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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 Alex DeCaria in Fall. Since its upload, it has received 44 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

ESCI 343 Atmospheric Dynamics II Lesson 12 Inertialgravity Waves Reference An Introduction to Dynamic Meteorology 3rd edition JR Holton AtmosphereOcean Dynamics AE Gill Reading Holton Section 75 INERTIALGRAVITY WAVES Inertial gravity waves occur when a statically stable ow is also inertially stable They are essentially gravity waves that have a large enough wavelength to be affected by the earth s rotation To study inertial gravity waves we need to include the Coriolis terms in the governing equations For simplicity we will use the incompressible continuity equation Therefore the linearized governing equations are 314 1 3p 7 i7 1 at 3 ax fv 31 1 ap39 7 i7 2 at 5 3y f aw 1 312 0 7 i77 3 at 5 dz 5 g 31 3quot al 0 4 3x 3y dz and p N2Az s Combining 3 and 5 to eliminate 0 gives 2 2 BEE 3 Zw 6 at 0 age Equations 1 2 4 and 6 are the governing equations for inertial gravity waves Assuming the usual sinusoidal solutions Mr Aeikxlywzewt 1 Beikxlywzeu wr Ceikxlywzeu Deikxlyrrziw and substituting into 1 2 4 and 6 results in the following algebraic equations for the coefficients A B C and D iaHfB ikED0 139 fA 1503 11 61 0 239 kAlBmC0 4 w2 N2Cmw7D 0 6 which yield the dispersion relation 02 fzmzN2k2lzk2lzm2 7a or w2 f2m2N2K KZ 7b Notice that if the effects of rotation are ignored f 0 then the dispersion relation becomes that for pure internal waves Since inertial gravity waves have long enough wavelengths to be effected by the earth s rotation we can assume that they are in hydrostatic balance This implies that m gtgt KH Therefore we can write the dispersion relation for inertial gravity waves as NZKZ w25f2 mzH 8 If we rearrange equation 7a to solve for m 2 we get 2 1V2 a 2 m w2f2 KH 9 From this we see that in order to have vertically propagating inertial gravity waves that f S a S N DISPERSION AND STRUCTURE OF INERTIALGRAVITY WAVES The phase velocity of inertial gravity waves is NZK K Ei f2 7 m2 K2 10 while the group velocity is 2 2 5g ikfzj KJI 11 m I f N KH m m Thus for inertial gravity waves the group velocity and phase velocity are orthogonal and upward propagating waves transport energy J J while J 39 waves transport energy upward The parcel trajectories for inertial gravity waves are ellipses From equation 4 we can see that 1 K 0 V 0 see exercises This means that the velocity is always perpendicular to the wave number vector ie there is no component of particle motion along the direction of the phase propagation Therefore the ellipses are always at 90 to the direction of phase propagation The particles move along these ellipses in an anticyclonic fashion in the Northern Hemisphere regardless of whether the wave propagation is up or down This makes sense since the Coriolis force must be directed toward the inside of the ellipse The figure below shows the direction of the particle trajectories for upward and downward propagating waves K The gures below show the 3rdjrnensional trajectories for a wave traveling in the positive 5 and z directions k gt 0 m gt 0 l 0 for Various Values of a The projection of the trajectories on each of the axes planes are also shown Notice that the traj ectory is a Vertical lines when a N as would be expected for a pure internal wave As a decreases the trajectories begin to slanL and also open up into ellipses At the lowest frequency possible a the traj ectory is a circle that lies completely in the horizontal a 80f 08N EXERCISES 1 Show that if m gtgt KH that equation 7 becomes equation 8 2 Show that the group Velocity for inertialrgravity waves is given by equation 11 3 Show that for inertialrgravity waves 5 o 5g 0 4 Use equations 1 2 4 and 6 to show the following phase relations between n V and w for a wave traveling in the xrz plane I 0 a u39iafv39 b u397mkw39 c v39ifmkaw39 d If w39cosk7cmz foot what areu and v 5 3 Use the results from 4a to determine whether the horizontal Velocity Vector will rotate cyclonically or anticyclonically with time Will this change if the wave is propagating upward Versus downward b Use the results from 4a to determine whether the horizontal Velocity Vector will rotate cyclonically or anticyclonically with height Will this change if the wave is propagating upward Versus downward 7 6 Show that K H7 0 is the same as equation 439

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