Atmospheric Dynamics 2
Atmospheric Dynamics 2 MET 4306
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This 6 page Class Notes was uploaded by Ida Auer on Monday October 12, 2015. The Class Notes belongs to MET 4306 at Florida Institute of Technology taught by Steven Lazarus in Fall. Since its upload, it has received 39 views. For similar materials see /class/221688/met-4306-florida-institute-of-technology in Meteorology at Florida Institute of Technology.
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Date Created: 10/12/15
Vertical Vorticity Dynamics OverviewSummaryBig Pic See Holton Section 44 Last semester we examined the kinematics of vorticity and circulation and have A de ned relative vorticity g V X a N019 that We SO Offlide 17110 the WV ticity vortex without really justifying or absolute vorticity Du D 29 motivating why we look at vorticity in the rst place Some ofthat will become evi mrcmatlon C E lt 39 dent over the course of the next week or so However it is worth mentioning that vorticity is not a concept intrinsic to dc 12 meteorology by any means it has its ori Clrcmatlon theorem E N 7 p gins in fundamental uid mechanics absolute circulation C a C 29146 Stokes theorem all r V X a lA coming this semester geostrophic vorticity actually haven t done this one yet A gt C related circulation and vorticity n 39 V X U llmA H OZ normal component of vorticity is the limit of CA for Agt0 for all ows vorticity and angular velocity normal component of vorticity is equal to twice the local angular velocity for solid body rotation Ve0rdrKr ave 1191 LKVJS giylrr lee Br r702K gt A A gt A A C CA gtC DrkA nr 014 since n k right hand rule curling fingers Today 1 vertical vorticity in natural coordinates curvature vs shear vorticity 7 V 3V g 7 R 7 3n 2 simple jet streak dynamics and vorticity With the exception of the circulation theorem above all of our circulationvorticity equations have been diagnostic Following today s lecture we are going to examine how vorticity changes evolves in time prognostic and what causes the changes The analog to this is the momentum equations and the associated forces responsible for uid ow Vorticity in natural coordinates see Holton pg 95 section 421 A gt Let s consider the vertical vorticity component only ie C k r 0 we have SW a r all curve is in xy plane A 7c gt7 C7 nrmikrmilzmAaozilzmlao Let s use our natural coordinate system again HOWEVER this time let sapply it to a snapshot ie streamlines of the ow rather than trajectories here the unit vector 71 is not the same 71 in Stokes Thm streamlines Let s look at the circulation around the ABCD segment the segment consists of streamlines lines and pieces J to the streamlines By convention we evaluate in a counterclockwise orientation So what d1rection 1s dl 1n wrt the various segments 9 AB all is in the same direction as a a BC all is in direction J to a a CD all is in opposite direction of quot gt DA dl Jto u So knowing this we have 0 0 C jvrqunmy AB BC CD DA Thus C lawman x 7 VWMSS lower curve upper curve We once again can use Taylor Series expansion to relate the velocity on the upper curve to that on the lower curve 3V Vupper Vlower E lower Plugging in for Vupper above we get 7 7 3V C 7 lawman dss 7 VWMSS 7 lawman dss 7 now a 5n H0T5s n lower Using the geometry from our gure 5n5 d5s 7 3V C 7 Vlower5n55ia 5n5s7HOT53 l n lower d5s or C 7 V5373 V5s5n HOT gt C 7 is 7 B VJSnSS HOT 311 fix an Note that the area of our enclosed curve is SB R8R 5B 5R 11 JR mgr619 7 lw 5R22R8R 751356 R 0 R 2 using RSB 5s and noting that 5R 5n A 25n 2665n 2 5n53 For an in nitesimal loop 5B N 0 so we have A N 5n5s and the circulation becomes C Vises 37A HOT Recall from Stokes theorem that we divide the circulation by the area and in the limit as the area goes to zero we obtain the normal component of vorticity in this case it is the vertical vorticity since we are in the xy plane V 7AHOT 7 C 7 53 aquot 7 5B 2 7 llmlAaoz 7 llmlAa0 7 Vaial What is EBB35s From our gure above we have RSB 5s or ESE5s lR where R is the radius of curvature of the streamline thus as 5n gt 0 we have BBBs fl and the vertical vorticity in natural coordinates can be written V 3V C 7 Holton Eq 49 R 371 curvature shear We can do the same for the X and y vorticity components and get a similar decomposition PRACTICAL APPLICATIONS Jet streak vorticity During the lecture I mentioned that the 4quadrant jet streak model applied to straight ow only ie RS approaches in nity pure shear vorticity only technically this is not correct The air parcel must experience acceleration as it enters and leaves the jet streak In order to accomodate this we assume a straight line geopotential through the center of the jet with curved ow off aXis as drawn in the gure below Note that there is a change in the geopotential gradient normal to the direction of motion however hence R is not constant for a parcel traversing the jet 39 CID ACID g 2 Q cycloni c sideofjet 300 mb A 3n 71 1s to leftof ow g lt 0 lticyglonic Side of jet 300 hPa hgts q3Aq3 I Recall that we can decompose the hz velocity into a geostrophic component and ageostrophic component u uag ug and v vag vg where it is assumed that vag ltlt vg and uag ltlt ug inviscid momentum equations can be rewritten as 13P dt gaxfv fvgfv fv vg fvag iv 12 dt pay f fugf f g fuag We can apply these equations to our jet streak known as 4quadrant model frictionless assume air moves thru jet Acceleration in the entrance region and decelleration in the exit region pro duce the following ageostrophic ow ID Mquot L I I 300 mb conlhr erk simking rising y A g I diye ence X a Vlt 0 I an I 5 l j 3 gt 0 dive genee con ence 39 39 rising H Wig s1nk1ng K V mAcD You should know which equation was applied above to get the ageostrophic wind vectors red arrows This is consistent with our 2D equations of motion in natural coordinates recall Holton 39 dV 31 dt 75 A parcel entering the jet will develop an ageostrophic wind component from high heights to low heights hence END3s lt 0 and dVdt gt 0 parcel accelerates The opposite occurs for a parcel leaving the jet Keep in mind that this is a highly idealized model and applies for a non evolving height field not reality where we ve assumed that the parcel travels through the jet streak i e the parcel is moving faster than the jet streak itself
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