin described how swinging a suspended glass containing oil on water created a great commotion at the water -- oil interface, tho the surface of the oil was perfectly tranquil. However, two fluids need not be as different as oil and water for internal waves to form. In 104, the noted oceanographer V. Walfrid Ekman confirmed mathematically that the passage of a sufficiently large ship through a layered region (fresh, lower-density water atop salty, higher-density water) generates great waves at the interface between the fresh and salt waters. This causes drag on the vessel, as the momentum of the ship is transferred to the waves that its entry to the two-layer region initiated. The mathematics required to study this phenomenon comes from what are called eigenvalue problems; that is, the motion may be modeled by a collection of modes (for example, corresponding to different frequencies), and the fluid state is computed by adding together the contributions from each mode.





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