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# Atmospheric Dynamics 2 ESCI 343

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This 4 page Class Notes was uploaded by Cordie Miller on Thursday October 15, 2015. The Class Notes belongs to ESCI 343 at Millersville University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/223521/esci-343-millersville-university-of-pennsylvania in Earth Sciences at Millersville University of Pennsylvania.

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Date Created: 10/15/15

ESCI 343 Atmospheric Dynamics II Lesson 10 Topographic Waves Reference An Introduction to Dynamic Meteorology 3rd edition JR Holton Reading Holton Section 742 STATIONARY WAVES Waves will appear to be stationary if their phase speed is equal and opposite to the mean ow 0 IZ Stationary waves will have a frequency of zero since they do not oscillate in time only in space For dispersive waves the wavelength of the stationary wave will correspond to that wavelength which has a phase speed equal and opposite to the mean ow None of the other wavelengths will be stationary An example of stationary waves is the stationary wave pattern that forms in a river when it ows over a submerged rock or obstacle The ow of the obstacle generates many different waves of various wavelengths but only the ones whose phase speed is equal and opposite to the ow will remain stationary If the ow speeds up or slows down the wavelength of the stationary wave will change DISPERSION RELATION FOR STATIONARY INTERNAL GRAVITY WAVES In the atmosphere ow of a stably stratified uid over a mountain barrier can also generate standing waves If we consider the ow over a sinusoidal pattern of ridges that are perpendicular with the x axis and with a horizontal wave number of k then we can analyze the structure of these waves Since these waves are internal gravity waves in the presence of a mean ow their phase speed and dispersion relation is simply Cz iL xk2m2 w kii k2 m2 remember that since these are linear waves the mean ow simply adds a term 12k to the frequency But we are only interested in the standing waves generated by the topography for which the phase speed and therefore frequency is zero For standing waves then we have aiLw xk2 m2 The horizontal wave number k is determined by the wave number of the terrain 1i and N are properties of the atmosphere Since these quantities are predetermined then there is only one value of vertical wave number m which can satisfy the dispersion relation Thus m is given by 2 N 2 2 m k 1 Equation 1 is the dispersion relation for stationary internal gravity waves VERTICALLY PROPAGATING VERSUS VERTICALLY DECAYING WAVES The vertical structure of the standing waves is determined by how N k and relate to one another Vertically prapagating waves If k lt N i then the right hand side is positive and m is real This corresponds to waves that propagate vertically since the sinusoidal form for the dependent variables is ezkxmz Along lines of constant phase kx mz the following relation holds kxmzb where b is a constant This means that phase lines obey the following equation z ixi m m This tells us that the lines of constant phase tilt upwind with height since they have a negative slope see figure below Since the wave is propagating upstream it has too since it is a standing wave the individual wave crests have a downward component of phase speed This means that the group velocity is upward so that topographically forced waves propagate energy upward so 100 l o Vertically decaying waves If k gt N a then the right hand side negative In this case the sinusoidal form for the dependent variables is eikxermz and the waves decay with height such waves are also known as evanescent In this case lines of constant phase are vertical This is illustrated in the figure below WAVES GENERATED BY AN ISOLATED MOUNTAIN RIDGE In the real world mountains are not pure sinusoids However through Fourier analysis we can approximate the real topography by its Fourier components with the component of wave number k having an amplitude of Hk Hk and hx are the F aurier transfmms of each other and are defined by the following equations hx THke kxdk Hk 7mede A very sharp or discontinuous function has a greater number of high frequency high wave number components in its transform A broad function has few high frequency components and is mostly made up of low frequency low wave number components Flow over a mountain will generate a whole spectrum of gravity waves Each wave component generated will either propagate vertically or decay vertically depending on whether its vertical wave number m determined from 2 N 2 2 m g k 1 is real or imaginary Based on the previous discussion of Fourier transforms we expect that a narrow mountain will generate a lot of high wave number gravity waves while a broad mountain will generate more low wave number gravity waves Since the narrow mountain generates mostly high wave number waves we expect that ow over a narrow mountain will likely generate vertically decaying waves while ow over a broad mountain will likely generate waves which will propagate vertically EFFECT OF VERTICAL WIND PROFILE In the previous discussion on mountain waves we ve assumed that the mean ow does not have vertical shear and that the static stability is constant with height These conditions are rarely met in the real world In the case of the mean wind speed increasing with height or for the stability decreasing with height it is possible to have ve11ically propagating waves in the lower levels with decaying waves in the upper levels This will lead to the waves being trapped near the ground as though they were in a wave guide illustrated below so i x In addition nonlinear effects can become important for severe downslope winds generated by mountain waves Nonetheless the simple linear treatment of these waves has shown us a lot of interesting and relevant features of this phenomenon EXERCISES 1 For an isothermal compressible atmosphere show the following remember that H R39T g 2 a c ng 13 N2 gH17 1 2 Assume that the Allegheny Mountains can be approximated as a parallel series of ridges approximately 25 km apart Also assume an isothermal compressible atmosphere with a scale height of 8100 m Calculate the critical wind speed below which topographically forced waves will propagate ve11ically and above which they will decay with height

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