ESCI 343 Atmospheric Dynamics II Lesson 14 Inertial/slantwise Instability
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1 ESCI 343 Atmospheric Dynmics II Lesson 14 Inertil/slntwise Instbility Reference: An Introduction to Dynmic Meteorology (3 rd edition), J.R. Holton Atmosphere-Ocen Dynmics, A.E. Gill Mesoscle Meteorology in Midltitudes, P. Mrkowski nd Y. Richrdson INERTIAL INSTABILITY Imgine n ir prcel in geostrophic blnce t speed v on the f-plne. The digrm below shows the blnce of ccelertions in the lterl direction. Note tht the direction of v is completely rbitrry. The coordinte r is directed to the right of the wind, while the coordinte n is in the direction of the geostrophic wind. The wind component in the trnsverse direction (long r) is denoted s u. n v r f o v The momentum equtions in this coordinte system re Du fv Dt (1) Dv fu. Dt () Imgine tht n ir prcel strts in geostrophic blnce. If the prcel is suddenly impelled lterlly in the direction of r t speed u, the blnce of ccelertions will chnge. Tking the time derivtive of (1) gives us n eqution for how the lterl ccelertion chnges with time, D Du D D fv. (3) Dt Dt Dt Dt The terms on the right-hnd side of (3) re evluted s follows: nd D u v u Dt t n (4) so tht (3) becomes D Dt Dv fv f f fu f u, (5) Dt
2 Du u f u (6) Dt or Du f u. (7) Dt The solutions to (7) will be oscilltory provided tht f. (8) In this cse, the prcel will oscillte round its originl line of motion, nd the flow is inertilly stble. The ngulr frequency of the oscilltions is f. (9) If insted, f, (1) then the trnsverse velocity will grow exponentilly with time nd the prcel will ccelerte wy from its originl line of motion. PHYSICAL INTERPRETATION OF INERTIAL STABILITY The physicl interprettion of inertil stbility/instbility is directly linked to how the pressure grdient tightens or loosens in the direction of r. The figure below shows the cse of the pressure grdient becoming tighter with incresing r, which implies tht. (11) The stbility criteri (8) tells us tht this cse is inertilly stble, so tht prcel displced ltitudinlly will return to its bse ltitude. To see why this occurs, refer to the digrm below. Imgine prcel in geostrophic blnce t Point 1. If the prcel is perturbed in the direction of, then there will lso be positive ccelertion in the direction of n due to Coriolis. This will increse the v component of the wind nd thus increse the
3 component of the Coriolis ccelertion in the direction of r. Since the prcel is lso moving into n re of weker pressure grdient, there is net ccelertion on the prcel towrd positive r. Thus, lterl perturbtion will result in restoring ccelertion bck towrd the originl line of motion. For the cse where the pressure grdient decreses in the r direction the physicl interprettion is little more complex. The digrm below shows this cse, where. (1) As before the prcel is perturbed in the negtive r direction t velocity u. There is still n ccelertion due to Coriolis in the n direction, which will increse the component of the Coriolis ccelertion in the r direction. However, the pressure grdient ccelertion is lso incresing s the prcel moves to Point. If the pressure grdient ccelertion is lrger thn the Coriolis ccelertion, the prcel will ccelerte towrd the negtive r direction. If, however, the increse in Coriolis ccelertion outweighs the increse in the pressure grdient ccelertion, the prcel will ccelerte bck towrd its originl line of motion. Thus, decresing pressure grdient with incresing r is not sufficient to produce inertil instbility. In order for instbility to occur in this cse, (8) shows us tht f. (13) Plots of geopotentil versus r for constnt pressure surfce re shown in the digrms below. For the first two digrms the tmosphere is inertilly stble, becuse the second derivtive of is either positive or zero. For the third digrm the second derivtive of is negtive, but instbility would depend on just how negtive the second derivtive is. 3
4 ABSOLUTE MOMENTUM AND INERTIAL STABILITY The stbility criteri (8) cn be written in lternte forms s follows: f f f fv g or v g f. (14) For ese of nottion we cn rewrite (14) s fr v g, nd defining quntity clled the bsolute momentum 1 s M f r v (15) g the condition for inertil stbility/instbility cn be written s M : Inertilly stble M : Inertilly neutrl M : Inertilly unstble Even though M is clled bsolute momentum, it is not exctly equl to the momentum s viewed from spce, but is equl to it within some function of r. The reson it is defined this wy is so tht its r-derivtive is equl to the bsolute vorticity vi M. (17) r Thus, we cn view inertil instbility of the flow s occurring if the bsolute vorticity of is negtive. This prtilly explins why we don t see negtive bsolute vorticity occurring on the synoptic scle, becuse if it does occur, inertil instbility will occur. (16) 1 Our derivtion of inertilly instbility nd bsolute momentum ws done in coordintes tht re completely rbitrry, with no preferred direction for the geostrophic wind. Mny bsic tretments of this topic do the derivtion for purely zonl geostrophic flow. In this cse, bsolute momentum is defined M f y u, nd the derivtives in (16) re tken with respect to y insted of r. insted s g 4
5 INERTIAL STABILITY OF A VORTEX The concept of inertil instbility cn be extended to curved flow (for detiled derivtion see Lesson 1 of the Tropicl Meteorology clss notes). In this cse the bckground flow is ssumed to be in grdient wind blnce. Also, it is the rdil grdient of the bsolute ngulr momentum tht is importnt, rther thn the grdient of the bsolute momentum. The bsolute ngulr momentum is given by M vr fr (18) where v is the tngentil velocity of the vortex, nd r is the distnce from the vortex center. The condition for inertil stbility/instbility of vortex is M : Inertilly stble M : Inertilly neutrl M : Inertilly unstble The reltionship between bsolute ngulr momentum nd bsolute vorticity is 1 M. (19) r Most tretments of inertil stbility mention tht negtive bsolute vorticity is inertilly unstble. While this is true for wek or stright-line flow, in strong vortexes it is possible to hve negtive bsolute vorticity nd still be inertilly stble. SLANTWISE/SYMMETRIC INSTABILITY Sttic stbility refers to n ir prcel s resistnce to verticl displcement, wheres inertil instbility refers to its resistnce to trnsverse-horizontl displcement. It turns out tht it is possible for prcel to be both stticlly (verticlly) nd inertilly (horizontlly) stble, nd yet be unstble with respect to digonl displcement. Such instbility is clled slntwise instbility, or symmetric instbility, nd my be n importnt instbility mechnism ner fronts or other broclinic zones. Mthemticlly, the condition for sttic instbility is M : Slntwise stble y M y : Slntwise neutrl M : Slntwise unstble y This looks just like the condition for inertil stbility except the derivtive is tken long n dibtic surfce, rther thn on horizontl surfce. 5
6 The figure below shows lines of potentil temperture (dshed) nd the lines of bsolute momentum (solid). The pressure surfces re lso shown with dotted lines. In this configurtion the tmosphere is stble with respect to horizontl motion (inertil stbility), verticl motion (sttic stbility) nd digonl motion long n dibt (slntwise stbility). A prcel moving dibticlly from Point A to Point B would be moving into region of higher bsolute momentum, so tht M / nd the tmosphere is slntwise stble. A B The next figure (below) shows wht cn hppen ner broclinic zone (front). In this cse there re regions where the dibts re more steeply sloped thn the bsolute momentum lines. In this region prcel moving in the positive r direction on n dibt from Point A to Point B would be moving into region of lower bsolute momentum, so tht M / nd the tmosphere is slntwise unstble. r A B 6
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