Introduction to Atmospheric Modelling

Introduction to Atmospheric Modelling Yazdan M.Attaei ABSTRACT An atmospheric model is a computer program that produces meteorological information for future times at given locations and altitudes. Within any modern model is a set of equations, known as the primitive equations, used to predict the future state of the atmosphere [2]. These equations (along with the ideal gas law) are used to evolve the density, pressure, and potential temperature scalar fields and the air velocity (wind) vector field of the atmosphere through time. The equations used are nonlinear partial differential equations which are impossible to solve exactly through analytical methods, with the exception of a few idealized cases [3]. Therefore, numerical methods are used to obtain approximate solutions. In this work, we study the Heat and Wave equations as two important aspects when studying meteorology and atmospheric modeling. We assume an idealized domain with certain boundary conditions and initial values in order to predict the evolution of temperature and track the wave propagation in the atmosphere. Keywords: Atmospheric model, Finite difference method, Heat equation, Wave equation. Introduction: An atmospheric model is a mathematical model constructed around the full set of primitive dynamical equations (equations for conservation of momentum, thermal energy and mass) which govern atmospheric motions. In general, nearly all forms of the primitive equations relate the five variables n, u, T, P, Q, and their evolution over space and time. The atmosphere is a fluid. Therefore, modelling the atmosphere in fact means the numerical weather prediction which samples the state of the fluid at a given time and uses the equations of fluid dynamics and thermodynamics to estimate the state of the fluid at some time in the future. The model can supplement these equations with parameterizations for diffusion, radiation, heat exchange and convection. The primitive equations are nonlinear and are impossible to solve for exact solutions and numerical methods obtain approximate solutions. Therefore, most atmospheric models are numerical meaning they discretize primitive equations. The horizontal domain of a model is either global, covering the entire Earth, or regional (limited-area), covering only part of the Earth [4]. Some of the model types make assumptions about the atmosphere which lengthens the time steps used and increases computational speed. Global models often use spectral methods for the horizontal dimensions and finite-difference methods for the vertical dimension, while regional models usually use finite-difference methods in all three dimensions. Since the equations used are nonlinear partial differential equations, in order to solve them, boundary conditions and initial values are required. Boundary conditions are specified by the assumptions related to horizontal and vertical domain of study. The equations are initialized from the analysis data and rates of change are determined. These rates of change predict the state of the atmosphere a short time into the future; the time increment for this prediction is called a time step. The equations are then applied to this new atmospheric state to find new rates of change, and these new rates of change predict the atmosphere at a yet further time step into the future. This time stepping is repeated until the solution reaches the desired forecast time. The length of the time step chosen within the model is related to the distance between the points on the computational grid, and is chosen to maintain numerical stability. Time steps for global models are on the order of tens of minutes, while time steps for regional models are between one and four minutes. The global models are run at varying times into the future. Approximating the solution to the partial differential equations for atmospheric flows using numerical algorithms implemented on a computer has been intensively researched since the pioneering work of Prof. John von Neuman in the late 1940s and 1950s. Since Von-Neuman’s numerical experimentation on the first general purpose computer, the processing power of computers has increased at a breath-taking pace. While global models used for climate modeling a decade ago used horizontal grid spacing of order hundreds of kilometers, computing power now permits horizontal resolutions near the kilometer scale. Hence, the range of the scales of motion that next-generation global models will resolve spans from thousands of kilometers (planetary and synoptic scale) to the kilometer scale (meso-scale). Hence, the distinction between global climate models and global weather forecast models is starting to disappear due to the closing of the resolution gap that has historically existed between the two [1]. In this work first we solve two-dimensional heat equation numerically in order to study temperature rate of change which is a part of the equation for the conservation of energy in atmosphere. Two different types of sources (steady state and periodic pulse) are applied to simulate the heat sources for a local (small-scale) domain and the results are illustrated in order to investigate results for the applied boundary and initial value conditions. In the second part of this study, two-dimensional wave equation is solved numerically using finite difference technique and certain boundary and initial value conditions are applied for the small-scale idealized domain. The aim is to study the wave propagation and dissipation along the domain from the results which are illustrated for different types of excitations (standing wave and travelling wave). Overall, the aim of this paper is to show the efficiency of numerical solutions particularly finite difference method for solving primitive equations in atmospheric model. Heat Equation: To study the distribution of heat in the domain, we consider following parabolic partial differential heat equation with thermal diffusivity a; Domain: The idealized 2D domain is a plane of the size unity on each side with the following initial values and boundary conditions; Boundary Conditions (BCs): Dirichlet boundary condition is assumed for all the boundaries except at the regions where the source with T=Ts is taking place; T (0,y)=0 , T(x,0)=0 (except at source) T(1,y)=0 , T(x,1)=0 Initial Values: At time zero, we assume temperature to be zero everywhere except at the region where the source is applied to; Finite Difference Scheme: Heat equation can be discretized using forward Euler in time and 2nd order central difference in space using Taylor series expansions and spatial 5-point stencil illustrated below; Figure 1: Five points stencil finite difference scheme which after simplifying it takes the form; If we apply equal segmentation in both directions so that and rewriting the equation in the explicit form we have; where . For stability of our scheme we need hence; Excitation: In order to observe the heat transportation in all directions, we assumed two different types of the source. First, we use a steady state source placed at the corner next to the origin with dimension of 5 grid cells with temperature amplitude Ts=10o . The second source will be the following pulse source applied for 5 time steps and removed for the next 15 time steps (period of pulse function = 20). This will help to visualize the ability of the scheme to evaluate the temperature at the source region when the source is removed (back-transport of the heat). Results: The following figures illustrate the results observed by applying the scheme, the sources described previously and thermal diffusivity of a=2 with grid cells of size (Ni=Nj=50 number of grid points in x and y directions); (a) (b) Figure 2: Distribution of temperature (a) t=0 sec, b) t=20 msec, steady state source of size 5 grid cells in each direction. It is observed that for t>0 while we have a constant temperature at the source, temperature is diffused along the domain in both directions and it will not diverge at any point when time increases since the stability criterion was already applied for the duration of time steps . Also, in the vicinity of the source temperature is remained almost constant or with small variations after a sudden large increase due to the adjacent source cells with Ts=10o and the nature of the scheme in which back grid points are included for approximation. When the steady state source is replaced by a pulse source with certain On and Off duration (period) as it is seen in Figure 3, diffusion continues even in the absence of the source at the whole domain including the source region as in Figures 3(b),(d). This is more visible in Figure 3(c) in the vicinity of the source but compared to the steady state excitation, there is a significant temperature drop due to the fact that the source has been Off for several time steps and temperature drops gradually with its maximum drop just before the source is applied again as illustrated in Figure 3(d). (a) (b) (c) (d) Figure 3: Distribution of temperature when Pulse source is applied (period=20 time steps). (a)Initial time, (b)At first Off state, c)Right after second On state, d)Before 24th On state The last parameter to study for the heat equation is the diffusion coefficient. It is the coefficient which affects the rate of diffusion. Figure 4 shows that during equal time period, by larger coefficient heat will diffuse in larger area (dotted circles) of domain compared to when the coefficient is small. (a) (b) Figure 4: The effect of thermal diffusivity on temperature distribution.(a) a=2, (b) a=0.25 Wave Equation: Similar to the heat equation, hyperbolic partial differential wave equation can be discretized by using Taylor series expansion. In this equation, c is the wave constant which identifies the propagation speed of the wave. Our goal is to study the reflection of the wave at the boundaries and the dissipation of the wave due to the numerical solution of the wave equation. Domain: We use the same idealized domain in studying heat equation but in addition to Dirichlet, we also consider Von-Neumann boundary condition in order to study the reflection of the wave at the boundaries. A proper set of initial values will be chosen since this differential equation is of second order with respect to time. Von-Neuman Boundary Conditions: At the boundaries we will assume the following conditions; Source region Initial Conditions: The following initial conditions are assumed since we will use central difference in time and two time steps (current and previous) are used to evaluate the value at the future time; ) Finite Difference Scheme: For the above parabolic differential wave equation, 2nd order central difference scheme in both time and space is used for discretization as follows; and with à ¢Ã‹â€ Ã¢â‚¬  x=à ¢Ã‹â€ Ã¢â‚¬  y=h and rewriting the equation explicitly; with , the CFL number which must be less than or equal to since the coefficient of should be a positive (or zero) for stability of the scheme. Hence; Now, back to the boundary condition, by using forward Euler difference for the left and bottom boundaries (i=1,j=1) we can write; and similarly using backward difference at right and top boundaries (i=Ni,j=Nj) ; As we numerically solve for the derived general finite difference equation and illustrate it, we will see that the above boundary conditions are the mathematical representation of full wave reflection at the boundaries. For the second initial value condition we use central difference at t=0 (n=1) and it is derived; Substituting in general difference equation we get; Now, we can apply second order central difference for both temporal and spatial variations for Von-Neumann boundary conditions. Excitation: In this work, in order to study propagation and reflection of the wave using numerical solution of the wave equation, two different sources are applied at the origin with the dimension of 5Ãâ€"5 grid cells for both Dirichlet and Von-Neumann domain boundary conditions; Travelling Wave: Stationary Wave: where and wave numbers . The wave constant c assumed to be c=1 for simplicity, therefore = 0.01 in both x and y direction. Results: For Dirichlet boundary conditions the following figures are obtained for Stationary and Travelling wave sources; (b) Figure 5: Dispersion of Stationary wave in domain with Dirichlet BCs (a) before reflection (b) after partial reflection In Figure 5(a) the wave which is scattered from a stationary source is dissipated through the domain since the source is stationary. In Figure 5(b) the reflections at the boundaries are seen to be weak because of the Dirichlet BCs. Infact, these ripples are mostly due to the nature of finite differencing. However, it is clearly observed in Figure 6(a),(b) that the magnitude of the wave at the boundary is kept zero by Dirichlet BCs. (b) Figure 6: Dispersion of Stationary wave in domain with Dirichlet BCs (a) before reflection (b) after partial reflection, 3D view Figure 7 illustrates the travelling wave propagating in the domain. The ripples have larger magnitudes since the wave itself is travelling and this reduces the amount of attenuation because of the scheme specially after the reflection at the boundaries the weakend ripples are magnified by continuously incoming waves. (b) Figure 7: Travelling wave propagates in domain with Dirichlet BCs (a) before reflection (b) after partial reflection For Von-Neumann BCs, it is expected that for both standing wave and propagating wave we observe full reflection by the boundaries as described during the discretization of these BCs. Figures 8 and 9 illustrate the application of such boundary conditions for standing wave source and travelling wave source respectively. (b) Figure 8: Dispersion of Stationary wave in domain with Von-Neumann BCs (a) before reflection (b) after partial reflection (b) Figure 9: Wave propagation in the domain with Von-Neumann BCs (a) before reflection (b) after partial reflection In the above figures, it is seen that at the boundaries the ripples are fully reflected back to the domain as well as the time when the wave is propagating forward from the source and is reflected at bottom and left boundaries. These would be more visible when showing the figures in three dimensions (Figure 10); (b) (c) (d) Figure 10: Wave propagation (a),(b)standing wave, before and after reflection (c),(d)travelling wave, before and after reflection To sum up, finite difference scheme which is used in this work provides numerical solution of the wave equation well and the results are close to what are expected for the wave propagation in such idealized domain with different boundary conditions. Conclusion: In atmospheric science, heat flow is related to temperature rate of change and the evolution of momentum and energy in atmospheric models are related the gravity waves as they transport energy. In the Earths atmosphere, gravity waves are a mechanism for the transfer of momentum from the troposphere to the stratosphere. Gravity waves are generated in the troposphere, propagate through the atmosphere without appreciable change in mean velocity. But as the waves reach more diluted air at higher altitudes, their amplitude increases, and nonlinear effects cause the waves to break, transferring their momentum to the mean flow. Therefore, numerical solutions of atmosphere primitive equations play an important role for studying the evolution of fundamental variables in atmospheric science especially since these equations are partial differential equations which cannot be solved analytically. In this paper, a brief study over the numerical solution of heat and wave equations was conducted as a basis for a bigger scale atmospheric modelling. The results demonstrate the efficiency of finite difference method to solve these equations (in small-scale domain) when they are compared to the theoretical expectations, therefore, solving primitive equations in atmospheric models by numerical techniques can be a following work to this paper. REFERENCES

Letting Nature Speak

Letting Nature Speak If you were walking in the woods and suddenly a tree started speaking to you, most likely you would either faint or start running the opposite direction. It would be pretty scary, to say the least. But nature does speak to everyone in a sense; we are just so busy with life that we do not take the time to listen. There is so much in nature that we can learn from and apply to our lives, but so often we only look at it for its face value and do not see the deeper benefits.Speaking of nature, as I stand outside on the back porch, the sun is shining and the birds are singing, the smell of freshly cut grass fills the air and the mild breeze feels so refreshing on my skin. In the background I can hear the faint sound of traffic on the highway, cars busily heading to their destinations. It has been breezy for a couple of days now, but the sun is shining and the clouds are moving. As the day progresses, the wind speed increases and the temperature steadily decreases makin g a visit to the porch a little less comfortable than it was this morning.The humidity level has steadily increased as well, making my clothing sticky and somewhat annoying, also causing my paper to become limp and not as easily manageable. The clouds seemed to be huddling together as if forming a mob, moving in slowing overhead creating a blanket between the sun and me. My pleasant sunshine has been taken away from me now and I am left with a gray blanket of cloud cover to observe, I am picking out different shapes and possible figures within them. As the clouds continue moving by, more ominous clouds replace their predecessors, making the world around me darker and darker.The temperature is cool and the breeze is stronger than it was earlier. I hear thunder rumbling in the distance, a normal precursor to a storm. The thunder seems like a would be stalker approaching from the darkness, only his footsteps are so loud it shakes the earth and rattles the windows, demanding its presenc e be known. Lightning flickers like a streetlamp attempting to turn on, but continually failing. One drop of rain lands on my cheek, another on my arm. As the rain increases in quantity, I head inside and continue watching from my window.Slowly the rain changes from a lawn sprinkler type shower to more like someone turning on a high-pressure water hose as if they were trying to douse a fire. I am now confined to my home, as if there is an army outside keeping me contained unless I want to endure their unrelenting siege. The troubles in life are much like a storm; there are always signs of it brewing but so often we are caught up in the beauty of the moment that we do not see the thunderheads rolling in behind us until it is too late.We are then caught off guard without an umbrella in the pouring rain. The rain soaking our clothes and in turn our body, is like the stress that comes with trouble forcing us to try and find shelter or something to protect us. When caught in a storm, we rarely see the beauty of it because we are focused on the damage it is causing. After a storm the grass is greener, the air smells so fresh, the sidewalks are washed clean, and there is a sense of calm and reassurance that we have made it through. The sun raises its head and always gives us a rainbow after the storm.There is a lot more to the sun then rainbows and illuminating the world as Ralph Emerson states in â€Å"Nature,† â€Å"Most persons do not see the sun. At least they have a very superficial seeing. The sun illuminates only the eye of the man, but shines into the eye and the heart of the child† (563). This is so true; often nature is only seen for its face value. Sunrise is a particularly beautiful, natural event to experience, and all to often we do not take the time to enjoy the wonderful events that unfold during a sunrise.As the sun is approaching the horizon, I hear birds singing and nocturnal animals scurrying back to their dens to sleep the day away. The birds seem to be calling to one another as if they are old men sitting at the local cafe, drinking coffee and discussing the day’s to-do list. The sky is no longer black but a deep ocean blue, like someone has turned on a light in another room, and the light is reflecting throughout the house. I hear the leaves rustle in the wind, and the trees sway as if they are stretching after a deep sleep.Slowly, things in the distance become recognizable and I can distinguish more shapes and figures. The sky becomes brighter and brighter, changing from a deep blue to a brighter shade as the sun moves closer to the horizon. Faster and faster light is filling the sky and illuminating the world around me. It is almost like opening my eyes when I awaken and taking in all the colors and objects around me. Suddenly the sun shows its bright and shining face, peeking over the horizon as if to say good morning to me.It rises slowly, becoming more and more visible, until its entirety is now shining down on me, demanding to be seen, demanding my attention. I feel the warmth on my skin, like a blanket pulled up over me. The sunrise is so beautiful but when the sun comes up all the way it doesn’t always seem as wonderful, especially if there is a lot of it. This last summer we experienced an enormous amount of the sun and the heat that comes with it and the effects seemed all negative. It caused droughts, crops to wilt, electric bills were high in the effort for the air conditioning to keep up with the heat and the list could go on.But there were some benefits to the high heat and drought. I was able to spend plenty of time inside my home this summer and I was able to downsize a lot of my belongings. My home stayed very clean all summer long because I did not want to be out in the heat and I took advantage of the time inside. I was able to catch up on my movie watching and shows that I was missing out on. I have to admit, I did miss taking my children to the park, but I was able to spend quality time with them when we were cooped up inside.Another advantage to the drought and high heat is the crime rate was lower this summer; criminals do not like the heat just like everyone else. But what is a thunderstorm or a drought in comparison to something as devastating and tragic as a natural disaster that kills thousands and leaves even more without a home. Hurricane Katrina ripped through New Orleans leaving in its wake, destruction and death. Causing many to wonder how could anything good come out of such devastation. At first there did not seem to be anything positive.Then as the clouds lifted and the water receded, people started to pull together and found in the midst of tragedy, a sense of community. Barely two months after the devastation of Hurricane Katrina, the New Orleans’ art community pulled together and reopened the doors of the Ogden Museum with an incredible turn out on opening night (Krantz). Before Katrina, the turnout staye d about 100, but over 600 citizens crammed the affair, an enormous result (Krantz). Nature has its own lessons, whether teaching us to be prepared or to look deeper and find the treasure beneath the rubble.Wearing its many different faces, nature will always put us to the test. Whether enjoying the beauty of a sunrise or the thrill of a thunderstorm rolling in, there is always something to walk away with; the beauty is in the eye of the beholder. Works Cited Emerson, Ralph Waldo. â€Å"Nature. † Sound Ideas. Ed. Michael Krasney and M. E. Sokolik. Boston: McGraw-Hill, 2010. 562-564. Print. Krantz, Susan E. â€Å"When Tragedy Inspires Recovery: Visual Arts In Post-Katrina New Orleans. † Phi Kappa Phi Forum 90. 2 (2010): 8-11. Academic Search Premier. Web. Oct. 25, 2012.

Comparison of Philosophies of Friedrich Nietzsche and the...

Science versus religion has always been a very controversial topic in this world and even more so in the United States. It seems that this topic in some way, shape, form, or fashion always finds its way into ones life through government, jobs, and most certainly in politics. When looking at the lives of two men who embody the two controversial ideals of science and religion one can look at the lives and views of His Holiness the Dalai Lama (views on compassion surrounding religion) and Friedrich Nietzsche (views on Morality as Anti-Nature surrounding science). Dalai Lama through his reading seems to establish the concept that compassion is a guide for ethical behavior, while Nietzsche strives to develop the concept that moral†¦show more content†¦In this selection the Dalai Lama establishes compassion as a feeling similar to that of empathy. Empathy as defined by the American Heritage Dictionary is, identification with and understanding of anothers situation, feelings, a nd motives. In establishing this type of compassion and empathy for our fellow man, then we establish a type of love and intimacy that is similar to one that a mother has for her only child. This compassion that one establishes allows one to enhance their sense of compassion, and in doing so then one develops an intense sense of responsibility toward another suffering in that they help the victim to overcome that suffering and the agent of that suffering. Even so, when we do this then we establish the concept of consciousness and we enter into an ethically wholesome life. If we can begin to relate to others on the basis of such equanimity, our compassion will not depend on the fact that so and so is my husband, my wife, my relative, my friend. Rather, a feeling of closeness toward all others can be developed based on the simple recognition that, just life myself, all wish to be happy and to avoid suffering. In other words, we will start to relate to other on the basis of their sentient nature. Again, we can think of this in terms of an ideal, one which it is immensely difficult to attain. (Dalai Lama 734) This total feeling of compassion, empathy, and an overall feeling of love create the feeling mentioned above. In

Animal Farm Comparison to Communism - 970 Words

In the book Animal Farm by George Orwell, a new â€Å"political party† is created by the members of their animalian society, which is not only comparable to Communism in theory but also in execution. This so-called political party goes by the name of Animalism; a name that is reminiscent of Communism due to the pronunciation. There is far more to Animalism than the name that brings the thought of Communism to mind. The idea of Animalism (the name would come later from a different source) was brought forth by one of the oldest and most respected members of the farm, Old Major. He relays his through a magnificent speech to the whole farm. Throughout his speech he speaks of a farm no longer controlled by humans and a world in which all†¦show more content†¦Anyone that was discovered to be a continued supporter of Trotsky was violently purged from the country with little mercy. Stalin and Napoleon both have massive similarities between each other. They both had rivals who m they forcibly removed from their midst. Soon after the removal of their rival, they began using that particular person as a scapegoat for anything bad that went on within the country or farm. They also used force to remove those that were still followers of their rival even after their removal. Napoleon had dogs to protect him with unrestricted force. These dogs were first used to remove Snowball from the farm and not long after that became Napoleon’s constant companions. The dogs weren’t only for protecting Napoleon, but they were also used to strike fear into the other members of the farm. Stalin used a secret agency known as the KGB for his protection. They KGB were used to remove Trotsky from the country and then were used to purge those that were against Stalin from the country. They were used for more than protection, however. They were also used to strike fear in the population beneath Stalin. Both Stalin and Napoleon used a group to protect themselves. These groups were used for numerous things; among them were the removal of enemies to the leader and also the protection of the leader. These similarities can easily be seen. Animalism and Communism are extremely similar in a number of ways. ChiefShow MoreRelatedGeorge Orwell s Animal Farm922 Words   |  4 PagesIn the novel Animal Farm, by George Orwell, the wisest boar of the farm, Old Major, mimics Karl Marx, the â€Å"Father of Communism,† and Vladimir Lenin, a Russian communist revolutionary. George Orwell introduces direct parallels between the respected figures through their mutual ideas of equality and profoundly appreciated qualities. Furthermore, his utilization of dialect and descriptions represent the key ideas of the novel. 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