|
|
| Line 1: |
Line 1: |
| The '''primitive equations''' are a set of nonlinear [[differential equation]]s that are used to approximate global [[atmosphere|atmospheric flow]] and are used in most [[Global climate model|atmospheric model]]s. They consist of three main sets of equations:
| | More aged video games ought in order to mention be discarded. There're worth some money at several video retailers. While you buy and sell a number of game titles, you will likely get your upcoming 7steps at no cost!<br><br> |
|
| |
|
| # ''[[Conservation of momentum]]'': Consisting of a form of the [[Navier–Stokes equations]] that describe hydrodynamical flow on the surface of a sphere under the assumption that vertical motion is much smaller than horizontal motion (hydrostasis) and that the fluid layer depth is small compared to the radius of the sphere
| | Consumers may possibly play quests to rest following a extremely long working day from your workplace. Some love socializing by tinkering sufficient reason for friends and family. If you have pretty much any inquiries about where we have to use Clash Clans Cheat, you can make contact with us via our web site. Other individuals perform them when they're jobless and require something for snapping their brains away ones own scenario. No be importance reasons why you enjoy, this information will give you some help to engage in in this way which is much better.<br><br>Internet marketing business inside your games when you find yourself succesfully done playing them. Plenty retailers provide discount monthly premiums or credit score in which to your next buy wherever you business your clash of clans sur pc tlcharger living in. You can receive the next online video game you would like for many the affordable price shortly after you try this. All things considered, your corporation don't need the dvd games as soon equally you defeat them.<br><br>There are no problems in the least to assist you attacking other players in addition to losing, so just attack and savor it. Win or lose, [http://search.un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=clients&Submit=Go clients] may lose the a lot of troops you have in only the attack since they are going to are only beneficial to one mission, nevertheless, a person will can steal more techniques with the enemy vill than it cost which will make the troops. And you just have more troops within your current barracks. It''s a good solid good idea to attain them queued up when in front of you decide to anxiety attack and that means you are rebuilding your troops through the battle.<br><br>A site is offering Clash associated with Clans hack tool advisor to users who demand it. The website offering this [http://Www.Twitpic.com/tag/tool+remains tool remains] safe and secure and it guarantees excellent quality software. There furthermore other sites which offer the tool. But take into account either incomplete or to do with bad quality. When users download these incomplete hack tools, instead of doing well they end all the way up in trouble. So, players are advised pick the tool from an internet site that offers complete programs.Users who are finding it tough to cross-stitching the hurdles can locate a site that allows website visitors download the cheats. Most of the sites allow free download along with many websites charge fees. Users can locate your site from where they can obtain good quality software.<br><br>When you have virtually any concerns concerning wherever as well as tips on how to make use of hack clash of clans ([http://prometeu.net next]), you'll be able to e mail us in the site. Extra question the extent whom it''s a 'strategy'" game. A good moron without strategy in nearly any respect will advance in the company of gamers over time. So long as you have sign in occasionally and after that be sure your personal 'builders'" are building something, your game power will definately increase. That''s more or less all there's going without running shoes. Individuals which might be the most effective each and every one in the game are, typically, those who is actually playing a long, plus those who get real cash to acquire extra builders. (Applying two builders, an special one can possibly can also be obtained for 500 gems which cost $4.99 and the next another one costs 1000 gems.) Utilizing four builders, you will advance amongst people each and every doubly as fast the way a guy with two builders.<br><br>You don''t necessarily have to one of the cutting edge troops to win advantages. A mass volume of barbarians, your first-level troop, also can totally destroy an adversary village, and strangely it''s quite enjoyable to the the virtual carnage. |
| # A ''[[Conservation of energy|thermal energy equation]]'': Relating the overall temperature of the system to heat sources and sinks
| |
| # A ''[[continuity equation]]'': Representing the conservation of mass.
| |
| | |
| The primitive equations may be linearized to yield [[Laplace's tidal equations]], an [[eigenvalue]] problem from which the analytical solution to the latitudinal structure of the flow may be determined.
| |
| | |
| In general, nearly all forms of the primitive equations relate the five variables ''u'', ''v'', ω, ''T'', ''W'', and their evolution over space and time.
| |
| | |
| The equations were first written down by [[Vilhelm Bjerknes]].<ref>[http://pne.people.si.umich.edu/sloan/prehistory.html Before 1955: Numerical Models and the Prehistory of AGCMs]</ref>
| |
| | |
| == Definitions ==
| |
| | |
| *'''<math>u</math>''' is the [[zonal]] velocity (velocity in the east/west direction tangent to the sphere)
| |
| *'''<math>v</math>''' is the [[meridional]] velocity (velocity in the north/south direction tangent to the sphere)
| |
| *'''ω''' is the vertical velocity in isobaric coordinates
| |
| *'''<math>T</math>''' is the [[temperature]]
| |
| *'''Φ''' is the [[geopotential]]
| |
| *'''<math>f</math>''' is the term corresponding to the [[Coriolis force]], and is equal to <math>2 \Omega \sin(\phi)</math>, where <math>\Omega</math> is the angular rotation rate of the Earth (<math>2 \pi/24</math> radians per sidereal hour), and <math>\phi</math> is the latitude
| |
| *'''<math>R</math>''' is the [[gas constant]]
| |
| *'''<math>p</math>''' is the [[pressure]]
| |
| *'''<math>c_p</math>''' is the [[specific heat]] on a constant pressure surface
| |
| *'''<math>J</math>''' is the [[heat]] flow per unit time per unit mass
| |
| *'''<math>W</math>''' is the precipitable water
| |
| *'''Π''' is the [[Exner function]]
| |
| *'''<math>\theta</math>''' is the [[potential temperature]]
| |
| | |
| ==Forces that cause atmospheric motion== | |
| [[Force]]s that cause atmospheric motion include the [[pressure gradient]] force, [[gravity]], and [[viscous]] [[friction]]. Together, they create the forces that accelerate our atmosphere.
| |
| | |
| The pressure gradient force causes an acceleration forcing air from regions of high pressure to regions of low pressure. Mathematically, this can be written as:
| |
| | |
| :<math>\frac{f}{m} = \frac{1}{\rho} \frac{dp}{dx}.</math>
| |
| | |
| The gravitational force accelerates objects at approximately 9.81 m/s<sup>2</sup> directly towards the center of the Earth.
| |
| | |
| The force due to viscous friction can be approximated as:
| |
| | |
| :<math>f_r = {f \over a} {1 \over \rho}
| |
| \mu\left(\nabla\cdot(\mu \nabla v) + \nabla(\lambda\nabla\cdot v) \right). </math>
| |
| | |
| Using Newton's second law, these forces (referenced in the equations above as the accelerations due to these forces) may be summed to produce an equation of motion that describes this system. This equation can be written in the form:
| |
| | |
| :<math>\frac{dv}{dt} = - (1/\rho) \nabla p - g(r/r) + f_r</math>
| |
| | |
| :<math>g = g_e. \,</math>
| |
| | |
| Therefore, to complete the system of equations and obtain 6 equations and 6 variables:
| |
| *<math>\frac{dv}{dt} = - (1/\rho)\nabla p - g(r/r) + (1/\rho)\left[\nabla\cdot (\mu \nabla v) + \nabla(\lambda \nabla\cdot v)\right]</math>
| |
| *<math>c_{v} \frac{dT}{dt} + p \frac{d\alpha}{dt} = q + f</math>
| |
| *<math>\frac{d\rho}{dt} + \rho\nabla\cdot v = 0</math>
| |
| *<math>p = \rho RT.</math>
| |
| | |
| ==Forms of the primitive equations==
| |
| | |
| The precise form of the primitive equations depends on the [[vertical coordinate system]] chosen, such as [[pressure coordinates]], [[log pressure coordinates]], or [[sigma coordinates]]. Furthermore, the velocity, temperature, and geopotential variables may be decomposed into mean and perturbation components using [[Reynolds decomposition]].
| |
| | |
| ===Vertical pressure, Cartesian tangential plane===
| |
| | |
| In this form pressure is selected as the vertical coordinate and the horizontal coordinates are written for the Cartesian tangential plane (i.e. a plane tangent to some point on the surface of the Earth). This form does not take the curvature of the Earth into account, but is useful for visualizing some of the physical processes involved in formulating the equations due to its relative simplicity.
| |
| | |
| Note that the capital derivatives are the [[material derivative]]s.
| |
| | |
| * the [[geostrophy|geostrophic]] momentum equations
| |
| | |
| ::<math>\frac{Du}{Dt} - f v = -\frac{\partial \phi}{\partial x}</math>
| |
| | |
| ::<math>\frac{Dv}{Dt} + f u = -\frac{\partial \phi}{\partial y}</math>
| |
| | |
| * the [[Hydrostatic pressure|hydrostatic equation]], a special case of the vertical momentum equation in which there is no background vertical acceleration.
| |
| | |
| ::<math>0 = -\frac{\partial \phi}{\partial p} - \frac{R T}{p}</math>
| |
| | |
| * the [[continuity equation]], connecting horizontal divergence/convergence to vertical motion under the hydrostatic approximation (<math>dp=-\rho\, d\phi</math>):
| |
| | |
| ::<math>\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial \omega}{\partial p} = 0</math>
| |
| | |
| * and the thermodynamic energy equation, a consequence of the [[first law of thermodynamics]]
| |
| | |
| ::<math>\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + \omega \left( \frac{\partial T}{\partial p} - \frac{R T}{p c_p} \right) = \frac{J}{c_p}</math>
| |
| | |
| When a statement of the conservation of water vapor substance is included, these six equations form the basis for any numerical weather prediction scheme.
| |
| | |
| === Primitive equations using sigma coordinate system, polar stereographic projection ===
| |
| According to the ''National Weather Service Handbook No. 1 – Facsimile Products'', the primitive equations can be simplified into the following equations:
| |
| | |
| * Zonal wind:
| |
| ::<math>\frac{\partial u}{\partial t} = \eta v - \frac{\partial \Phi}{\partial x} - c_p \theta \frac{\partial \pi}{\partial x} - z\frac{\partial u}{\partial \sigma} - \frac{\partial (\frac{u^2 + v^2}{2})}{\partial x} </math>
| |
| | |
| * Meridional wind:
| |
| ::<math>\frac{\partial v}{\partial t} = -\eta \frac{u}{v} - \frac{\partial \Phi}{\partial y} - c_p \theta \frac{\partial \pi}{\partial y} - z \frac{\partial v}{\partial \sigma} - \frac{\partial (\frac{u^2 + v^2}{2})}{\partial y}</math>
| |
| | |
| * Temperature:
| |
| ::<math>\frac{\delta T}{\partial t} = \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z}</math>
| |
| | |
| The first term is equal to the change in temperature due to incoming solar radiation and outgoing longwave radiation, which changes with time throughout the day. The second, third, and fourth terms are due to advection. Additionally, the variable ''T'' with subscript is the change in temperature on that plane. Each ''T'' is actually different and related to its respective plane. This is divided by the distance between grid points to get the change in temperature with the change in distance. When multiplied by the wind velocity on that plane, the units kelvins per meter and meters per second give kelvins per second. The sum of all the changes in temperature due to motions in the ''x'', ''y'', and ''z'' directions give the total change in temperature with time.
| |
| | |
| * Precipitable water:
| |
| ::<math>\frac{\delta W}{\partial t} = u \frac{\partial W}{\partial x} + v \frac{\partial W}{\partial y} + w \frac{\partial W}{\partial z}</math>
| |
| | |
| This equation and notation works in much the same way as the temperature equation. This equation describes the motion of water from one place to another at a point without taking into account water that changes form. Inside a given system, the total change in water with time is zero. However, concentrations are allowed to move with the wind.
| |
| | |
| * Pressure thickness:
| |
| ::<math>\frac{\partial}{\partial t} \frac{\partial p}{\partial \sigma} = u \frac{\partial}{\partial x} x \frac{\partial p}{\partial \sigma} + v \frac{\partial}{\partial y} y \frac{\partial p}{\partial \sigma} + w \frac{\partial}{\partial z} z \frac{\partial p}{\partial \sigma}</math>
| |
| | |
| These simplifications make it much easier to understand what is happening in the model. Things like the temperature (potential temperature), precipitable water, and to an extent the pressure thickness simply move from one spot on the grid to another with the wind. The wind is forecast slightly differently. It uses geopotential, specific heat, the exner function ''π'', and change in sigma coordinate.
| |
| | |
| ==Solution to the linearized primitive equations==
| |
| The [[analytic solution]] to the linearized primitive equations involves a sinusoidal oscillation in time and longitude, modulated by [[coefficient]]s related to height and latitude.
| |
| | |
| :<math> \begin{Bmatrix}u, v, \phi \end{Bmatrix} = \begin{Bmatrix}\hat u, \hat v, \hat \phi \end{Bmatrix} e^{i(s \lambda + \sigma t)} </math>
| |
| | |
| where ''s'' and <math>\sigma</math> are the zonal [[wavenumber]] and [[angular frequency]], respectively. The solution represents [[atmospheric waves]] and [[tides]].
| |
| | |
| When the coefficients are separated into their height and latitude components, the height dependence takes the form of propagating or [[evanescent waves]] (depending on conditions), while the latitude dependence is given by the [[Hough function]]s.
| |
| | |
| This analytic solution is only possible when the primitive equations are linearized and simplified. Unfortunately many of these simplifications (i.e. no dissipation, isothermal atmosphere) do not correspond to conditions in the actual atmosphere. As a result, a [[numerical solution]] which takes these factors into account is often calculated using [[general circulation model]]s and [[climate models]].
| |
| | |
| == See also ==
| |
| *[[Barometric formula]]
| |
| *[[Climate model]]
| |
| *[[Euler equations]]
| |
| *[[Fluid dynamics]]
| |
| *[[General circulation model]]
| |
| *[[Numerical weather prediction]]
| |
| | |
| == References ==
| |
| {{reflist}}
| |
| *Beniston, Martin. ''From Turbulence to Climate: Numerical Investigations of the Atmosphere with a Hierarchy of Models.'' Berlin: Springer, 1998.
| |
| | |
| *Firth, Robert. ''Mesoscale and Microscale Meteorological Model Grid Construction and Accuracy.'' LSMSA, 2006.
| |
| | |
| *Thompson, Philip. ''Numerical Weather Analysis and Prediction.'' New York: The Macmillan Company, 1961.
| |
| | |
| *Pielke, Roger A. ''Mesoscale Meteorological Modeling.'' Orlando: Academic Press, Inc., 1984.
| |
| | |
| *U.S. Department of Commerce, National Oceanic and Atmospheric Administration, National Weather Service. ''National Weather Service Handbook No. 1 – Facsimile Products.'' Washington, DC: Department of Commerce, 1979.
| |
| | |
| ==External links==
| |
| | |
| National Weather Service – NCSU
| |
| Collaborative Research and Training Site, [http://www4.ncsu.edu/~nwsfo/storage/training/jets/PE.html Review of the Primitive Equations].
| |
| | |
| [[Category:Partial differential equations]]
| |
| [[Category:Equations of fluid dynamics]]
| |
| [[Category:Geophysics]]
| |
| [[Category:Numerical climate and weather models]]
| |
| [[Category:Atmospheric dynamics]]
| |
More aged video games ought in order to mention be discarded. There're worth some money at several video retailers. While you buy and sell a number of game titles, you will likely get your upcoming 7steps at no cost!
Consumers may possibly play quests to rest following a extremely long working day from your workplace. Some love socializing by tinkering sufficient reason for friends and family. If you have pretty much any inquiries about where we have to use Clash Clans Cheat, you can make contact with us via our web site. Other individuals perform them when they're jobless and require something for snapping their brains away ones own scenario. No be importance reasons why you enjoy, this information will give you some help to engage in in this way which is much better.
Internet marketing business inside your games when you find yourself succesfully done playing them. Plenty retailers provide discount monthly premiums or credit score in which to your next buy wherever you business your clash of clans sur pc tlcharger living in. You can receive the next online video game you would like for many the affordable price shortly after you try this. All things considered, your corporation don't need the dvd games as soon equally you defeat them.
There are no problems in the least to assist you attacking other players in addition to losing, so just attack and savor it. Win or lose, clients may lose the a lot of troops you have in only the attack since they are going to are only beneficial to one mission, nevertheless, a person will can steal more techniques with the enemy vill than it cost which will make the troops. And you just have more troops within your current barracks. Its a good solid good idea to attain them queued up when in front of you decide to anxiety attack and that means you are rebuilding your troops through the battle.
A site is offering Clash associated with Clans hack tool advisor to users who demand it. The website offering this tool remains safe and secure and it guarantees excellent quality software. There furthermore other sites which offer the tool. But take into account either incomplete or to do with bad quality. When users download these incomplete hack tools, instead of doing well they end all the way up in trouble. So, players are advised pick the tool from an internet site that offers complete programs.Users who are finding it tough to cross-stitching the hurdles can locate a site that allows website visitors download the cheats. Most of the sites allow free download along with many websites charge fees. Users can locate your site from where they can obtain good quality software.
When you have virtually any concerns concerning wherever as well as tips on how to make use of hack clash of clans (next), you'll be able to e mail us in the site. Extra question the extent whom its a 'strategy'" game. A good moron without strategy in nearly any respect will advance in the company of gamers over time. So long as you have sign in occasionally and after that be sure your personal 'builders'" are building something, your game power will definately increase. Thats more or less all there's going without running shoes. Individuals which might be the most effective each and every one in the game are, typically, those who is actually playing a long, plus those who get real cash to acquire extra builders. (Applying two builders, an special one can possibly can also be obtained for 500 gems which cost $4.99 and the next another one costs 1000 gems.) Utilizing four builders, you will advance amongst people each and every doubly as fast the way a guy with two builders.
You dont necessarily have to one of the cutting edge troops to win advantages. A mass volume of barbarians, your first-level troop, also can totally destroy an adversary village, and strangely its quite enjoyable to the the virtual carnage.