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by: Demarcus Schaden V


Demarcus Schaden V
Texas A&M
GPA 3.65


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Class Notes
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This 4 page Class Notes was uploaded by Demarcus Schaden V on Wednesday October 21, 2015. The Class Notes belongs to ATMO 336 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/225954/atmo-336-texas-a-m-university in Atmospheric Sciences (ATM S) at Texas A&M University.

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Date Created: 10/21/15
Integrating the Hypsometric EquationiThe Trapezoidal Rule Atmo 336 Fall 2007 Getting Started In class we showed that the thickness of the layer of air between any two pressure levels pa and pb is given by the hypsometric equation 2 7 21 g T In 12 1 R where implicitly we7ve let pa be the top of the layer lower pressure and pb the bottom The T term in 1 is the log weighted average temperature of the layer de ned by fideUnp ltTgt fpib dump m where dlnp dpp Typically we use these formulae in class by de ning some simple function Tp to represent the temperature of the atmosphere as a function of pressure and then integrating this function as in 1 and 2 to get the thickness or distance between the pressure levels of the layer in question But in the real world we never actually know the function Tp What we know instead is a series of discrete observations giving T at a nite set of pressure values between pa and p5 To get the thickness of the layer we then need to approximate the hypsometric equation using only this nite set of observations A Simple Test Case We7ll start by de ning a simple test problem for which the integral de ning T can be evaluated by hand This is still dynamics class after all We7ll use our answer to this problem later when we test our methods for integrating the hypsometric equation using only a nite set of values a First let7s put the hypsometric equation into a form that7s more convenient for computation Verify that substituting 2 into 1 reduces the equation to Pb Zn 7 Zb E imp dp 3 g p 0 where again we7ve let pa be the top of the layer lower pressure and pb the bottom b Now to de ne the tem erature ro le Let7s su ose p p pp Tltpgt T 7a 1 7 woos gs lt4 where TS is the surface temperature7 pS is the surface pressure7 and Oz and B are speci ed constants This looks a bit complicated7 so lets rst plot the function to get a basic idea of what it looks like Start by making a function m le of the form function y Tdefp Ts bet g y Ts alph1 pps betcosppi2pspps and saving the le as Tde m The values of the constants will of course be lled in later Then to make the plot you would type on the command line or in a separate m le gt pa g gt pb g gt p pa2pb gt T Tdefp gt p10tT7p gt axis ij where the last command ensures that the pressure decreases along the vertical axis rather than increases Plot the temperature pro le between pa 500 hPa and pb 1000 hPa for the values TS 300 K7 pS 1000 hli a7 and Oz 73 K First plot the case B 0 for reference and then overlay as a red line using plotTp r the case B 40 K Save your result as Tpro lejpg and copy to the gure directory to turn in c Ok7 now for the derivation Show that substituting our temperature pro le 4 into 3 results in the layer thickness being given by Oz p7r 5 Pa zaizb E Tsiozlnl pbipasin i 9 pa 19 7T pg 2 where of course the last term is evaluated at pb and pa What does the 1000 to 500 hPa thickness turn out to be for the values given above with B 40 K Record your answer to 2 decimal places or six signi cant digits well use this as a baseline for testing the approximation methods that follow lt5 Approximating the Integral Now suppose that we dont actually know that the temperature pro le is described by It still isiwe just dont know that Instead all we are given is a set temperature measurements at some nite set of pressure values between pa and pb as returned from a radiosonde or some other intrument7 for example But our boss really wants to know the 1000 to 500 hPa layer depthiand to within 1 m So what do we do Well7 we approximate the integral of course The simplest way to see how to approximate the integral 3 is to recall that an integral is nothing more than the sum over a number of thin slices in the limit that the slice thickness decreases to zero That is7 symbolically N Pb E T0919 AP 6 Th dp 9 kl pk 9 p p in the limit that the slice thickness Ap becomes very thinior equivalently in the limit that the number of slices N between pa and pb becomes very large To appoximate the integral well just turn this around That is7 the integral is approximated by a sum over N slices in the limit that the number of slices is large Now to be more precise suppose that we have N 1 observations of temperature equally spaced in pressure between pa and p5 Suppose that the observations include values at both pa and p5 These N 1 observation points can then be used to de ne a set of N sublayers between pa and p17 as illustrated below x 171717quot 1 l l l l l l l quot Having broken the pa to pb layer into sublayers7 the question then just becomes how do we approximate T and p for each layer Well7 the simplest approach is just to average the the values at the top and bottom of the layer That is7 for layer 1 we approximate p p1 p2 2 and T T1 T2 27 and so on Making this approximation for each of our sublayers and then summing gives 9 a 10 9 k1 10k 10k1 Pb N R Th dp R Z T10k T10k1 AP 6 where Ap pl7 7 pa N This scheme for approximating the integral given a set of N 1 observations is known as the trapezoidal rule for numerical integration Computations Ok7 now to try out our method For each of the following compute the layer depth between 1000 and 500 hPa given the temperature pro le 4 with the values of TS7 p57 04 and B as given above with B 40 a Compute by hand the trapezoidal approximation for two sublayers That is7 assume that you have observationsireturning temperature values as in 47at the pressure levels 5007 750 and 1000 hPa How good is your approximation in this case Record your answer in terms of the fractional error E i Dt39rue 7 Dapprom Dt rue where D is layer depth and Dime is given by Have we satis ed the boss with this calculation b Now lets try the trapezoidal approximation for N 4 In this case computing the approximation by hand will be a bit too tedious for me anyway so we7ll instead write a little program to do it for us which you should save as an m le N 4 pa 500 pb 1000 dp pbpaN p paidpipb T Tdefp sum 0 for k1N sum sum end truval err truval sumtruval What is the fractional error in this case Has increasing the layers improved the approximation c Compute the approximation for N 2 3 4 6 8 10 16 32 and 64 and record the fractional error in each case Then plot the errors as a function of Ap in MATLAB Save your result as erronplotjpg and copy to the gure directory to turn in ls the trapezoidal approximation approaching the true integral for Ap a 0 That is does the fractional error go to zero in this limit And have we satis ed the boss


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