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Atmospheric Dynamics 2

by: Cordie Miller

Atmospheric Dynamics 2 ESCI 343

Cordie Miller

GPA 3.65

Alex DeCaria

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Alex DeCaria
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This 5 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 Alex DeCaria in Fall. Since its upload, it has received 31 views. For similar materials see /class/223515/esci-343-millersville-university-of-pennsylvania in Earth Sciences at Millersville University of Pennsylvania.


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Date Created: 10/15/15
Escl 343 7 Atmospheric Dynamics u Lesson 6 7 shallowwater Surface Gravity Waves References AVL Im39rua39umuyl tn Dymmzc Metem39ulugy 3 edltloh LR lloltoh Nummml Predzcttmt ml Dymzmlc Metem39ulugy 2 edltloh 3 H altlner and RT wllllarhs Wave m Fzmds J nghthlll Reading ll olton 732 GENERAL whldT ls nglty These waves are farrulrar to all ofus as the waves on the ocean or a 13 5 HT lv Md lsa useful prelude to the study of other nglty wave types In the atmosphere We wlll frrst lrrrut our study to surface nglty waves on the free surface of a Thls 1 h alet rat r than u mm Thus we are lrrruted to erther Vexyrlong wavelengths or very shallow water Thls ls approxrrhatloh THE SHALLOWWATER MOMENTUM EQUATIONS n t ht dehsltles The dashed llne shows the posltloh of the lhterface lfthe flulds are uhdlsturbed The solld llne shows the lhterface dlsplaced The depth ofthe lower fluld ls H lnterface undisturbed interface the dashed line in the figure remains constant at a value of p01 If the lower uid is in hydrostatic balance then the pressure at any point in the lower uid is proportional to the weight of the uid above it Therefore at the point shown in the diagram the pressure will be p p0 M H z plgn p2gn and the horizontal pressure gradient force will be ialp1p2 in 01 ax 01 ax and since h H 7 then 8h 87 3x 3x so ialp1p2g ah 01 ax 01 ax The quantity gp1p2g 01 is called reduced gravity and the momentum equations for the lower uid are written as Du Bh i g ifv Dt 3x 1 an if Dt 8 3y If the two uids are greatly different in densities such as air and water then 0 1 p2 E p1 and g E g note that the prime on g does not refer to a perturbation Since we ve assumed that the lower uid is in hydrostatic balance we ve constrained our analysis to motions whose horizontal scale is much greater than the vertical scale the depth of the uid For this reason we refer to equation set 1 as the shallowwater momentum equations Note that the pressure gradient force at any point in the lower uid is independent of depth This means that the uid motion is also independent of depth Therefore the lower uid is barotropic 1 We have to assume that the upper uid is deep in order to assume that p0 is constant This is because a displacement of the interface upward results in either divergence or convergence in the upper uid as it adjusts to the change in interface height This means that there would be horizontal ow in the upper uid which would require a horizontal pressure gradient in the upper uid By constraining our discussion to a very deep upper uid there is minimal convergence or divergence in the upper uid since the amount of mass replaced is minimal compared to the overall mass in the uid column and therefore minimal horizontal ow in the upper uid THE SHALLOWWATER CONTINUITY EQUATION The continuity equation in the lower uid is Bu 8v 3w 7 7 7 3x 3y dz If we integrate the continuity equation from the bottom of the uid to the interface we get h h 81aljdz Ia7wdz 0 0 3x 3y 0 az 0 which becomes wh w0 ha7u x a By The vertical velocity at the bottom of the uid is zero Also WUL DZ D7h Dt Dt so the shallow water continuity equation is Dh Bu 3v 7 h 7 7 Dt 3x 3y SHALLOWWATER GRAVITY WAVES The dispersion relation for shallow water gravity waves can be derived directly from the linearized form of the shallow water equations of motion To linearize the shallow water equations we substitute the following for the dependent variables u u vvV hHn and ignore products of perturbations to get L I in 17 L at 3x dy 31 731 7 31 37 g39a7nfv39 f17 3x at Maxv dy g g fu fu a7n a7n1737n H 9131 at 3x dy 3x 3y In the absence of a mean ow and ignoring Coriolis we have Bu39 37 at 8 3x ai 3997 at 3y 3774 L39hriv at 3x 3y Assuming a sinusoidal disturbance such as Mr Aeikxlyrw 1 Beikxlyru 7 Ceikxlyru and substitute these into the equations of motion we get the following matrix equation a 0 g39k A 0 0 a gl B 0 0 kH H a C In order that A B and C not be zero then 0 g7lt 0 a g l 0 kH H a and solVing this for 0 gives a llkz lzlg71 The total wave number in the direction of propagation is given by K 2 k2 l 2 so we get the following dispersion relation and phase speed wKg39H a 07 H K g EXERCISES 1 P Squot 9 Find the dispersion relation for one dimensional x direction shallow water gravity waves with a non zero mean ow in the zonal direction ie 12 0 17 0 3 Find the dispersion relation for two dimensional x and y directions shallow water gravity waves with a zero mean ow but including the Coriolis parameter these are known as shallowwater inertialgravity waves b Find the group velocity and phase speed of these waves Are they dispersive The general dispersion relation for one dimensional surface gravity waves not restricted to shallow water traveling in the xdirection is a kigktanth a What is the phase speed for these waves b Are these waves dispersive P For very short waves or for very deep water kH gtgt 1 Show that in this case the dispersion relation for surface gravity waves is a 12k 4 Jg This is known as the shortwave approximation or deepwater approximation Note that for x gtgt 1 tanh x 51 5quot What is the group velocity for these waves c Are these waves dispersive P For very short waves or for very deep water kH ltlt 1 Show that in this case the dispersion relation for surface gravity waves is a 12k i k gH This is known as the longwave approximation or shallowwater approximation Note that for x ltlt 1 tanh x a x b What is the group velocity for these waves P Are these waves dispersive Calculate the speed of a shallow water surface gravity wave for a uid having a depth equal to the scale height of the atmosphere 81 km assume zero mean flow How does this compare with the speed of sound


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