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| The '''omega equation''' is of great importance in [[meteorology]] and [[atmospheric physics]]. It is a [[partial differential equation]] for the vertical velocity, <math>\omega</math>, which is defined as a [[Lagrangian]] rate of change of pressure with time, that is, <math>\omega = \frac{dp}{dt}</math>.
| | If you present photography effectively, it helps you look much more properly at the globe around you. It is thus, on these grounds that compel various web service provider companies to integrate the same in their packages too. Should you go with simple HTML or use a platform like Wordpress. 2- Ask for the designs and graphics that will be provided along with the Word - Press theme. You can customize the appearance with PSD to Word - Press conversion ''. <br><br>Choosing what kind of links you'll be using is a ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines. WPTouch is among the more well known Word - Press smartphone plugins which is currently in use by thousands of users. Well Managed Administration The Word - Press can easily absorb the high numbers of traffic by controlling the server load to make sure that the site works properly. E-commerce websites are meant to be buzzed with fresh contents, graphical enhancements, and functionalities. You can also get a free keyword tool that is to determine how strong other competing sites are and number of the searches on the most popular search sites. <br><br>Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. Whether or not it's an viewers on your web page, your social media pages, or your web page, those who have a present and effective viewers of "fans" are best best for provide provides, reductions, and deals to help re-invigorate their viewers and add to their main point here. You can allow visitors to post comments, or you can even allow your visitors to register and create their own personal blogs. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration. <br><br>It has become a more prevalent cause of infertility and the fertility clinic are having more and more couples with infertility problems. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. If you adored this article and also you would like to receive more info with regards to [http://playing.co.in/profile/rodockery wordpress backup plugin] generously visit our own web-site. Websites that do rank highly, do so becaue they use keyword-heavy post titles. A whole lot worse, your site will likely be useless as well as your merchandise won't sell if no one has the endurance to wait for the web pages to load. Now all you have to do is log into your Word - Press site making use of the very same username and password that you initially had in your previous site. <br><br>There is no denying that Magento is an ideal platform for building ecommerce websites, as it comes with an astounding number of options that can help your online business do extremely well. Being a Plugin Developer, it is important for you to know that development of Word - Press driven website should be done only when you enable debugging. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. Thus, Word - Press is a good alternative if you are looking for free blogging software. Definitely when you wake up from the slumber, you can be sure that you will be lagging behind and getting on track would be a tall order. |
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| The equation reads: | |
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| :{{NumBlk|:|<math> \sigma\nabla^2_H\omega + f^2\frac{\partial^2\omega}{\partial p^2} = f \frac{\partial}{\partial p} \left[ \mathbf{V}_g \cdot \nabla_H (\zeta_g + f) \right] - \nabla^2_H \left( \mathbf{V}_g\cdot\nabla_H \frac{\partial \phi}{\partial p}\right) </math>|{{EquationRef|1}}}}
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| where <math> f </math> is the [[Coriolis parameter]], <math> \sigma </math> is the static stability, <math> \mathbf{V}_g </math> is the geostrophic velocity vector, <math> \zeta_g </math> is the geostrophic relative vorticity, <math> \phi </math> is the [[geopotential]], <math> \nabla^2_H </math> is the horizontal Laplacian operator and <math> \nabla_H </math> is the horizontal [[del]] operator.<ref>Holton, J.R., 1992, ''An Introduction to Dynamic Meteorology'' Academic Press, 166-175</ref>
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| ==Derivation==
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| The derivation of the <math>\omega</math> equation is based on the [[vorticity equation]] and the thermodynamic equation. The [[vorticity equation]] for a frictionless atmosphere may be written as: | |
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| :{{NumBlk|:|<math> \frac{\partial \xi}{\partial t} + V \cdot \nabla\eta - f \frac{\partial \omega}{\partial p} = \left( \xi \frac{\partial \omega}{\partial p} - \omega \frac{\partial \xi}{\partial p} \right) + k \cdot \nabla\omega \times \frac{\partial V}{\partial p} </math>|{{EquationRef|2}}}}
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| Here <math>\xi</math> is the relative vorticity, <math>V</math> the horizontal wind velocity vector, whose components in the <math>x</math> and <math>y</math> directions are <math>u</math> and <math>v</math> respectively, <math>\eta</math> the absolute vorticity, <math>f</math> the [[Coriolis frequency|Coriolis parameter]], <math>\omega = \frac{dp}{dt}</math> the individual rate of change of pressure <math>p</math>. <math>k</math> is the unit vertical vector, <math>\nabla</math> is the isobaric Del (grad) operator, <math>\left( \xi \frac{\partial \omega}{\partial p} - \omega \frac{\partial \xi}{\partial p} \right)</math> is the vertical
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| advection of vorticity and <math>k \cdot \nabla\omega \times \frac{\partial V}{\partial p} </math> represents the transformation of horizontal vorticity into vertical vorticity.<ref>Singh & Rathor, 1974, Reduction of the Complete Omega Equation to the Simplest Form, Pure and Applied Geophysics, 112, 219-223</ref>
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| The thermodynamic equation may be written as:
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| :{{NumBlk|:|<math> \frac{\partial}{\partial t} \left( - \frac{\partial Z}{\partial p} \right) + V \cdot \nabla \left( - \frac{\partial Z}{\partial p} \right) - k\omega = \frac{R}{C_p \cdot g} \cdot \frac{q}{p} </math> |{{EquationRef|3}}}}
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| where <math> k \equiv \left( \frac{\partial Z}{\partial p}\right) \frac{\partial}{\partial p} \ln\theta</math>, in which <math>q</math> is the supply of heat per unit-time and mass, <math>C_p</math>the specific heat of dry air, <math>R</math> the gas constant for dry air, <math>\theta</math> is the potential temperature and <math>\phi</math> is geopotential <math>(gZ)</math>.
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| The <math>\omega</math> equation ({{EquationNote|1}}) is then obtained from equation ({{EquationNote|2}}) and ({{EquationNote|3}}) by substituting values:
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| :<math>\xi = \frac{g}{f}\nabla^2 Z </math>
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| and
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| :<math>\hat k \cdot \nabla\omega \times \frac{\partial V}{\partial p} = \frac{\partial \omega}{\partial y}\frac{\partial u}{\partial p} - \frac{\partial \omega}{\partial x}\frac{\partial v}{\partial p}</math>
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| into ({{EquationNote|2}}), which gives:
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| :{{NumBlk|:|<math>\frac{\partial}{\partial t}\left(\frac{g}{f}\nabla^2 Z \right) + V \cdot \nabla\eta - f \frac{\partial \omega}{\partial p} = \left(\xi \frac{\partial \omega}{\partial p } - \omega \frac{\partial \xi}{\partial p} \right) + \left(\frac{\partial \omega}{\partial x}\frac{\partial v}{\partial p}\right)</math>|{{EquationRef|4}}}}
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| Differentiating ({{EquationNote|4}}) with respect to <math>p</math> gives:
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| :{{NumBlk|:|<math>\frac{g}{f}\frac{\partial}{\partial t} \nabla^2 \left(\frac{\partial Z}{\partial p} \right) + \frac{\partial}{\partial p} (V \cdot \nabla\eta) - f \frac{\partial^2 \omega}{\partial p^2} - \frac{\partial f}{\partial p}\frac{\partial \omega}{\partial p} = \frac{\partial}{\partial p}\left(\xi \frac{\partial \omega}{\partial p } - \omega \frac{\partial \xi}{\partial p} \right) + \frac{\partial}{\partial p} \left(\frac{\partial \omega}{\partial y} \cdot \frac{\partial u}{\partial p} - \frac{\partial \omega}{\partial x}\cdot \frac{\partial v}{\partial p}\right)</math>|{{EquationRef|5}}}}
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| Taking the Laplacian (<math> \nabla^2 </math>) of ({{EquationNote|3}}) gives:
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| :{{NumBlk|:|<math>\nabla^2 \left(-\frac{\partial Z}{\partial p} \right) + \nabla^2 V \cdot \nabla \left(-\frac{\partial Z}{\partial p} \right) - \nabla^2 k \omega = \frac{R}{C_p \cdot g} \cdot \frac{\nabla^2 q}{p}</math>|{{EquationRef|6}}}}
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| Adding ({{EquationNote|5}}) and ({{EquationNote|6}}), simplifying and substituting <math>gk = \sigma</math>, gives:
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| :{{NumBlk|:|<math>\nabla^2\omega + \frac{f^2}{\sigma} \frac{\partial^2\omega}{\partial p^2} = \frac{1}{\sigma} \left[ \frac{\partial}{\partial p} J(\phi,\eta) + \frac{1}{f}\nabla^2 J \left(\phi, -\frac{\partial \phi}{\partial p} \right) \right] - \frac{f}{\sigma} \frac{\partial}{\partial p} \left( \frac{\partial \omega}{\partial y} \cdot \frac{\partial u}{\partial p} - \frac{\partial \omega}{\partial x} \cdot \frac{\partial v}{\partial p} \right) - \frac{f}{\sigma} \frac{\partial}{\partial p} \left( \xi \frac{\partial \omega}{\partial p} - \omega \frac{\partial \xi}{\partial p} \right) \frac{R \cdot \nabla^2 q}{C_p \cdot S \cdot p}</math> |{{EquationRef|7}}}}
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| Equation ({{EquationNote|7}}) is now a linear differential equation in <math>\omega</math>, such that it can be split into two part, namely <math>\omega_1</math> and <math>\omega_2</math>, such that:
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| :{{NumBlk|:|<math>\nabla^2\omega_1 + \frac{f^2}{\sigma} \frac{\partial^2\omega_1}{\partial p^2} =\frac{1}{\sigma} \left[ \frac{\partial}{\partial p} J(\phi,\eta) + \frac{1}{f}\nabla^2 J \left(\phi, -\frac{\partial \phi}{\partial p} \right) \right] - \frac{f}{\sigma} \frac{\partial}{\partial p} \left( \frac{\partial \omega}{\partial y} \cdot \frac{\partial u}{\partial p} - \frac{\partial \omega}{\partial x} \cdot \frac{\partial v}{\partial p} \right) - \frac{f}{\sigma} \frac{\partial}{\partial p} \left( \xi \frac{\partial \omega}{\partial p} - \omega \frac{\partial \xi}{\partial p} \right)</math>|{{EquationRef|8}}}}
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| and
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| :{{NumBlk|:|<math>\nabla^2\omega_2 + \frac{f^2}{\sigma} \frac{\partial^2\omega_2}{\partial p^2} =\frac{R \cdot \nabla^2 q}{C_p \cdot \sigma \cdot p}</math>|{{EquationRef|9}}}}
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| where <math>\omega_1</math> is the vertical velocity due to the mean baroclinicity in the atmosphere and <math>\omega_2</math> is the vertical velocity due to the non-adiabatic heating, which includes the latent heat of condensation, sensible heat radiation, etc. (Singh & Rathor, 1974).
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| ==Interpretation==
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| Physically, the omega equation combines the effects of vertical differential of geostrophic absolute vorticity advection (first term on the right-hand side) and three-dimensional Laplacian of thickness thermal advection (second term on the right-hand side) and determines the resulting vertical motion (as expressed by the dependent variable <math>\omega</math>.)
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| The above equation is used by meteorologists and operational weather forecasters to assess development from synoptic charts. In rather simple terms, positive vorticity advection (or PVA for short) and no thermal advection results in a negative <math>\omega</math>, that is, ascending motion. Similarly, warm advection (or WA for short) also results in a negative <math>\omega</math> corresponding to ascending motion. Negative vorticity advection (NVA) or cold advection (CA) both result in a positive <math>\omega</math> corresponding to descending motion.
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| ==References==
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| <references/>
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| ==External links==
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| * [http://amsglossary.allenpress.com/glossary/search?id=omega-equation1 American Meteorological Society definition]
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| [[Category:Atmospheric dynamics]]
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| [[Category:Partial differential equations]]
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