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# FLUID DYNAMICS II ME 631

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This 104 page Class Notes was uploaded by Vidal Goyette on Friday October 23, 2015. The Class Notes belongs to ME 631 at University of Kentucky taught by Raymond Lebeau in Fall. Since its upload, it has received 18 views. For similar materials see /class/228234/me-631-university-of-kentucky in Mechanical Engineering at University of Kentucky.

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Date Created: 10/23/15

Surface Gravity Waves Assume ir r o ra rional incompressiblezl InVISCId flow 84 ad Velocity potential u w 3x 32 au 3w32 92 0 F 3x 32 8x2 822 8 Dn BCw 02 H E wnz n p 0z n Dn Klnematlc BC w 2 Dr 11 11 Small motions ltlt 11 ltlt 1 uv small x LEE 2 32 3t 32 211 211 Surface Gravity Waves B Bn a 2 32 ar E 2 x w 91 mm 32 2 32 20 322 20 32 20 377 19 0 j 31 32Z DynamicBC p0277 FH UnsteadyBernoulli iuz w2 g2ft 50 3t 2 p 3 3 Lt t I Ft t e062 XZ at at f are unchanged also assume uZ w2 N small 3x 32 aw in am 2 02 a p g2 g at 2 at 20 Bernoulli Equation For inviscid ow with conservative body forces Duj 3 17 8G 8G th Bx paxj fl Bx gta uVuu ugtltoo 1VpVG at p dXVdxdydz p p p p p lezVId p 2 a uVJd Puu Gugtltm p p at p Bernoulli Equation irrotational if irrotational VI Pt 39I auVJdpuuGuXw0 eoci39ry oen ria at p Since nqu0consistenttoletuV VXV 0 34gt dp 2V lV V G 0 at ID 2 I I i 31 dp 1 2 V V GFt at lp 24gt 0 Unsfeady Bernoulli Cons ranT Surface Gravity Waves Wantn acoskx wt BC 2 0 gt N sinkx wt 3 x z t fz sinkx wt 82 82 2 92f 0 k 8x2 322 3 f 82 2 0 2 fz Aekz Be z 3 Aekz Be kz sinkx wt BC Bi 0 3Ake kH BkekH0 3 Ae 2k az 27H 877 a 3amsinkx wtkA Bsinkx wt at 92 20 3awkA B Note get same thing if assume 2 k77ltlt1 gtekquot 1 82 271 Surface Gravity Waves am ame 2kH am 72m BA Ae 3A k lea 8 2 lea 8 2 kz 7kz2H am e e smkx mt k 182kH Fez111 kz 7kz2H am e e smkx mt k eikHekHeikH ekH eikH ekzH 87mm sinhkH T coshkz H 2 2 q coshkz H Sing mt k smhkH 3 u amwcosa x mt w amth ZHsinkx mt sinhkH sinhkH Standard Wave Formulations Compare to standard wave form d39AlemberT39s soln 02 7 fx ctgxct BIZ 3x2 3 7 a sin 96 00 asinkx at Wavenumber k E 2 Period T i Freq v l l c T gtclv 505vasz c Phasespezd 3D 7asinkxlymZ atasinKx at y C a Z KK k212m2 121 I K cwKc ic 39ckwiw KK x 2 k 1 DopplerShift c0 cUKa0 aUK j k m For Gravity Waves G E coshk0 H at k sinhkH 2 m2 gktanhkH 2 m 1 gktanhkH Dispersmn Relation c 9 tanhkH g xtanm k k 27 7 Pressure Variation coskx mt ga coskx mt 31 p 31 p39 31 03 P P PgZat p P82 at p P Pat 3 p39 p pamz t k s1nhkH MW k sinhkH COShWZ H kx t P811 coshkH cos D coskx mt coskx mt Particle Paths for GW Let xx0 zzo ltxoy0rgt4ltxmymrgt 2 u w E amcoshkzo H coskx a a sinhkH 5 asinl xo at amsinhkzo HSjnkx ml at sinhkH 3 g awcomo gt sinhkH 2 2 1 coshkznHlz sinhkznHz 1 sinhkH 1 sinhkH Fluid particles move in ellipticalpaths with the axes decreasing with depth sin2 ozcos2 011 2 Streamlines for GW Streamlines 31 amwcosbcm Bz s1nhkH an sinhkz H kx I F I 2 y k SinhkH cos a x 8 1 w aa Smhkz H sinkx at Bx s1nhkH coskx at Gz z k s1nhkH F x I Gz I H I 0 if streamline is for wave motion only so 1 0 coincides with a 0 For I 0 this yields lltX sinhkzHcoskx lIOZ H lI0kxin0l2m lt gt7olt coskx0 Crests gt motion in direction of propagation Troughs gt motion opposite directionof propagation Energy Consideretions for GW Kinetic EnergyNavelength Ek J0 JH u2 w2dzdx E pmz Ja2coszkx coldfocosh2kzHdz k ZsmhzkH iJ397a2sin2ac cozdxj390 sinh2kzHdz 7 0 7H 7 7 7 5 a2 coszkx max 5 a2 sin2kx cotdx 51 nzdx n2 1 sinh2kH kH I0 cosh2kz Hdz iJ39kH cosh2udu 7H k 0 4 2k PT 1 sinh2kH kH E 4 E 02 kgtanhkH kg 2kg sinh2 kH sjnh2 kH sinhkH coshkH sinh2kH p 2kg 2 lsinh2kH I kH I lsinh2kH kH 1 2 2 sinh2kHL 39 2ng I0 sinh2kz Hdz iIkHsinh2udu 7H k 0 Ek k 4 Zkk 4 2k Energy Considerations for GW pgquot39 pg pg1quot pg12 Ep 7J0 Lszzdx 7J Lszzdx 7J J minke 7 7 dx EP pg7 2Ek eequipaltition of energy EEp Ek pg72 pga2 ifsinusoidaln Energy ux gt pressure work across x 0 FE Q pu dzgt lt M dzgt pguJ0H 2 dz I dz 2 3 0 cos2kx at amp COSh2kZHdZ gtksinh2kH Li 1 2 C 1 2 2 1 2pga f sinh2kH 2pga Cg Deep Water Approximation c 2 quotg tanhkH fgtanh k k 272 xi H gt m tanhkH gt 1 kH 175 tanhkH 0941 3chgforHgtiixi028xi k 4k 87239 Example To N 10 seconds 00 N 3 3 N 0628 3 1 cean 2 0 0 Assume deep water k tanhkH N k N 00402 gt xi 156m 3 H gt 437 m 3 Most of ocean has deep water waves coshkzH ekzekH 87k287kH ekz e kze 2kH sinhkH ng gkH 1 8 2 H gt m coshkz H 81a s1nhkz H 8k s1nhkH s1nhkH Deep Water Approximation 52 C2 2 2 kz 2 3 13E C ae 2a ZO0 aekz 12 ate2 12 Circular particle paths freq of 03 size of the wave amplitude at the surface u amekz coskx0 mt w amekz sinkx0 mt 3 1u2 w2 lug 1r202 iawe2 2 3 Me iawe2 2 2 2 2 5 H gt m P39 pga COShkZ H coskx mt pgaekz coskx mt cosh H Z 4 k7 27I 87k 2 000187 gt 0187 of surface amplitude Shallow Water Approximation xi For shallow water waves are quotlongquot or F gtgt 1 Zmomentum ia p a p f ppgz edynamicpressure Dt p 32 32 ScalingargumentsBC w3W p03 1gh Continuity a ua WOUN 3x 32 HT Xmomentum a p g hgfwi Dt 3x T xi lgH Zmomentum a pZg hiilt inconsistent Dr 32 T H 12 H Z p z 0 3 p pg2 pressure is hydrostatic 2 Shallow Water Approximation Likewiseirrotational ow a u a WODEZ3iia uz 32 3x H 7 H2 7 32 cJtanhkH kH 21H ltlt1tanhkHzkHcgH Within 3 accuracy if H lt 0077 for small values coshx 2 l sinhx z x 0 3 E sinkxo mt C a l coskxo mt 2 L2L gtgtlz 3A gtgtAC l l2 Mug U Thin elliptical paths depth of ellipse linearly approaches zero as 2 gt 0 am u coskx mt wa CH 0 20 1 lax mt HJSIIK 0 H independent of z 7 ltlt13 p39 pga coskx mt pgn Wave Refraction Surface Tension and GW Laplace39s equation Ap pa p Z 2 2 2 2 2 iz z 1391 m lming r5 1anax232 3x2 2 Bx 3x2 Bx 32 32 3Ap0 7 p 0 7Z77 ppa0z77 Bx Bx a p M 0327 DynamchCEFgz033V gnz0 Solve as before to get a quotgki1 0 itanhUcH c J 1 39k ZtanhkH J 1 4fzajtanh 2 H k 95 27139 1 98 xi Surface Tension and GW kl N lor eater pg gr 39 So surface tension is important if k 2 p g 7 S 27E 1 20017mz2cmforwater 5 Pg Assume deep water find minimum phase speed c l 414 21 7 pg 712 zozl amp1 4120 21ZI0 2i 22 7227 2 d 2 27 Mg 27 M3 2c 27 M3 pg 4 14 Cmin 8quot l1 j z 23 cms for water pg p ripples 7 lt 7 cm gt surface tension important capillary waves 7 lt 4 m gt surface tension dominates Stand mg 8 u rface Waves Non propagating waves can be made by superimposing identical waves moving in opposite directions T a coskx 02 a coskx col 2a coskxcoscol 2 T 0kx i n 012 9 nodes 2am sinhkz H cc kx COZ kxcoz kx cut 11 cos cos k sinhkH sm sm Contained body of water bathtub lake bay can form re ecting standing waves Such a standing oscillation in a lake is called a quot seiche EA Forel 1890 a Swiss seismologist Let lake dimension be L then Bw coshkzH kL n1TE 2 39kx39 2 0 0L 32 am sinhkH sm Slnm u x X2Ln1 C0 Wtanh 9 natural frequencies of the lake u Grou p VeIOCIty Consider two waves of slightly different wavenumber k2 k1 dk 77 a cosk1x all a sink2x 6021 2a cosk2 k1x a2 mzlcosgacz k1x a2 01 k1 k2 De nezk wdkk2 kldww2 wl 2 77 2a cosg dk x dalcoskx at a 1 1 da coskx atac cos7dkx 3dwlacg E This arrangement yields wave groups and wave packets da 0 ZkH Gravi acketsa1 ktanth c 1 typ g 5 dk 2i Sinh2kHi kH gtoltgtcg 3 kH gt0cg cenondispersive capillary cg 3c2 c 3w 8quot Bki F 13 L 231 E cg Zpga2 sinh2kH Group Velocity amp Wave Dispersion Wavetrain of quot gradually varying wavelengthquot T axlcos8x 1 ax lcosk1x coll k1xl 01xl 9 local values that change gradually as ax I as as Bkl 329 am 329 Bkl Bcol ml gt 0 Bx Bl Bl BxBl Bx BxBl Bl Bx Assume existence of dispersion relation 0 0k then Bcol dcol Bk Bk dcol Bk Bk Bk c k1 0 Bx dkl Bx Bl dkl Bx Bl g Bx cg is the speed at which kl is advected dx c E 2 x x0 00 to 9 lines of crest motion cg x x0 cgl lo 9 lines of constant and C0 Group Velocity amp Wave Dispersion What if the depth varies such that C0 gk tanhkHx gt C0 0k x 50 kxawk xjc akamakam 539 Bk 53932 3sz a a ka m 020 a kc a m a mc a C0 0 a 3x gal gax az 83x dx cg 2 x x0 c z zoe lines of constant 0but notk dz 539 3D 3 0 Vco0 a 539 Finite Amplitude Nonlinear Effects But what about wave steepening zi wave breaking uid drift These are nonlinear effects due to nite amplitude Previously we assumed that for T smallH a a 32 a r 3i 2 Bi 1 Hal Z zT1 Z 20 BZ 20 Z 20 8n at 0 3 at 32 Z But what if the amplitude is suf ciently large for this to fail A number of interesting things Geophysical Fluid Dynamics Example of rotating ow system Three key effects e Rotation e Stratification e Thinness T 0C 10 20 30 01 1020 1023 0 p kgm3 N rads Geophysical Fluid Dynamics Boussmesq Approxnmatlon For a rotating system with Boussinesq approximation Vu 0 E2Qgtltu 7Vp EkF o Dt Po p0 Dt Key assumption for geophysical ows is that the vertical scale of the problem does not result in significant changein density This scale is of order 02 g 200 km for ocean 10km for atmosphere The typical approach is to look a perturbation about a local point with z 72 which satisfies the hydrostatic law Look at perturbation form of momentum equation 17 iz p xt p zp xt EZQXU 7V7p 7ppg kF Dt P0 P0 i kvp Ek EkF 7Vp39 ampRF Po dz P0 P0 1 0 1 0 Geophysical Flow For the moment we will assume no friction effects 39 inviscid F O U W W H Scahng of continuity suggests 7 7 2 7 7 L H U L Although the natural coordinate system for planetary scale geophysical motions ltlt1 typically quot thin is oblate spherical coordinates if the horizontal scale of the motion is small relative to the radius of the Earth can adopt a local cartesian plane in which the xy plane is tangent to the surface 2 is normal to the surface Positive directions are typically assumed to be northward and eastward The rotation of the 2390quoter ON 1 planet translated into this system becomes Corioli Q Oi 2cos 9j 2sin9k 2 ZQXu 2 Ziwcos Q vsin 9ju s Q ku cos z zifuj 2 2u cos6k wheref2 2sin 9wltltv u f2 2 2467 60 1 45 X10 Geostrophic Flow Also note that the vertical coriolis component is generally small compared to other terms leaving the following equations of motion amp rial amp u rial Qrii g Dt p0 Bx 7 Dt p0 By 7 Dt po 2 p0 Depending on the horiztonal scale one can use the full definition of df 29 cos 90 f 29 sin 9 the plane approximation of f f0 y W R or the f plane model of f f0 constant A common approximation is to assume steady ow and linearize the horizontal momentum equations yielding 1 B 1 B fv ii fan77p Po ax Po 3y 2 This linearization isvalidif W U IL 2 Geostrophic Flow For geostrophic ow horizontal ow is along the isobars as opposed to across the isobars uVp 1 a ia ja iaijo gt uin Pof ay ex ex 9y This creates rotating ows about high pressure and low pressure regions as shown This is called geostrophic balance as opposed to cyclostrophic balance where centripetal acceleration balances the pressure gradient see 1 3p 2 7 if a v 7 solid body rotation p Mi 717 paf 3 Br fgtogtovgto u u p 39 u Geostrophic Balance Cyclostrophic Balance Thermal Wind If the ow is strongly strati ed vertical motions will be suppressed leading to hydrostatic balance With geostropic balancethe governing equations become 1 3p 1 3p 3p 7i 77 oi 190 9x f 190 By 32 gp 2 3iipgtizlapgip 32 0 3x 32 0 f 3x32 0 f 3x 2 gt gt 81 1 a72 7g 37p 32 I70 3 32 Pof ayaz Pof Egty If assume an ideal gas along an isobar the horizontal density gradient is proportional to the temperature gradient 17 3V g 3T 3u g 3T P W W So vertical wind shear is proportional to the horizontal temperature gradient TaylorProudman Theorem The thermal wind requires a horizontal density gradient and is generally a baroclinic phenomena The simplest barotropic version is for a homogeneous uniform density uid In that case g 9720 3u g 9190 32 0 f 3x Further can show i Lipi Lip alzziw 5 3y fv po3x3xfu p03y f3y3x O f32 Therefore 37quot O lt Taylor Proudman theorem Z 32 Pof And since w O everywhere on surface this implies w z 0 everywhere Strictly speaking this also implies that u v O everywhere as well based on no slip boundary but assumptions break down in boundary layers ShallowWater Equations Assume a thin layer such that g ltlt 1 lt consistent With Taylor Proudman Therefore u uxytv Vxy mm Q small B2 B2 Assume that the thickness of the layeris hx y H 17x y t 17 ltlt H Hydrostatic then implies 2 Bl pg 34371 pg Bl 7 Bx Bx By By L H Incompressible J W V u Odz H 17 31 H 17 i w17 O o Bx Egty wnH17H17ampuvi o Dt Bx By Bt Bx By Rearrange and linearize to get Bl HEEL O Bt Bx By Bu an Bv 917 S t t I 7 i i ame o momen um gives at z g ax at f 3 By Vorticity and ShallowWater Return to the non linear equations for SW halhalalualval02 ahum0 Bx By Bt x By Bt Bx By ailtuaiuvaiu z 3717L uELVVifu g3717 Bt B By Bx Bt Bx By By Approximate f9 29 sin 9 z 29 sin 90 29 cos 90 9 90 2 f0 y i in i amp in 5 Dr fV ng BxDtfu gBy 3y 3ijiuivijiuvhjfoij v0 Bt Bx By Bx Bx By By Bx By Bx By iiu 1 iquot i De ne ax 3Dtfoaxay v o i 1 D11 Bu Bv Mass conservation if 7 i h Dt Bx By Conservation of Potential VortICIty D ma 2 0 Dr h Dt Use DifalualvaifVal Wfo gtgt y Bt Bx By By Dlt flt fogh0g 5f 0 Dr h Dt Dt h This is the conservation of potential vorticity It is a fundmental condition of geophysical uids essentially consistent with a vortex tube concept except the vorticity is now the absolute vorticity Note that the addition of f gives a bias to the solution since on a planet it increases with polarward motion decreases with equatorward motion This can be seen in hurricane motion which tends to be westward and northward in the northern hemisphere and the jet stream whose wind is eastward countering the phase speed of the westward Rossby waves Frequency Regimes Start with linearized SW equations alH 311 0iu gal7 avfuga7 at Bx a By Bx 3 By a Bu an Bzu av a Bu av i 7 7 i Hi i i Eula fV ng 5 gm th g axaxay 3 av an 9 Bu 3 Bu Bv i 7 7 i Hi 7 BtBtfu gBy 5 at th g ayaxay Eliminaten 2 3 2 2 a Bufiunga 91 livwaiunga 91 Bt atz Bt By Bx By 3 at2 Bth Bx 7 9y 3 2 Qf faltngi ali gH a alav 3 Bt Bx Bx By Bth Bx By Frequency Regimes Now construct a vorticity equation a ahv an 5 321 am 3217 a E g 3y g Bxay 3 av 377 32v Bu 3277 axiafrfu gdy 2 Blaxf3x gaxay azu 32v 3v Bu 3f 3 Bu 3v 3v Bu 77 7720 222 22 0 azay azax f 3y3xj Vay 23 3y ax f0 ayax v Combining the two equations a a Bu 3v 3v Bu Hiii i 77 O 5 g axiaziay Bx f iay 3x v 33v 3v 3 Bu 3v 32 Bu 3v 7 7 H7 77 H7 77 33 f if az g Bx 3x Byji g ByBl Bx 3y 3 2 LV gHiV vf02 3V 3v 2 H 20 313 at t g ax Frequency Regimes As with the w equation assume wavelike solution form v Oe w ly m 2 m3 iagHk2 lz if02a igH k 0 2 w3 wc2K2 f02w c2 k O This equation has three reairoots two where a gt f one where a ltlt f High frequency waves a gtgt f 2 w3 wcz K2 2 O 2 a iK gH Moderate frequency a f 2 603 0X22 K2 fozws O 2 a i igHKZ f02 Low frequency wltlt f 2 wcz K2f02w cz c s O gH k csz fez Note that high frequency waves are unaffected by coriolis low frequency waves affected by gt a Gravity Waves Start with shallow water form 7 3x By 317 317 Bu 3V 0 at Bu 3 av E I gai7 Efu g 7H and assume standard waveform MM 17 a ameww m 2 iw12 3 ikg z39w 9 2 ilg iw iHk12l9 0 Eliminate the velocities and cancel the de ection to get dispersion relation wz f2 gHk2 12f2 nglt2 2 l H 03 figHnondispersiveif f2 ltlt gHK2 or iltlti K K2 K f 20 gHwsK gH gtgtf Oceanzf1074s 1 H1000mglOms2 222ltlt27 gH 106m So surface gravity waves in the ocean are typically in the high frequency regime G raVIty Waves Although originally found by Kelvin these waves are called Poincaire39 or Sverdrup waves The dispersion relation is isotropic in terms of k 1 so simpler to orient wavenumber vector along the X axis 1 0 Assume 17 1 coskx wt 1 Reei0 m Corresponding velocities u if coskx wt v kH kH So uid parcels move in horizontal ellipses with an axis ratio of wf When w gtgt f sinkx wt v gt O and motion is like the non rotating shallow gravity wave As w gt f K gt O and 1 gt O In this cae the equations of motion become al z0 i fu0 ali0withalO gtw0 at at Bx By a The simplest solution form is u q cosftv q sinft This corresponds to inertial motion in circular orbits w f with an inertial radius r q f and an inertial period of 2711f Kelvin Waves If these waves are bounded by a surface channel coast such that v 0 277 aquot aquot 2 7 277 A A iUocimt i Hi07 i i I 7 7 at ax at ax fu g By use u 17 u me ia ikg gal ia in72O W wz ng20gt w ng2 cgH Rossby Radius H Nextassume znoe fy znoe yA A i of Deforma hon n noe fy coskx ct u 170 56 coskx ct These waves are typically called Kelvin wavesThe development of a tranverse velocity v is impossible due to the shore The wave can only propogate such that the boundary is to the right in the northern hemisphere so that e ify M decays away from the wall Internal Waves Now use the linear non shallow equations for strati ed uid B oussinesq 2 LL91 g wz 3x By 32 at g in ryg sway alriilh at p0 Bx at p0 By at po 2 p0 Can get w equation using same procedure wnon rotating IW 2 2 LV2wN2V wf2 a w OwhereNNz Btz 322 Again with wave like solution u v w 0 eikxly m 2A 2 2 2 2 22 2d wN a k l Wzo d wmzltzgt 0 dzz 02 f2 dzz m2 lt O 2 exponential decay horiztonally propagating surface wave m2 gt O 2 internal waves with frequency range f lt a lt N Internal Waves Further solution of this problem is traditionally done with a Wentzel Kramers Brillouin WKB approach in which both N and m vary only slowly in z The assumed solution form then becomes w Azel z Substitute this into the w equation and equate real and imaginary parts 2 2 2 dim mz 0 z m w dz 2 dz dz dz dz 2 For slowly varying value can treat first term as small get solution z l 2m 7 ijmdz A AO N w A0 nZ dZeZWHy quot0 which with more work and settingl 0 leads to u A015 cos kxijzmdz wt v sinkxiJzmdz wtj A0 may d at W7COS quot1 Z J Internal Waves The result is elliptical motion but in a tilted plane at 9 tan 391 m k k2 N2 z w2 w2 f2 2 2w2 f2LN2 w2 lettan9 m2 k The dispersion relation is m 2 z 2 12 f2 sin2 9N2 cos2 9 This leads to three wave regimes l Highfrequency 1 N wgtgt f 2 a Ncos 9 non rotating solution 2 Low frequency a f wltlt N 2 02 sf2N2 cot 9 hydrostatic solution 3 Midfrequency f ltlt 1 ltlt N 2 02 s N2 cot 9 both hydrostatic and non rotating Lee Waves A particular type of interal waves occurs in the quotleequot of a mountain in which the mean ow and phase speed exactly counteract each other to create a standing wave To see this assume that a gtgt f Then w2N2cos29 i k2 m 2 In a moving ow with velocity U the effective frequency is 10 0 K U Fora standing wave 10 O 2 a K U kU 2 U N k2 m2 U is defined by the wind k by the moutain shape so the above determines the downward propogation of the phase Q uaSIGeostrophy Quasi geostrophy QG expands the geostrophic balance to include the beta effect Specifically we will look at the effect on the PV equation D 4 f D Bk 7 O 2 hi 7 0 Di h DIGquot f 4 fDl Assume that the height is the combination of an undisturbed and disturbed layer hH77 8fl8y 84 84 84 877 877 877 2H 777 7770 miaz ax V8y v 4 f0 8t ax V8y Assume small perturbations linearize by dropping all HOT 84 877 2 H 7H 7 O 87 v f0 8 Assume approximately geostrophic balance for velocity 2 2 3377 V2534 alalj Lm 8x 8y f0 3x2 ayz Rossby Waves 2 2 Wm Vial 5 iuimm f0 ay7 Nfo ax ax 3 fo 3x2 Byz 2 2 2 3 Lam 3H 91f0970 f0 at 3x2 Byz f0 3x at 2 ajzogg in at 3x2 ayz gH ax at 3x2 ay2 02 8x Note that we are violating strict geostrophic balance by leaving in the BnBt term This approximation is consistent with the low frequency regime corresponding to waves with only a few oscillations around a planet ie the jet stream To get waves assume 1 eiu wly Solving for the dispersion relation in the usual way gives k w k212f02c2 O Ross by Waves The phase speed of Rossby waves is i c i phase progation is westward k k2 12 f02 lcz To generate a standing wave superimpose on an eastward ow c U k212f02c2 The simplest exampleis if I 0 ow is barotropicso f02 02 ltlt k2 cC 02Uk 2 Friction in Geophysical Flows For a rotating system with Boussinesq approximation Vu 0 gnaw in ampkF o Dt I70 I70 Dt The friction term is not typically laminar or isotropic in geophysical ows In particular the vertical viscosity is typically of a different scale than the horiztonal viscosity In stratified conditions VH 1000 VV In areas with strong vertical convection VV can become much larger So as a first approximation zar arxyar x ax 3 32 H 3x2 Byz V 322 37x 8139 372 2 2 2 F y7w ysz H VVBV y 3x By 32 3x2 ayz 322 any any 3139 32w 32w 32w FZ77 vH 77 VV7 ax 3 32 3x2 Byz 322 Ekman Layer Look at near surface ow assuming the velocity is horizontally homogeneous 2 2vd7 fu VL dz dz 2 Can solve with a complex velocity V u iv 2 d Z I39LV dz V The boundary conditions vary depending on the problem Two classic cases are the ocean and the ow over a solid surface Ocean pv r 0z0 uv gt0z gtoo dz 2 d tz5z7t rz5 zrc2v ll 79 cos 77 V 7e SIn i5 7 PM 5 4 PM 5 4 f NoslipsurfaceuU v0z gtoouv0z0 Pofa l u U1 e z 5 cosz5 v Ue z 5 sinz5 Ekman Layer Differences from the traditional boundary layer 2v 1 5 inot5f f x 2 v 02 ledz U5e cross ow transport Geostrophic Turbulence Large scale geophysical turbulence is different because 1 Vertical motions are suppressed 2D ow 2 Vortex stretching to mixing energy cascade is absent Result is known as quot2Dquot turbulence eddies cascade upscale rather than down in direct constrast to the turbulence doggerel quot Big whirls have little whirls that feed on their velocity and little whirls have littler whirls and so on to viscosityquot This behavior is actually re ected in the performance of Rossby wave triads which we will not discuss in depth here Key feature is enstrophy or the mean of the square of the vorticity g 2 If we represent an energy spectrum as S K where K is the wavenumber then the following relations hold i7 JSKd1lt P fK25KdK with SK having units of m3s2 Geostrophlc Turbulence In an effectively unbounded effectively inviscid system just as large scale geophysical phenomena both of the quantities are conserved d w d w 2 7 SKdKO 7 KSKdKO mi dzio H Given this imagine an initial disturbance that has all of its energy in a tightly bound gaussian about the wavenumber K 0 As time passes we expect that non linear effects not viscosity but Reynold Stress like terms would cause the energy to spread away from this initial configuration To make the illustration simple assume that after some time all the energy has effectively spread to two different wavenumbers K and K where K lt K0 lt K By conservation of energy amp enstrophy this requires s0 s s K350 K35 KZS Doing some algebra one can determine that SK K0 K K0 K35 K3 Kf Kg S KO K1 KOK1 x 2 K K3 Geostrophic Turbulence These relations imply that as one ratio increases the other decreases Again to further simplify assume that K 2K0 and K K0 2 Then the above ratios become 4 and 14 respectively The result is that 80 of the energy cascades upscale mm quotquotlv and 80 of the enstrophy cascades downscale K7 K0 K In K This upscale energy motion eventually breaks down at about the Rhines length L R M where due to the beta effect and Rossby wave propogation the eddies become stretched into waves or jets Further for 2D turbulence to occur the Rhines length must be considerably longer than the Rossby radius of deformation LD NH f Jg39iHf which is not typically the case on Earth However this is true on the jovian planets leading to both the zonal jets and vortex features Turbulent Eddy Conductivity Energy Equation for 2D incompressible steady laminar thinshear ow BT BT BZT Bu 2 pc u v k u p Bx By ayz By For turbulent ow f uuta U 3 a k kta T ay ay c Turbulent Prandtl number Prt 2 2M1 l Reynolds 1874 I Use Pr fPr or Pr I constant 2 07 10 Thus kt cput Prt Turbulent Eddy Conductivity Assume that near wall U z 0 V x 0 a v v a g IO kkt Pcp 8y E Pr 5 y1 6b T 3610O 1 Vt JTOPdeTPCPTo T ng TE 10 3TETO T y dy c u T O 1 V p p ET vhf If assume a model for u I can integrate numerically N 237 Thermal sublayer T Pr y CPrl 13pr 7 P901 CPr 358Pr13713W 2121nPr P Thermal log layer T zilner CPr K Compressibility in Thin Shear Flows Assume Thin dpdx gtgt dpdy S ready 2D Laminar Perfecf Gas Compressible BoundaryLayer equations pszT c zchT CPTCVTR PIZJCPk B B 0 axon aymv au 813 all 812 Bx By dx Byk By By i alwdpmi u W 390qu poy dx By PrBy pu Compressibility in Thin Shear Flows 2 Total Enthalpy H hu7 v2 ltlt u2 Inviscid Bernoulli dp X 2 pl ah ah 1 8p 90 14th p Bx uth 1i 8 Bx By By kPr By By Pr By Equa rlon Pr 13H constant is solution 3 0 a h ua u ay ay Bh At surface u 0 3 0 gt adiabatic wall 3y Cr occoBusemann Relation 1 Pr Ladiabatic wall gt H h 7 constant Compressibility in Thin Shear Flows afan 3 jpuaupvaua BMW 2 0 1 dagBur 3x 3y 3y 11 W Alternate case Pr 1 a 0 gt h hu 3 2 2 1h C1uC2 u hhWu03C2hw hw hw ui2 u hzhw uzuw 3C1 co Cr occoBusemann 2 Relation 2 3H hujhw Hw hw1 ueo Compressibility in Thin Shear Flows Add assumption that cp constant gth CpTC 2 2 3TTw T ui Tw i L 20p uw 20p u2 h Adiabaticwall hw 2H 2h 4 Taw 27W 2 CF 71 112 szTwTaw Twii uw 20p Sqwzkwal Taw Twkw TawTwkwTw By W quotwe MW Mao1 qw 1 Ch Reynolds m ZPI Analogy Sim Soln for Comp Laminar Flow Continuity a W pu a W pv 3 0 By Bx Bx By By Bx WM I my G gtfngt man um f n Sub into X momentum equation force DE to be indep of E Illingworth Approach ax fpumumommomx may joypdy 2 dues pm 2 PM 2 C 0 C f umd kp f pmm No slipfreestream BC f0f 00 f oltgt10 Incompressible Check umpu constant gtf 0 gtB1asiusot1 C 1d 0 puconstant gtf 2 1 f 20 gtFalkner Skan C1um xm m1 2 Lia va uu00 61 Va u Bernoulli u00 61 ldp Bx By dx ayz dx p dx f0f 00 1 2 0 3f 0ltffB f3 fmgt1gtw 3d 12 duw d 2 V wcwmm oa V dx B3V20 B dxuw xquot uwx m1 2n Sim Soln for Comp Laminar Flow Energy Equation hx y 2 he g77 C I 5 6m x u 3 7 i Pr g fg h co H d 2 f 2g fa le 2 2 2 MM 1uw C 7 y 1Mi y7f u 2 Perfect Gas f M CPTM 7m V Low Speed Perfect Gas g Pr fg39 0 Boundary Conditions Adiabtic Wall g0 gwg 0 0gltgt 10 Heat Transfer g0 gwg0 i 0g 10 Sim Soln for Comp Laminar Flow V n G fn u m um f39n h n hm gn x um y ax jopmmumrxmmrxmx may EL pay 2 Cf 2g ip f 2i 0 um d k D J C I I I E M I 2 I2 2 2 2 4 C Mg fg Hm d fg hoof hm f ForSimilarity CPrconstorfg pifg P Ed Hi amp dH w const const h const small or Pr 1 x x u Sim Soln for Comp Laminar Flow PerfectGas 0 2L0 zimy 1 pm Tee hm gm since py z constant nil a n h h n 2 ng77 T Aiizi g 1C77 P hm ee the Air nz 23C z g m Pr z constantfor gases in ll E zgmgm l Zm HOO if 3 1 MM dd 49 H dd Common Assumptions H X const j 0 m 0 Flat Plate u const FP Pr z 1 many gases uw m3 eo Q 0 Stagnation hw gtgt uw Also need gw constant gt HW TW 2 const Sim Soln for Comp Laminar Flow Cf 39 if 6f 2 g fg H i f 2gf 2 Cf 2 WM G fn um more mm hexagon ax joxpmommmmomx may 3 joypww B Recovery Factor on an Adiabatic FP 0 TW constant Cl gtuT 3f 0 g Prfg Bf 2 B Pr7 1Mfo f0 f 0 g 0gt 0 f ltgt goo 1 3 f Blasius Solution 77 N N 77 N 77f 772d77 N g gw AJO G77d77 BJO Gumbo GWJdn 77 N N G exp Pr JO f77d77 If adiabatic wall h 0 g 0 0 A gw Taw TM 2 V 2 c 3Taw Twr 3 r2prj G07 WW m7 d 2gp 0 0 G07 Recovery Factor 0 Tw constant C 1 2 f 0 g Prfg Bf 2 B Pr7 1M M Pr recoveryfactor Pr 07 gt r z 085 TS Tw 01ltPrlt 30 rz JE 30lt Pr gtrzl905Pr13 115 Why is T aw i T S 7 Dissipation heating Pr Ladiabatic wall 3 Taw TSmgw Taw TSmg qw 0 FE Conduction loss 2 Dissipation gain Pr i 1 adiabatic wall 3 Taw i TSmgw Taw z TSmg q0 0 Recovery factor measures the difference between the temperature of an inVisid uid decelerated adiabatically to zero velocity and of a Viscous uid decelerated to zero at an adiabatic wall Flat Plate Flow C 0 Chapman Rubesin parameter 0 Cf r 0 Blasius wvariable p 3 15cm Cof39ltogtjexp j 5 d d77 Near wall C x CO z constant 2 2 f f0 f 077 f 0gt 77 z 1 Cojyvncoltf g Co 13 w quot2 1 1 lo ml 2CO lo dnldn3f 0 H3 Reference Temperature 1 6CO 3 1 048 C zC 0 r oz X0 204696 0 of3f0 33f f3m 139 f 0 gt c z C C C adiabaTic walls 0 f Re h h MO 0 MT hoTcold walls X More generally can use reference temperature W at plate equations gtslt 713 0 390 I z T TquotTMwTOlt RefTemperature Poo1w Tm T t 2 c a zllqlpr 7 1M3m LOZZM szr 23 Too 2 p u Rexw Chm coco Eckert z 05 00391 05 0 x x Flat Plate Flow Van Driest Solution for Flat Plate based on alternate similarity approach 32 AuzPrO7i iiz Z 75S 0 To TS T 7 1 2 u2 3 z1Pr w l forT Ta To i 2 l l u 0 0 gJngzJCODm2O9LX7 MM a0 Momentum Integral Equation 5 c ist 2Huidpw 1 dx um dx p0o duoo pm dx 2 H21 pwuw 9 5 w1LLd 9 wLL1Ld J O pwuw y Opwuw um y quotide d 2 Mi 9 if adiabatic freestream poo use 5 if assume g p 2 negligible 3 page J O pdy 2 0 3 d9 9 duos 3 dx H M L y y Gruschw1tz u ug pp 2 2 5 2 JAG duos A 2 A idy zw Tm v00 dx A 0pm A 315 52 du z wA25 gtA P P V dx eu d6 T 2 4v A F x xi 2 M F x 2 z w 1 Vw I 1ltgt TO 00 gm of A 7 as F 231 AA 2 023sex 54x 1 A 15 240 p Pr072714 2 F2 171 53M 207Sexp 157 2 9 A 2160 53 798048 4656A 758A2 7A3 A 4324320 CompressibleTurbulent BL Flow Assume Thin dpdx gtgt dpdy S ready 2D Turbulenf Perfecf Gas Compressible BoundaryLayer equa rions 7 7 5 7 E RTdh cpdTcp zconstPrcp k Bu 6 B BuB puv u 3lt7gt3ltgt 0 Bx p By W p Bx By dx By By By imltlt apltygzw Eva prm Bum By By dx pm Ty dx Bx 7 i a Bi p Bx pv By dx By By pv By W By CompressibIeTurbulent BL Flow Let pu v pa H Z522 3y 3y pV 7kt3T 3y ilt5ugt3ltpvgt0 via 531 d 3rltumgta W Bx By x a a E Byk By 5531453121 i h Bi prtz Bx By By Pr Prt By kt y02 pv0fTo6po y6ltyoogtz aumltxgtfwwltxgt apmltxgt Turbulent CroccoBusemann Relation of BH B17 B BLT P P zl z03 r rt dx 390 Bx pv By By W lat By pu pv B7 BE 1 Bx By By Mya j ay Bx d1 By d1 By By d1 1 C1C2i 2C2 Energy gtM0mentum 2 2 3h C1 C217 u gtsz0 Ta0 T0i 2 um 20p Cr occo Busemann Rela rion r z Pr 3 for gases Integral Equations amp gdumf lu ltdpoltl 1d 5 Cf 1100 dx k2H pm dumjupmui dkamS JOPWJ 2 2 50 3 Cf pmui g7 Compressible Turbulent Law of the Wall Trends 1 Increasing Mach number drives velocity profile below incompressible log law 2 Cold wall raises velocity profile above incom pressible loglaw opposite is true for a hot wall 2 T Tt51i lmzky 71 6b 2 C BR pTOi 1Ta0L Joe iii p T0 TO M ZcmeTOkum 2 1 Tao 1 u y lMazTi L TO M 2 00T0 um If C x to can integrate above Van D est 1950 Van Driest Effective Velocity ulln j C uisin1 2aZ u b sin1 E Me K a Q Q q 2 Too V0 T a y 1Maolt b O 1 QVb24a2 u erp 2 T O T 0 N0 heat transfer at wall adiabatic b 0 Q 2a T E a2 1 m um 1 Sueq sin a ltgtltgt T110 u ForfullBLusewall wake1aw eq zilny C sin2 1 9 K K 2 8 7 12 u Note that e ecm 2 IO d17 poo Van Driest Flat Plate 0 2 2E G 2 cell Use previous VD forms f dx 0 pm umk um 1 1 swzmmg Rex cfmulj17 cfmaao TO 1 M0 2 2a b B b A Vbz 4a2 V172 4a2 1 z 41510gRex cf 17 ch Mmz0 gt Van Driest Flat Plate Interpret result as incompressible solution modified by compressibility turbulent Viscosity 1 1 c m 2 0 Re a FR c Reece FR F f Fc fmc x ex Fc flquot ex 0 Fcz aOTV 2 FRchz sin Asin B X No Cfinc SpaldingampChi1964 cfmc fRex lt FReX cfjcam 2 FC Turbulent recovery factor Dorrance 1962 115 2 rt z Prt Pr Prt a PR mm a PrlZ um Instability Consider instabilities from two Viewpoints Normal modes and growing instabilities Assume that there are small disturbances in the ow of the 6 X70 yeikxmz6t7 where 039 can be complex k m 2 real If these small disturbances grow exponentially the ow is unstable If the disturbances are decreasing the ow is stable Mathematically 56 16 unstable or gt 0 stable 6r lt 0 neutrally stable or 0 For the neutrally stable or marginal state the imaginary term is important If there is no imaginary part 039i 0 there is a stationary pattern of motion cellular convection secondary ow If there is a non zero imaginary part the instability is oscillatory with growing amplitude oscillations Vortex Instability View Vorticity Assume the ow has small vortices The arrangement of vortices can either suppress or enhance cross ow motions In the later case this encourages instability F u 0 9 7 727 NH gt 1TH Fz iL1nz zo 2 Stable Unstable in gt 11 1 B nard Problem Start with the Boussinesq equations including thermal energy conservation 3 D N I 1 L9PLgi szui axi Dr p0 axi po 13 p0 1 altTquot To gt1 g am g xvzf Dt 10A1 z d fr07rz Lg T0 B nard Problem Let L7 Ouixt f zT39xt p7PZpxt Bu Du 1 3 2 P 1 0LTT T vV u axi Dr p0 axi P 0g Bin 3 TT KV2TT at 3x Basic or mean state 2 orig 1 ocltf Togtg 018 P0 Bxi dz 2 Kd f 03Tclzcz dz BC z d2 T0fzd2 TO AT T TO rzd2 B nard Problem Subtract basic state from full equations to get perturbation equations Dui LiP P LBP Dt th p0 3x 1 0L T39 TOgi 1 ocf TOg VV2ui ia p OLT39gi VV2ui PO axi 3T uiTT KV2TT KVZT at x 33iwa Tuiai ai wfuiaiKV2T39 3t 32 th at 3x Linearize perturbation equations 3 u ia p OLTgi VV2ui ai wl KVZT 3 p0 3x a B nard Problem Scaling arguments TT39rdgtw KT K d Pd2 7 Therefore the ratio of the final two terms buoyancy Viscousin the wr KV2T39V2T39 momentum equation compare as xT gi xT g de 4 g VV2ui vw d 2 VK Next create w equation39 from mometum Ra lt Rayleigh Number Laplacian of z direction 3V2w LV2 3717 gotV2T39vV4w at 30 32 Divergence of vectorequation 3 ii in gxaivV2 at Bxi p0 Bxi 3x1 3x1 3x 2 r 2 r 20LV23 a iv i a 1 2 20 7V pg0 g P0 32 P0 32 322 P0 32 322 B nard Problem 2 Combine szw ga gaV2T vV4w gaV T vV4w Z BC 2i wT O a For rigid no slip plate boundaries u v O and MI O gt O 3xi 32 i i i d2 K Nondimensionalize t gtitxi gtxidw gtgwT39 gtFdT39 K ai l wl 72T gt i VZ T w at at 4 3V2wgavgr39vv4w gt i3 V2 Vam vgr39 at Pr at W waiwT O2ii gtw T39Ozil 32 2 32 2 Benard Problem Nonnal mode analysis w zeikxlygta T39 f2eikxlygta 3 A 7wme hlygtmm V w k2lzw K2w I V2wL K2wD2 K2w d2 Updating the equations g VZJT39W gta D2 K2f vW 4 i3vzjvzw vgr Pr 3t K39V gtP1 D2 4202 K2W RaK2f 1 walT 0zi l gtWDWZA Ozii 32 2 B nard Problem Kundu proves 039 is real 0391 O we will take that as a given This implies that the marginal state 039r O which means the onset of instability is transition to a different steady state cellular convection This is described as the principle of exchange of stabilities39 So examine this marginal state more closely 039 0212 K2f W D2 K22W RaKzf Eliminatef f D2 K23W RaK2W D K BC W DWD2 K22WO 2 Sixth order homogeneous DE with six homogeneous BC eigenvalue problem Fdz A qz d q qz Can solvebyassunnngform WiwWoe diWoe que qW K 2 D2 K23W RaK2W gtq2 1lt23 RaK2 Benard Problem D2 K23W RaK2W altq2 K23 RaK2 Let RaK2 RaK2gilt2n4gtn 2 12 K2 2 RaK213e2n7113 ei2n7113 Lli 71 2 2 2 z 2 13 2 2 3 2 2 Rd W Low K K 16 Al 13 12K21l aim5 2 K4 gt q iiqo q1 qf ltaib a z39b 12 12 13 13 q0 Ki 1 q1 K1Ej It 145 7 7 7 2 W 61191402 6128 1402 a3eq z a4e 4 2 aseq z a e 4 z B nard Problem Mathematical recombination fie a cos9 ai sine iar sine ai cos9 b cos9 7 sin 9 i br sine b cos9 a 12 z39a ibicos9 a b iar z39brsin9 aeie be Acosh9Bsinh9 gage Ae 3e Be 04 Bee A Be aee 1259 Using the above can rewrite W W Ae cosq02 Be coshq12 Ce coshqikz A0 sinqoz Bo sinhq12 Co sinhqu Typically look at even vertical motion symmetric about centerline gravest lowest mode one cell and odd vertical motion antisymmetric gravest mode two cells modes separately B nard Problem Apply the boundary conditions to the even mode solution W DW D2 K22W 0 z i DW Aq0 51110102 Bq1 sinhq12 qu sinhq1z o D2 K2 W Am KW cosmz Bq12K22 coshmz 2 Cqf K22 coshq1z W o 2 A cosq0 B coshq1 C coshq71 o DW o 2 Aq0 mm Bq1 sinhq1 qu mm o D2 K2 2W o 2 14013 22 c056 30112 K22 coshq1 2 Cqf 22 coshq1 o B nard Problem Writein Ax 1 form 10 11 11 cos 7 cosh i cosh i A 2 2 2 qo sin i q1 sinmqi qf sinh B o 2 2 2 2 mi K2gt2cosltq7 gt q K2gt2coshltizlgt q K2gt2coshltq71gt c 2 detA O 2 K fRa defines the line of marginal stability Ra Ramin 1708 K 312 A y m 2d KEV The above approach creates rectangular top View xy plane cells Benard39s classic result is hexagonal cells which requires a different assumptionin terms of the form of w T39 In general need to assume form w Wzgtfxygtequot T fltzgtfltxygte quot 2 2 Wherefxysatisfies Bigaij2fa2f O 11 K Bx By B nard Problem Stress Free surfaces layers of different density liquids Bu 3w 3V 3W BC T O 0 77 0 w EO Maz ax0 M32 30 E 01 0 w0f0rallxyatwall 3x0 Byo 2 31 2i 02lti0 2 330 320 320 323x 32 O So the problem becomes D2 K23W RaK2W BC WD2WD2 K22WOzi D2 K22W 0 D4 K2D2 K4W 02D4W 0 Benard Problem All even derivatives vanish on boundaries so odd modesolution W Asinrm39z 112752 K2 3 K2 Ra 2 2 2 2 2 3 Gravest mode n 1 KZW WO dK K K L 5 Can also solve for rigid bottom stress free top K 268 Ra 1101 KC 222 Ran 754 657 The linear theory predicts horizontal wavelength at onset of instability not cell shape magnitude of ow or direction of ow DoubleDiffusive Instability Whatif h d jisdl d tont n in gradients typically due to temperature and concentration of a solute like salt r3po1 ocltf TogtBltE sogt This type of system can be unstable even with a stable density gradient if the diffusivity of the concentration is much less than thermal diffusivity Jumping to the appropriate non dimensional linear normal mode equations D2 K2f W ampD2 K2 W K D2 K22 RaK2f Rs K2 BC f K S 0zrl K 2 4 4 Razgord dTdz 1ng dSdz VK VK DoubleDiffusive Instability Simplify by eliminating to D2 K2f W D2 K22 Rs RaK2f Rs RS39L S Same as Benard convection stress free solution 4 Rs Raamp dS3dT 657 v KS d2 K d2 Can get two unstable modes 39salt fingers39 and oscillating instability Taylor Problem The Taylor Problem is the instability of Couette ow between rotating cylinders The instability is generated by an adverse momentum gradient with centrifugal forces powering advection The stabilizing force is again Viscosity The basic equations assume axisymmetry and are 3 r r3 z D3N3N3 0 ur uz 3r r 32 Dr 31 3x 32 N N2 N N N N N 1 Dur u9lpvvz r ur Due MVV2 e 2 r r Dt r p r r2 Dr N N 2 2 lapvvz z V2alia Dt p 32 arz r 3r 322 Go through the same steps as before to get pertubation equations Separate into background plus perturbation L7 U ui p7 P p Taylor Problem 2 Backgroundstate Ur Uz 0 U9 Vr V pdr r 9 Rz Q R2 Q 9 R2122 VrAr A 2 211152122212 Substitute the seperated variables neglect nonlinear terms subtract the backgroundstate to get perturbation equations 3 a zo l9 13PV Vzur ur 3r r 32 at r p 3r r2 a M 4 dVK urzv Vzue e auz la pVV2uz at dr r r2 at p 32 Numerous approaches to solve from here including multiple non dimensionalization approaches Kundu follows the approach of Chandrasekhar to get Taylor Problem D2 k2 csD2 42 l0x e D2 k2 me Tak2a a Da ae 0 x o1 r R1R2 uwue araere z u1 xquotdR1 dR2 R1 Ddi 1 r 4 2 2 4 24149161 24921123 Qlel 91d v R2R1 v Taylor assumed that the margan state was the stability boundary created a determinant equation to relate Ta and k using r Z bn n Kr7 e 20quot Dr Knr 1 Knr e Bessel Function quot1 nl The result is that there is a minimum marginally stable Taylor number Ta i kc 3127tc 22d 121QZQl kc Taylor Problem A somewhat different approach by Yih assumes a non dimensionalizing of rarRl 222131 tat 21 and a solutionforrn of u Q r coskzeot ue e r coskzem with z and f1 following the same forms z with a sinkz This yields D2 2 i k2 oRD2 2 i k2a 2sz A1 as r2 r2 r2 12 2 i2 k2 6R e 2RA112r r a Da ae Orl0L0LR2R1 B 4191 1 27 RQIR12 1 QlRl V This set can be solved as well and is closer to Taylor s original approach Since viscosityi the rnost L is if the ow is inviscid orR gt oo Taylor Problem Then 5R gtgtDZ2 i2k2 r 2 5a2 g i2 k2 r 2k2A1 r 2 2 D2 2 i2 k2ae 4k 11 A1B 21 a9 o r r 5 r l V Bl We can define the rotational frequency as no A1 2 r r And using D l lDr F 210V ZTEOJOrZ Dz 2030 r r d d rz di4n2A1r2 15212 16n2A12r3 AlBlr 1612A1r3030 r r 4 A 2 D1Dr e k21 jrae o r r czr Taylor Problem 2 2 D1Dr e WE dF rae o r r 415262r4 617 dl z gt O 52 elt Orealstable Sturrn Louisville system 12 all 2 i d lt O o egt 0 one root is gt Ounstable r So condition for stability is that the square of the circulation increases everywhere with r This is Rayleigh39 s stability criterion and is consistent with the simple physical explanations used by Rayleigh 18881916 and Von Karrnan 1934 Taylor Problem Consider two rings of equal mass with steady velocitiesThe balance that keeps the incompressible uid on a circular path is 2 2 3 1 U9p r 7r2mUe 3r r 412r3 So for the inner and outer rings the KB is E U92 r12 r22 1 7 2 2 8Tl392r12 8Tl392r22 2 2 1 F1 F2 3 E12iE1E2 2 2 2 8T r1 r2 Now imagine that the inner ring shifts in and the outer ring shifts out corresponding to a secondary ow By Kelvin s theorem circulation must stay constant so Taylor Problem r2 r2 1 r2 r2 Comparing the energy results AE Elzf Ew 2022 1 12i2 i2 87 r1 Only AE S O is physical satis es conservation of energy Since r2 gt r1 this will only occur if 2 F12 gt F22 3 dbl lt O unstable r Can also achieve this result considering pressure gradients To maintain the circular uid paths the following balance is necessary U 2 2 p grad centripetal accel 3 3 p p 9 p r 3r r 472 3 r2 Taylor Problem As the a ring moves from r1 to r2 r A r t 472r13 472r23 472r13 472r23 To stabilize the ow the pressure gradient inward force must exceed p grad the centripetal acceleration outward force or 2 2 2 F F df 2 1 3 F22 gt F12 3 d gt0 stable 472r23 472r23 A final note Instability similar to the Taylor problem can occur in curved ow in which centripetal acceleration is significant One is curved channels the other is boundary layer ows on concave walls This latter case gives rise to Gortler vortices KelvinHelmholtz Instability Assume two 39in nite39 layers of uid of different densities and different uniform velocities creating a discontinuous shear across 2 0 Since the background state is irrotational in each layer Kelvin39 s theorem says that for inviscid ow the ow remains irrotational or N N 2 N 2N 1 U1x17 2 U2x27 V 1 V 2 0 Which creates a peturbation problem of V21V22 0 1 gt0Z 2 gt0z gt oo The linear boundary conditions for the peturbation at the interface are aq D CU E a 2 U2z0 x W quot 32 Dr 32 3t ax 32 at a The dynamic boundary condition is again similar to the wave problem and is derived starting With the unsteady Bernoulli equation at z C KelvinHelmholtz Instability amp at 31 1 1 V 12 p1gCC1 Vz2 p2gCC2 at 2 pl 2 p 2 1 P 1 P VU1x2 1 gC C1 VU2x2 2 gC C2 2 P1 2 P2 Tohave a pressuiein the state P1 P2 at Z O 2 p1ltU12 Cl p2ltU C2 Introducting the decomposition of the total state and removing the background state assuming p71 fiz at Z 0 31 31 32 32 3 U U z0 P1 at 13x gC I32 at 2 ax gC Assume wave like perturbation modes C Cltzgteikltx quotgt 1 1ltzgteikltx gt 2 132ltzgteikltx quotgt where k is positive and real and c c ici where unstable ows have c gt O KelvinHelmholtz Instability Solving the Laplace equations requires 1 Aeikz 2 Be The kinematic BC gives A iU1 of B z39U2 a The dynamic BC yields p1ikU1 cA g 72 ikU2 cB g5 kP2U2 of kp1ltU1 cgt2 We pigt 212 U U U U 62172 2 I71 iigpz I71p1p21 2 kp2pi I72 I71 I72 I71 Term in bracket will lead to imaginary roots if lt 0 Let9lt0then 1 2 ii 9 c So if the bracket is less than zero there will be an unstable root where ci gt 0 2 U1 U2 gt I72l71 I72 171 k I72 171 if U1 U 2 there exists some k such that ci gt 0 so shear ows are unstable 6 gt 0gt xvi72E 2 gm 912 lt kmin U22 to short wave disturbances but if too small viscosity will keep under control KelvinHelmholtz Instability If the densities are equal the equations for a velocity discontinuity gives c U1 U2 i U1 U2 7 or unstablefor all wavelengths Critical balance in K H instability is the destabilizing effect of shear overcoming the stabilizing effect of strati cation The energy for this comes from the kinetic energy of the shear Uz U1 2L h PZ U Z Note that momentum is conserved between the two states Now examine a continuously stratified case KelvinHelmholtz Instability The key number for K H instability in continuous stratification is the Richardson number Rz39 N 2 dU dz2 lt stratification vs shear Taylor conjectured that Ri must be less than 14 for instability Several others suggested other numbers but ultimately Miles 1961 and then Howard 1961 proved Taylor39 s conjecture Following Howard start with the standard deconstruction 17xzl Uz uxzt 171xzl 0 wx zl xzl Pz pxzl g3xzl z pxzl Applying these to the Boussinesq equations and linearize and subtract the background ow the perturbation equations are Bu 3w D17 1 Bi Bu Bu BU 1 Bl V O gt77O 7 77 gt7U7w7 7 3x Bz DZ p0 3x 31 3x Bz p0 3x DiwiaiLggtalUalialLg Dz p0 Bz pg 81 3x p0 Bz p0 2 EOaaipUaippoNiwo N2 DZ 31 3x g p0 dz KelvinHelmholtz Instability 3141 31y Replace the velocity with a streamfunction u 7 w 7 Bz 3x 3214 3214 314 an 1 ap z 7 77 77 x mm m 3232 3va ax 32 p0 ax 2 2 z momenlum U ial p7g 8sz 3x2 P0 32 P0 2 39incompressibilily39 37p U 373 0 at 3x g 3x Wave nodal mode form p W 32 i7z1zequotkltx gt B A BU A 1 A A 1 BA gt U cgt7 7w 7p k2U cgtw 77p p7g Bz Bz p0 p0 Bz pg 2 U C ampl 0 g Solve for an equation involving only the streamfunction KelvinHelmholtz Instability a BA BUA 1A A 18A A FwW 47w 41 U w4 g 2 P0 5 dz p0 Bz p0 3211 B2U A 2 A 3g N2 2U ci iw k U cwi 7W Bz2 3z2 P0 U C 32 2 A B2UA N2 A U c i k w iw7w0 3 J 3z2 U C BC wgi0z0dgt1110z0d x This is the Taylor Goldstein equation which represents an eigenvalue problem in which the solutions come in complex conjugate pairs or for every wavenumber k there is a pair of solutions cIIcII Therefore unless c 0 there is an unstable mode either 0 or 0 has a c gt O which as with the two layer case indicates instability Richardson Number To get to the Richardson number criterion do a variable transformation W A I w0aqgt0z0d VU c Subsititue into T G eqn multiply by If integrate between 0 and d and manipulate to get 2 1 2 1 BU 2 31 N 7 7 dz U 7 JU c 43z Jq Ilt cgt 3z 2 a 2 2 k2 2dz 1 3 glo zdz Z Last term is real Imaginary part of first term from 1 U c p4 2 2 Equatei parts a N2 l l2dz of k2l l2 dz U 0 2 4 Hz 32 2 2 k2l l2 dzgt0alwaysJ 1 2 N2 la l l2dzgt0 32 U c 4 dz It Richardson Number Equateiparts ciJUj62N2 aa J 2dzciJ a 2 2 2 2 3 2 2 J k lq Jclzgt0alwaysJ Ulc2 N Jo dzgt0 dz 2 if N2 J gt O In this case 0 O is only Viable choice or Z 2 k2 2dz Bz 2 N72 Rz39 gt i gt stable flowRz39 lt l gt not always unstable BU IBz 4 4 Howard s SemICIrCIe Theorem Howard developed a more general result to characterize the behavior of a complex phase speed in an inviscid parallel shear ow To wit A A 2A 2 2 F w kww gm uUcaizagara UF U c Bz Bz Bz 322 322 Bz Bz 322 Subsititue into T G eqn 2 U cU ca F2alai k2U cFN2F0 322 Bz dz 2 iKUCV 3i Ic2U c2F172F 0 dz Bz Multiply by F 9 integrate between 0 and d and manipulate to get 2 IU c2de JN2lFl2dz Q ii k2lFl2 gt0 Z Equate real and imaginary parts JU c2 c3de JN2lFl2dz ciJU cde 0 Howard s SemICIrcle Theorem Equate real and imaginary parts JKU cr2 02de JN2lFl2dz ciJU crde 0 If unstable 0 t O the integral must be zero which means that at some point U cr O gt Umin lt c lt Umax The real part of c is the phase speed of the distrubance so if U is the same sign everywhere the distrubance wave must propogage in that direction A similar analysis see Kundu reveals that the growth rate k0 is bound by k or awn Umio2 c s Umax Umm gt12 gt kci lt 5mm Um The first equation is Howard39s Semicircle Theorem as it bounds the complex wave velocity of the disturbance in the c plane to a semicircle Umax Umrn Instability of Parallel Flows Start with incompressible viscous equations with negligible gravity effects 311 Diil 1 3 311 0 77v 3x1 DZ 0 3x1 ijax Assume the basis state and perturbation are 171 Uyuvw i Pxp Bu 3v 3w 777 3x By 32 Bu Bu 3 Bu 1 a 2 7 U 7 7U 7 77 P vV U at Wax Vay u Waz pax P u 31U uiviwaiv ialW2v at 3x By 32 0 3y ELMUuall ELM142le iaipIV2w at 3x By 32 p 32 13P 32U 77v7 1LPNVZU x 0 3x ayz Background state O i p B Instability of Parallel Flows Perturbation equation 31U u alviU u wait ialVV2u Bl Bx By B2 0 Bx Linearize 8194mm 31U31V3 i3l Bx By Bz Bl Bx By 0 Bx alUi ialIV2v ELMU87W ialIV2w Bl Bx 0 By Bl Bx 0 B2 Non dimensionalize these equations by a characteristic length L and a W2u characteristic velocity U0 This leads to the following scalings u u U0 U39UU0 x x L z zU0 L p ppU P PpU Explicitlyplugging these into the X momentum equation U2 U2 U2 U2 U 70 3 70U 3 70v39 3U 70 317 VJV2u L Bl L Bx L By L Bx L2 a u U aiv 3U i 317 Bt Bx By 0 Bx U L inu39 Rei Re 1 Instability of Parallel Flows Use similar approach on remaining equations and dropping primes to get dimensionless perturbations equations alalalo alUalvaiU aliV2u Bx By Bz Bl Bx By Bx Re alUaiV aliv2v alUaiw aipiv2w Bt Bx By Re Bl Bx Bz Re Clearly inviscid ow corresponds to Re gt oo strongly viscous ow to Re gt 0 Assume wave like perturbation form u yeikxmzik gt A 2 A ik alimw 0 ikU caoLU ikfaiaJ K2a K2 k2 m2 By By Re ay2 3 1 3 2 1 3 2 ikU c3 iii K O ikU cW imf7ii K W By Re ay2 Re ay2 The 39 again are that 1 are real and positive wave speed is complex and that unstable solutions correspond to c gt O 20 Squire s Theorem Squire showed that for every unstable 3D distrubance there is a more unstable 2D distrubance Apply Squire39s Transformation IEKVk2m2E k mw Ecvo k1 e A A Bi m kA 2uwi 7w k k to the normal perturbation equations This yields ik iim 1 0 ilg aiv 0 ay ay 1B1 13 2 1a 1329 72 iU 7c0 7777 K 0 gt iU c7 77 77k a k By kRe ayi k By kRe ay2 Sum the X z mometum equations and apply ikU c WO ikmf7ii W K2 i By Re ay2 Squire s Theorem gt iU cl m k 2 a ay kkm 1 32 7 A 27 A 1f 77ku m kw K ku m kw kRe ay2 7 7 1 32 7 iU ck Oa ikp7u k2u By Re By A f7 m k 32w 2 zU cmkw zm7km77 7 K a k kRe ayi Second row is simply z momentum equation multiplied by m k k so the final transformed equations are 13213267 7 BO 7 27 ikui0 iU cv Ti k V 3y k By kRe ayi 7 AaU 7 1 32 7 27 iU ckuv7 jk 47 ku gt 3y p Reay2 1 21 SqUIre s Theorem So these equations have the exact form of the first set if m w O This in effect represents a two dimensional problem created by rotating the X z coordinate system of a three dimensional problem so that X is in the direction of the motion More critically since I W 2 k this implies that the effective Reynolds number for our 2D representation is E Re S Re k or the 2D version is associated with a lower Re than the 3D one So 2D disturbances should lead to instability before strongly 3D ones More directly the time variations of the 2D and 3D disturbances are L7 expl El vs u expkclbut since I gt k E c the 2D version grows faster than the 3D version OrrSommerfeld Equation Return to perturbation equation but based on Squire39s Theorem assume a 2D problem A 2 A ik i 0 ikU c Oa ikfaiaJ k2 By By Re ayz A 2A ikU co al iQ k By Re ayz B B Use streamfunction to satisfy continuity u l v 1 By Bx A A B A A A Given a wave perturbation 1 lle k 5 2 u in z39k1 By A 3A A gtikU c ik1p ikjaiM k2M By By Re ay3 By A 8 1 3219 kZU ii k cw By Rez By ik3y9 22 OrrSommerfeld Equation 2A A 2 A ikU ca Vikalal iky9L U ikalal ay2 ay By By2 ay By A 4A 2A ikalialk2 M 3y Re By4 By2 A 3i 1 23217 4A k3U kiik 7 k z cw 1 By Re ayz 1 2A 2 4A 2A ikU c a ZI kzzfl z39k1fL2iL sz l15 ay ay Re ay 8y lo 2 4A 2A U c a W kzzfl 473 U 1 L V ZkZL WWW ay2 ay2 ikRe ay ay2 This is the Orr Sommerfled equation a fourth order ODE This is effectively a vorticity equation since pressure has been eliminated On a wall where u v O the boundaries become 17 1 0 y y1 yyz y InVISCId Instability of Parallel Flows Inviscid Orr Sommerfeld Re gt oo 2 A 2 U c 37 2111 111372 O lt Rayleigh Equation ay ay Twowalls BC IIIOy y1y2 For the above eigenvalue problem if there exists a solution corresponding to an eigenfunction III and an eigenvalue 006 then their complex conjugates A gtllt 111 0 also form a solution Therefore any solution With a non zero 0 must have a growing and a decaying solution The only stable solutions are for real values of c 23 Rayleigh s Inflection Point Criterion 2 A 2 2 A A 2 U c aW kzy 1f37UOM k V a7U0 Byz Byz Byz U C ayz Consider the unstable mode cl gt O U c t O multiply the Rayleigh equation by 17 integrate from y1 to y2 and apply the wall BCs This will yield 2 1 BZU 2 k2 p2 dyJlii A dy 0 l H H ayz w The first term is real the imaginary part of the latter term is lle BZU 77d 0 cI Uc 2 ayz y To get a solution when cl t 0 then the integral must be zero which requires that aw y somewhere in the interval y1 lt y lt y2 the U yy term must zero and that U yy must change sign which corresponds to an in ection point However the existence of the in ection point does not guarantee instability Fjortoft s Theorem Fjortoft built on Rayleigh 70 years later by looking at the real part of the previous analysis A 2 Illll ltUcgt BZU dyj U clz By2 By If a point of in ection does existin a profi at the velocity Uy1 UI then 2 k21i2dylt0 W2 BZU lt U gt 0 6 I Ignaz ay2 dy lt0 0 WWCJBZU W BZU 2 7dyc U 7 dy l U clz ay2 1 Ila CV ayz A 2 UU 2 IW 2 iyigdylt0 gtUyyU UIltOforpartofthe oW U c Note that the vorticity of the base ow is 0 ayl a ay 24 Fjortoft s Theorem 3 2U So y Uyy 0 corresponds to a maximumminimum of vorticity Therefore Fj ortoft39s Theorem may be stated as follows For instability the absolute value of the vorticity of the primary ow must have a maximum in the domain of the ow if the velocity is monotonic Fjprtoft39s Theorem like Rayleigh39s ln ection Point Criterion is only a necessary condition for instability not a sufficient one Critical Layers lnviscidparallel ows satisfy Howard39s semicircle theorem Umm lt c lt Umax For neutral modes ci 0 then or U y yc somewherein the ow field if instabilities are possible At that point there is a discontinuity in the eigenfunctions that solve the Rayleigh equation but not the full Orr Sommerfeld equation Ucazlk2q A E 1 2k2 k4m ay2 ay2 sze ay ay2 In the vicinity of this critical layer the streamfunction observed moving With the phase speed is T JU Cdy AWltygteikx Taylor expand this about y yc 1 y yc 2 Av yc coskx l 7 2 By Y The sketch of these streamlines in the critical layer reveals What is known as Kelvin39s cats eye pattern anindication of potential instability 25 Brief reVIew of 08 solutions The Orr Sommerfeld equations are challenging to solve key breakthroughs were the analytical work of Tollmien 1929 and th numerical work of Kurtz and Crandall 1962 A quick summary of this history may be foundin Schlichting and Gersten A brief review of results is given here Free shear ow Profiles with in ection points characterized by low critical Re39s generally unstable to long wave disturbances instability has form similar to K H with growth of voriticity 39blobs39 Plane Couette ow Linear profile no in ection 39points39 linear analysis of viscous ow is unconditionally stable experiments eventually go unstable probably due to finite disturbances Pipe ow lnviscidly stable no in ection points linear viscous analysis also unconditionally stable reality is that typically instability occurs for Re 3000 but careful experiments have pushed this up to Re 50000 Brief review of 08 solutions Boundary Layers lnviscidly stable no in ection points For zero pressure gradients linear viscous analysis indicates instability for Re5 gt 520 with transitional range lasting until around Re5 950 Instability region corresponds to lower k39s as Reincreases eventually shrinking to a zero range of k as Re gt oo For favorable pressure gradients behavior is similar For adverse pressure gradients instability occurs at lower Re covers a large range of wavelengths at higher Re Note that since the ow is stable in inviscid analysis unstable in viscous one viscosity appears to make this ow more unstable Plane Poiseuille ow No point of in ection inviscidly stable Linear viscous analysis indicates instability if Re gt 5780 Non linear analysis indicates instability for Re gt 2510 The latter agrees better with experiments Viscosity destabilizes this ow the causative waves are TS waves 26 TS Waves and Instability TS waves or Tollmien Schlichting waves or Tollmien Schlichting Schubauer Skramstad waves are two dimensional waves that developin parallel or near parallel shear ows due to the destabilizing effects of viscosity To see how viscosity can destabilize a ow Kundu amp Cohen present the following argument starting with the non linear perturbation N S equation aain au Uj au Wj BU ialv 3 I ij ij ij pri ijaxj Multiply by ui to get the mechanical energy equation and integrate over a region corresponding to no slip walls and an integral number of wavelengths Then the terms of the above equations become au d u Bu 1 a 2 1 2 IuiEdV EIYdV JuiqujdV Ejaxijmiu dV Ejmiuj lj O Bu 1 B 2 l 2 Juin dVEjaxiuindVEJuindAj 0 1 TS Waves and Instability J39u aldV Maw JpudA 0 Bxi Bx Bzui B Bu Bu Bu Bu Bu u dV 7 u V I I V I I V I Jij ij Jij ij Jij Bx 2 1 idyjuuyvj dVjuudV dl 2 Jij Bx Bx 18x Reduce to a 2D disturbance for a shear ow ijlmz v2dV J39qudV dl 2 By The first termis the KB of the disturbance The viscous dissipation I will reduce the KB of the disturbance stabilizing The second term is related to the Reynolds stress of the ow which for inviscid ow without in ection points is zero u and v are 900 out of phase so the average over a wavelength is zero But viscosity causes the phase relationship to shift from orthogonality so this term becomes non zero and can evenutally cause the KB of the disturbance to increase 27 TS Waves and Instability A more qualitative argument from Sherman Lighthill assumes the formation of a train of oppositely oriented vortices along the critical layer where U c To maintain the no slip condition at the wall a counter vortex must be created at the wall as each vortex in the train passes These counter vortices diffuse away from the wall and are advected by the mean ow but still have significant vorticity as they reach the train As they reach the critical layer the counter vortices may constructively or destructiver interfere with the critical layer train Constructiveinterefence can lead to growing disturbances Vorticity eqn 887 u Vm m Vu VVzm TS Waves and Instability A standard picture of transition is that the TS waves grow eventually leading to three dimensional spikessecondary instabilities further non linearity and then turbulence Some transitional ows appear to skip the TS waves in the quotTS pathquot example Plane Poiseuille ow and have been dubbed39bypass transition39 However recent work Reshotko 2001is the piece suggested by K amp C has shown that bypass transition may be connects to a TS wave phenomena related to the non orthogonal natureof theOSand Ll39 39 leadingt 1 39t39 1 1 a transient growth in both space and time which may lead to a TS path in these cases 28 Summary of Instability Type Stab Destab Unstable Form of Remarks Force Force region instability Benard Visc Buoy Ra gt 1708 Cells rolls Critical Ra depends Ra gt 657 on EC Double Visc Buoy RsRa gt 657 Fingers Needs two sources Diffusive convecting of density gradient layers Taylor Visc Cent Ta gt 1708 Toroidal If lengf 1 11919 cells Kelvin Strat Shear Always Waves to Unstable to short Helmholtz Unstable vortex sheet waves large k Continuous Strat Shear Ri lt 025 Like KH Necessary but not sufficient for instab Inviscid Adve Shear In ection Catseye to Necessary but not Shear ction point KH like sufficient for instab Type Stab Destab Unstable Form of Remarks Force Force region instability Viscous Visc Shear Re gt small Like KH Can be uncond Free Shear IF unstable Blasius BL Visc Shear Re8gt520 Transitional Pressuregrad Visc TSwaves dependent Plane Visc Shear Re gt 5780 Transitional Bypass transient TS Poiseuille Visc TSwaves waves Pipe Flow Visc Inlet Always Transitional Exp Re gt 3000 up effects stable linear to 50000 Couette Visc Shear Alwa s Transitional Exp unstable for stable linear large Re finite disturbance effect 29 Vorticity and Circulation Bw Bv Bu Bw Bv Bu nszu 1 J k By B2 B2 Bx Bx By 1 ijk sequential mi k 3 Si a J Bijk 1 anti sequential Xk xj xk 0 other 2 2 2 2D coza Va 3 2 3239vzw ax By 3X By Laplace39s 2 2 Equation if irrotational co 0 9 le 8 H V2 S rokeS39s Bx B 2 Theorem Circulation Fzsgudlz LVgtltundAJAcondA Vortex Lines and Vortex Tubes Li Li i gt vortex line analogous to streamline x y tangential everywhere to the vortex vector A group of vortex lines can form vortex tube analogous to streamtube dFmdAlt gtdQudA Strength of tube is the circulation F Helmholtz Theorems for inviscid conservative barotropic non rotating ow 1 Vortex lines move with the uid 2 Strength of the vortex tube is constant along length V V X u V1 0 lt vorticity vector divergence free no source or sink of vorticity in the ow compare to continuity Vm02Jl VdeJ mndS0 Gauss sTheorem V s Vortex Lines and Vortex Tubes OveravortextubeJ mndSJ mndS0 on00nsides S1 S21 JmndS F1 I na73391 2gt1 21 1 qA1w2A2 S1 S2 3 Vortex tube cannot end in uid Must either be a loop or end on a surface No sources or sinks implies that ow is divergence free So vortex lines must form closed loops or hit a surface 4 Strength of vortex tube remains constant in time 1 and 4 rely on Kelvin s Theorem 39 7 denslhemem For inviscid ow with conservative body forces Du ap 9G ag Dt 3x ij 3x Lagragian change in F Du D dx E2uw 4hMrLA Br Br Br Br Dde Dx 3x 3xj d di 7d0 5 d Dr Dr 3 uk Bxk Mk 1quot 1 Dr Du 1 9 BG 5 K Dt dxjujduj Ejdxjdejujduj d 1 d FpdGEdujujj E fj if barotropic p fp dp f pdp E 3 0117 0 Dr p Irrotational Vortex 1 9 r19 9 r 3r r 90 C6 lI Cnr Fz C6 z39C1nr z39C1nr 8quotquot z39C 1112 Vu 0 27 1 1 r jurrdg 2711C 2 C Fz i lnz 0 27 27 F If center at 20 Fz z lnz 20 27 a r 0 except at center where oz gt 00 z r 3r r 90 to satisfyStokes39s Theorem uds J VXudA3FJmdA C A A Irrotational Vortex WEquotH e w gzr a Bui Bull azui b F 97 V t 7 7 X u w ij ij ij Bxi BxJij 14 01 so the net Viscous force on a uid element is zero for the irrotational vortex Applying the inviscid Euler equations 2 2 u 9 3p 317 pF y 0 7 2 d 2 dr dz r 3 32 pg p 47 2 r 3 pg I72 P1 gu 2 u 1 Pgz2 21 2 2 u u igz1 g22 Bernoulll 2 p 2 2 2 2 F 1 1 5 22 21L91L92 7 7 isobars 23 23 S zg r12 r22 SolidBody Rotation l u0 u9w0r5wr 2 Vu0 lau 3 Me 7 7 7 0 Try r 39 rBrE r1 Applying the inviscid Euler equations 2 quota 3p 3 P 0 Pg r Br 32 2 dpgjdrajdzlw2rdr pgdz r 32 4 1 172 171 Pw2 22 V12 l7322 21 l 2 1 2 5 p2 Epu92 pg22 171 EMgl pg21 Nof Bernoulll 1 02 gt 22 zl r22 r12lsobars Vector in a Rotating Frame Assume any vector P H il in a frame of reference R gt x1 x1 x2 x3 rotating at an angular velocity Q 91 jl relativeto a flxedframe F gtXl X1X2X3 Relative to the xed frame the rotating unit vectors change direction dr d dr dil sz dil 7PiiziPzi 7 137 dz F dz dz dz dz R dz To compute the change of the unit vector consider a vector of constant magnitude in a rotating frame that is rotated through an angle A9 in a time At Pz Az Pz AP n P sin 0mg 0A92 Where n AP iAP is the unit vector in the direction of the change in P Q XP which must be perpindicular to both P and Q so n 7 iQxPi Vector in a Rotating Frame Now divide by Ar and take the limit as it goestoO AP dP d9 QXP 1m Psma AHo At dt dt leP and Qgtlt H lQHPlsina 3 g exp Replace P with unit vector i ldil sinad93 sina sina dt 3 2gtlti dt 3P PQxiQxP QgtltP dt dt R dt R ReplacePwithr Qgtltr3uF uRQgtltr dt F dt R Vector in a Rotating Frame d d ReplacePwithuF HF HF QgtltuF 3 dt F dt R du F dt uR erR QgtltuR er du R 9x j QgtltuRQgtltQgtltr dt R dt R d9 aR aR 2 2gtltuRQgtltQgtltr 0 dt 9 Q rRr R rcosa RQr 9 ll l l l lz QT QxQgtltrQgtltQgtltRQgtltQgtltQ QgtltQgtltR l lz agtltbgtltc acb abc QXQXR 22R aR aR 2 2gtltuR 22R Vector in a Rotating Frame D 1 D So u VpVV2uFg u ZQXuR QZR Dr F 0 Dr R Solve for acceleration in rotating frame 3 21R 2 inVV2uF gQzR ZQgtltuR p l VpVV2uR geff fgtltuR 1n rotatlngframe 0 Note VZuF VZuR V2Qgtltr VZuR QXV S VZuR Coriolis force gt ZQgtlt u R Centripetal Acceleration gt 22 R Replace g eff with g drop the R subscripts and apply Boussinesq approximation to get 22gtltu LVp ampk1V2u DI P0 P0 Vorticity Equation Bu 1 2 NSconstantpu Vu0 EuVu EVPVV u 22 1Vu u ugtltVgtltu V szu p VGaa a39Va ax2vXa VxltV gt0 VX a uVuu ugtltVgtltu V V2u at P 30 2 E quxvaV n anxbz aVbbVa aVbbVa 21 uVcowVuV2w VVXa0 Vorticity Equation in Rotating Frame Assume Boussinesq Equations so V 11 0 VectorIdentity Van 0 3 V1 Vqu 0 So votticity is non divergent even if compressible andor unsteady Conservatlon laws a VF V62 21 Du 1 2 7 a X V X a Vu 0 20Xu VpVV uge Dt p Ea uuVua UVluu UXVXUiVluu uxm Dr 3 at 2 at wax VVa a V2uVVu VXVXu me ogtltu uxo Vza 3 3 Vuu uXm 2ux Vp V eff VVXO 3 3 Vuu e uxm20 Vp VVXO Vorticity Equation in Rotating Frame Takecurl Vx a uVluu uXm29 iV vim at 2 eff p p aVb7bVa 1 V V 0 VXVEuu e 04 M VXaXb 7aVbbVa Vxuxm29 u Vm2om2oVu V gtlt u Va 0 20 Vu WW VX Vx in iVpr V i priVPXVP WV p p p p2 VgtltVgtlta ivza VxmeVVm V2m V2m 2 1 UVm m20Vu i2VprpvV2m p 3 D m20Vu LVpXVpVVZOJ Dt p2 Vorticity Equation amp Physical Meaning Vortlclty equatlon m rotatmg frame Viscous Diffusion D l m 0 29Vu VpgtltVp VV20 DI p2 Vortex sfrefching Bar oclinic Generation Vortex Stretching Use a coordinate system where s n m correspond to along a vortex line in the direction of the radius of curvature i and in the third normal perpindicular to the other two Then m i 1v w a Vu a i EH ii a u wiu J 3s quot 311 m 3m 3s smcem im in 0 a 15 w Stretching effect Bu 3uAv 3un 3um ngzw as 1Jw as 1nw S m Tilting effect mvuw3us Duquot 0 uquot 1 D0 w3um Dr Dr as Dr as Dr as Vorticity Eq uation amp Physical Meaning Rotating System Vortex Stretching Set 2 axis in direction of Q 9 9k 262V 2 2931 Stretching effect 1 32 Isolating this term ng 63ft Du E29312Dw22 37W Dwx2 Bl y29i Dr 32 Dr 32 Dr 32 Dr 32 So stretching is in the direction of 9 not in direction of vortex lines If a column of uid is stretched vertically it gains vorticity vice versa if it is shrunk Kelvin s Theorem in rotating system Dr 1 02ra i m2odAr2iodA t A A Geophysical Fluid Dynamics Example of rotating ow system Three key effects 7 Rotation 7 Strati cation 7 Thinness T c 10 20 Depth 2 km 3 T b l i an m Eu 2m 23 2m 2m 2m 1020 1023 1026 0 001 H p kgm 3 N rads Geophysical Fluid Dynamics Boussinesq Approximation For a rotating system with Boussinesq approximation Vu 0 gnaw in EkF 0 Dt Po p0 Dt Key assumption for geophysical ows is that the vertical scale of the problem does not result in signi cant change in density This scale is of order 02 g N 200 km for ocean N 10 km for atmosphere The typical approach is to look a perturbation about a local point with z 72 which satis es the hydrostatic law Look at perturbation form of momentum equation p z p39x r p 172 pix r EZQXU iV7p39 7ppg kF Dt P0 P0 ikvp Ek EkF in kF Po dz P0 0 1 0 1 0 Geophysical Flow For the moment we will assume no iction effects inViscid F 0 U W W H Scalin of continui su ests N N 7 2 7 N 7 ltltl icall quotthinquot g ty g L H U L typ y Although the natural coordinate system for planetary scale geophysical motions is oblate spherical coordinates if the horizontal scale of the motion is small relative to the radius of the Earth can adopt a local cartesian plane in which the xy plane is tangent to the surface 2 is normal to the surface Positive directions are typically assumed to be northward and eastward The rotation of the 1quot Plane Corio planet translated into this system becomes Comol Q 0i 2cos 9j 2sin9k 2 2Qgtltu 22iwcos Q vsin 9ju s39 Q ku cos z zifuj 2 2u cos6k where f 29 sin 9 w ltlt v N u 2 39 f 2W 145gtlt10 Geostrophic Flow Also note that the vertical coriolis component is generally small compared to other terms leaving the following equations of motion ampfVLBL ampfai 1 917 W Dr p0 Bx Dt p0 By Dt po 2 p0 Depending on the horiztonal scale one can use the full de nition of d 29 9 f 29 sin 9 the plane approximation of f f0 y or the f plane model of f f0 constant A common approximation is to assume steady ow and linearize the horizontal momentum equations yielding 1 B 1 B fv ii fan77p P0 ax P0 3y 2 This linearizationisvalidif N U L Geostrophic Flow For geostrophic ow horizontal ow is along the isobars as opposed to across the isobars uVp 1 aipiaipjaipiaipj0 gt uin Pof ay ex ex 9y This creates rotating ows about high pressure and low pressure regions as shown This is called geostrophic balance as opposed to cyclostrophic balance where centripetal acceleration balances the pressure gradient see Geostrophic Balance Cyclostrophic Balance SolidBody Rotation l u0 u9w0r5wr 2 Vu0 l3u 3 Me 7 7 7 0 Try r 39 r3r r1 Applying the inviscid Euler equations 2 quota 3p 3 P 0 Pg r 3r 32 2 dpgjdrajd2lw2rdr pgd2 r 32 4 1 172 171 Pw2 22 V12 Pg22 21 l 2 1 2 5 p2 Epu92 pg22 171 EMgl pg21 Not Bernoulli 1 02 gt 22 21 r22 r12lsobar s Thermal Wind If the ow is strongly strati ed vertical motions will be suppressed leading to hydrostatic balance With geostropic balance the governing equations become 1 3p 1 3p 3p 2 7i 2 77 0 7 190 9x 190 By 32 gp 2 g i yiip g amp3Piip 32 0 3x 32 0 f 3x32 0 f 3x gt al 1 32p g 87p 32 I70 3 32 Pof ayaz Pof Egty If assume an ideal gas along an isobar the horizontal density gradient is proportional to the temperature gradient 7 3V g 3T 3u g 3T I 3 W So vertical wind shear is proportional to the horizontal temperature gradient TaylorProudman Theorem The thermal wind requires a horizontal density gradient and is generally a baroclinic phenomena The simplest barotropic version is for a homogeneous uniform density uid In that case i 3 LP 2 0 iquot 3 LP 32 pof ax 32 pof 3 Further can show 9 Lil 3 Lil iiquot 91 gi I70 axiaxifu I70 Byi fEBy Faxj O f Therefore 37quot 0 lt Taylor Proudman theorem Z 0 And since w 0 everywhere on surface this implies w z 0 everywhere Strictly speaking this also implies that u v 0 everywhere as well based on no slip boundary but assumptions break down in boundary layers ShallowWater Equations Assume a thin layer such that g N ltltl lt consistent with Taylor Proudman Therefore u ux y t v Vx y t M 0 N small 32 32 Assume that the thickness of the layer is hx y H 17x y t 17 ltlt H dp H Hydrostatic then 1mp11es 7 pg 2 J dp pgd2 Z 2 5 17H77 I7Z PgH77Z De ne pH 17 0 2 72 pgH 17 2 Take X and y derivatives Bl pg al 3717 pg Bl 3x 3x 3y 3y Integrate the incompressible continuity equation noting that u v f 2 H H J quotVu0dzJ quot31iaiw0d2 0 0 3x 3y 32 3u 3v H gH wH WV 0 ShallowWater Equations Kinematic BC wH 17 Dt HnalHnialualval 0 Bx By Bt Bx By Rearrange and linearize 17 u v smallto get Bl H 31 i 0 Br Bx By In a similar manner linearize the momentum equations Du 1 Bp Bu B17 thv pr Bt ng DV 1 B17 Bv 7 i 7 i 7 Dr fquot pr Bt fquot ga So the linearized shallow water equations are 117 91 0 mfbgin L pgaj Bt Bx By Bt Bx Bt By PV and ShallowWater Return to the non linear equations for SW de ne h H 17 Bu Bv B17 B17 B17 H i H 77 7 70 max may Bt quotax VBy amHhaluanHhivamH0 Bt Bx Bx By By BlBhuBhV h Biui 0 Bt Bx By Dt Bx By Bu Bu Bu Bh BV BV Bv Bh Squot11777 7777 7 wary Bt quotax VBy fV gax Bt quotax VBy fquot gBy Approximate f9 29 sin 9 z 29 sin 90 29 cos 90 9 90 2 f0 y 29cos9 y9 90R T D Bh Dv Bh Fifo yvg Efo yug PV and ShallowWater Sum cross derivatives of the momentum equations B Du Bh B Dv Bh E fo yV g fo yu g 2 ualv fv viampi uivifu BthByBxByO BthBxBxByO 32h 32h gByBx ngBy 3iv zitiuivBVJ EEuvf0ij v0 x BthByBxBngyBxBy By Bv Bu BEEN Bu De ne i 7 i ii Bx By Bt Bx By Bt B Bv BV B Bu Bu i uivi i uivi Bx Bx By By Bx By PV and ShallowWater 3 av 3v 3 Bu Bu g EJL W3 gag 81 M 81 gag BxBxBy ByBxBy BxBxBy ByBxBy u v all Bx By Bx By 2 uvujfogj v0 D4quot Bu Bv 7 i i 0 2 DIfoaxay3V Massconservation Ilih aiuaiv 02 lD7h ali Dt Bx By th Bx By Conservation of Potential Vorticity Diawghwgo DI h DI Use uvv v f0gtgt y DffoDihmfDih DI h DI h DI 1Df fDh Df D 1 D f h D Wald lil o l 7 H E Z Dr Dr h This is the conservation of potential vorticity It is a fundmental condition of geophysical uids essentially consistent with a vortex tube concept except the vorticity is now the absolute vorticity Note that the addition of f gives a bias to the solution since on a planet it increases with polarward motion decreases with equatorward motion This can be seen in hurricane motion which tends to be westward and northward in the northern hemisphere and the jet stream whose wind is eastward countering the phase speed of the westward Rossby waves Frequency Regimes Start with linearized SW equations 317 Bu 3V Bu 31 av 31 7H 77 07 7 7 7 at Bx By a fV gax at fquot gay e au an M av e au av i i 7 7 i Hi i i aria fV gaxl 5 at2 far g BxEBx 1j 3 av an 9 au e au av i i 7 7 7 Hi 7 i atlafr gay 5 at th g ByEBx i By Eliminaten 3 37 By Bx By g 37 g 9y 3 2 2 Q ngHg 91 ngL 91 3 at 3x 3x By 9th 3x By 2 3 2 2 3 VfaugHiaiu 2 a Vfa ugH a 3mg 1 Frequency Regimes Now construct a vorticity equation 2th iv 5 a By Bx Blay 3y Bxay 3 av 377 32v Bu 3277 axiafrfu gdy 2 Blaxf3x gaxay azu 32v 3v Bu 3f 3 Bu 3v 3v Bu 7 7 2 2 2 2 2 2 0 azay azax af x Vay Bl 3y Bx af x v Combining the two equations a a Bu 3v 3v Bu Hiii i 7i 0 5 g axiaziay Bx f0 3y 3x v i 33v 3v 3 Bu 3v 32 Bu 3v 7 2 H2 22 H 72 33 fOisz g Bx 3x Byji g ByBl Bx By B3 3 v 3v 3v 2 7 H2V2 27 H 70 33 g a HV f az g ax Frequency Regimes As with the w equation assume wavelike solution forrn v Ge 2 m3 iagHk2 lz if02a igH k o 2 w3 wc2K2 f02w c2 k0 zkxlyiat This equation has three real roots two where a gt f one where a ltlt f High frequency waves a gtgt f 2 w3 wcz K 2 s 0 2 a iK gH Moderate frequency a N f 2 w3 wcz K 2 f02 ms 0 2 a i igHKz f02 Low frequency wltlt f 2 wc2K2f02wc2 c s 0 gH k Note that high frequency waves are unaffected by coriolis 2a low frequency waves affected by Gravity Waves Start with shallow water form i I gi ifu Bl alHEBJijO at 3y at 3x By and assume standard waveform u v 17 a v WWW z39wt2 fx3 z39kg z39wamp t z39lg iw iHk72l3 0 Eliminate the velocities and cancel the de ection to get dispersion relation w2 f2 gHk2 12f2 nglt2 2 l H 03 f7gHnondispersiveiffz ltlt gHKzor iltlt g K K2 K f c gHwsK gH gtgtf Oceanf10 4s 1H1000mg10ms2 21ZKlltlt27 JgH 106m So surface gravity waves in the ocean are typically in the high frequency regime Gravity Waves Although originally found by Kelvin these waves are called Poincaire39 or Sverdrup waves The dispersion relation is isotropic in terms of k 1 so simpler to orient wavenumber vector along the X axis 1 0 Assume 17 1 coskx wt 1 Re iwim Corresponding velocities u coskx wt v sinkx wt kH kH So uid parcels move in horizontal ellipses with an aXis ratio of wf When w gtgt f v gt 0 and motion is like the non rotating shallow gravity wave As w gt f K gt 0 and 1 gt 0 In this cae the equations of motion become al z0 i fuo aliqwithai7 at at Bx By a The simplest solution form is u q cosft v q sinft This corresponds to 0 gtw0 inertial motion in circular orbits w f with an inertial radius r q f and an inertial period of 2711f Kelvin Waves If these waves are bounded by a surface channel coast such that v 0 alHal0 al gal fingain at 3x at 3x By use u 17 eikquot z w ikg g iw in72 0 wz ng20gt a ng2cgH H Rossby Radius NeXt assume 1 nOeTyC noe yA A i i f Deformaflon f f gt n noe fy coskx ct u 170 56 coskx ct These waves are typically called Kelvin waves The development of a tranverse velocity v is impossible due to the shore The wave can only propogate such that the boundary is to the right in the northern hemisphere so that e Ty M decays away from the wall Internal Waves Now use the linear non shallow equations for strati ed uid Boussinesq 2 alalal0 ai poN w0 3x By 32 at g Bu 1 3p 3v 1 3p 3w 1 317 pg 7 vii imp 77 vii 7 at p0 Bx at p0 By at po 2 p0 Can get w equation using same procedure wnon rotating IW 2 2 a V2wN2V wf2 a w0whereNNz Btz 322 Again with wave like solution M V w 3 vizeikxly 2 A 2 2 2 2 2 A N k l A A dw OX wzo dwm2zw0 dzz 02 f2 dzz m2 lt0 1 39 39decay 39 quot l l 39 surfacewave m 2 gt 0 2 internal waves with frequency range f lt a lt N Internal Waves Further solution of this problem is traditionally done with a Wentzel Kramers Brillouin WKB approach in which both N and m vary only slowly in z The assumed solution form then becomes w Azel z Substitute this into the w equation and equate real and imaginary parts 2 2 2 diAA m2 0 z AM0 dz 2 dz dz dz dz 2 For slowly varying value can treat rst term as small get solution z z if m dz A A0 N w A0 My mel y which with more work and setting I 0 leads to A Z A 2 u 012 cos kxijmdz wt v sin kxijmdz wtj w icos kxirm dz wt M Internal Waves The resultis elliptical motion but in a tilted plane at 9 tan391 m k k2N2z w2 wz f2 2 2w2 f2LN2 w2 1ettan9 m2 k The dispersion relation is m 2 z 2 12 f2 sin2 9N2 cos2 9 This leads to three wave regimes 1High frequency a N N a gtgt f 2 a Ncos 9 non rotating solution 2 Low frequency ow f wltlt N 2 02 sf2N2 cot 9 hydrostatic solution 3 Midfrequency f ltlt 1 ltlt N 2 02 s N2 cot 9 both hydrostatic and non rotating 20 Lee Waves A particular type of interal waves occurs in the quot leequot of a mountain in which the meanm ow and phase speed exactly counteract each other to create a standing wave To see this assume that a gtgt f Then w2N2cos29 i k 2 m 2 In a moving ow with velocity U the effective frequency is 10 a K U For a standing wave 10 0 2 a K U kU N U is de ned by the wind k by the moutain shape so the above determines 2U the downward propogation of the phase QuaSIGeostrophy Quasi geostrophy QG expands the geostrophic balance to include the beta effect Specifically we will look at the effect on the PV equation D 4 f D Dh 7702h7 70 Di h DIG f 4quot fDZ Assume that the height is the combination of an undisturbed and disturbed layer hH77 3f3y 34 34 34 377 377 3 2 H iuivi iuivi 0 mi 3 3x 3y v g f a 3x 3y Assume small perturbations linearize by dropping all HOT 3 4 317 2 H 7 H 7 0 a v f0 3 Assume approximately geostrophic balance for velocity 2 2 53777 V3534 g 429231 Lam 3x 3y f0 3x2 ayz 21 Rossby Waves N gen Vial 2 alji N iiy N f0 ay f0 ax ax 3 f0 3x2 Byz 2 2 3 Lam gH imfoin0 f0 at 3x2 ayz f0 ax at 2 2 2 2 2 2 3 L LTLLoU 972023 L LTLLo 910 at 3x2 ayz gH ax at 3x2 ay2 02 8x Note that we are violating strict geostrophic balance by leaving in the 317 at term This approximation is consistent with the low frequency regime corresponding to waves with only a few oscillations around a planet ie the jet stream To get waves assume 17 eiwlly Solving for the dispersion relation in the usual way gives k w k212f02c2 Rossby Waves The phase speed of Rossby waves is ox i phase progation is westward k k 1 f0 c To generate a standing wave superimpose on an eastward ow c U k212f02c2 The simplest exampleis if I 0 ow is barotropic so f02 02 ltlt k2 cC 02Uk 2 22 Friction in Geophysical Flows For a rotating system with Boussinesq approximation Dr The friction term is not typically laminar or isotropic in geophysical ows In particular the vertical viscosity is typically of a different scale than the horiztonal viscosity In strati ed conditions VH N 1000 VV In areas with strong Vu0 g2 2gtltu in EIF 0 D2 0 I70 vertical convection VV can become much larger So as a rst approximation a1 2 2 2 F297 xyar Hau Bu au v 77 7 x ax 3 32 3x2 Byz V 322 3139 8139 3139 2 2 2 Fy y 7W y H vVaV 3x By 32 3x2 Byz 322 2 3x By 32 a a1 a 2 2 2 F 7 4 rvHa w3w aw Ekman Layer Look at near surface ow assuming the velocity is horizontally homogeneous 2 2 2vd7 fuvL dz dz 2 Can solve withacompleX velocity V uiv 2 d E ilV v The boundary conditions vary depending on the problem Two classic cases are the ocean and the ow over a solid surface Ocean pv r 0z0 uv gt0z gtoo dz dz r 25 27 r 25zn2v ll 79 cos 77 V 7e s1n 775 7 PM 5 4 PM 5 f NoslipsurfaceuU v0z gtoouv0z0 Pofa l u U1 e z5 cosz5 v Ue z5 sinz5 23 Ekman Layer Differences from the traditional boundary layer l 5 27Vnot5fx 2 v 02 ledz U5e cross ow transport Geostrophic Turbulence Large scale geophysical turbulence is different because 1 Vertical motions are suppressed 2D ow 2 Vortex stretching to mixing energy cascade is absent Result is known as quot2Dquot turbulence eddies cascade upscale rather than down in direct constrast to the turbulence doggerel quot Big whirls have little whirls that feed on their velocity and little whirls have littler whirls and so on to viscosityquot This behavior is actually re ectedin the performance of Rossby wave triads which we will not discuss in depth here Key feature is enstrophy or the mean of the square of the vorticity g 2 If we represent an energy spectrum as S K where K is the wavenumber then the following relations hold i7 JSKd1lt P JOMKZSKdK with SK having units of m3s2 24 Geostrophlc Turbulence In an effectively unbounded effectively inviscid system just as large scale 39 A both of t quotquot are conserved J 391 irisme o iJMK ZSKdK 0 dz 0 dz 0 Given this imagine an initial disturbance that has all of its energy in a tightly bound gaussian about the wavenumber K0 As time passes we expect that non linear effects not viscosity but Reynold Stress like terms would cause the energy to spread away from this initial configuration To make the illustration simple assume that after some time all the energy has effectively spread to two different wavenumbers K and Kr where K lt K0 lt K By conservation of energy amp enstrophy this requires SD 5 5 K0250 K35 Kis Doing some algebra one can determine that s K K0 K K0 K35 fo K S KO K1 K0K1 K35 2 K02Kf Geostrophic Turbulence These relations imply that as one ratio increases the other decreases Again to further simplify assume that K 2K0 and K K0 2 Then the above ratios become 4 and 14 respectively The result is that 80 of the energy cascades upscale and 80 of the enstrophy cascades downscale K Kg 19 1nK This upscale energy motion eventually breaks down at about the Rhines length LR N u where due to the beta effect and Rossby wave propogation the eddies become stretched into waves or jets Further for 2D turbulence to occur the Rhines length must be considerably longer than the Rossby radius of deformation LD N NH f N Jg39iH f which is not typically the case on Earth However this is true on the jovian planets leading to both the zonal jets and vortex features 25 Instability Consider instabilities from two Viewpoints Normal modes and growing instabilities Assume that there are small disturbances in the ow of the ype X70 yeikxmz6t7 where 039 can be complex k m 2 real If these small disturbances grow exponentially the ow is unstable If the disturbances are decreasing the ow is stable Mathematically 66ric5i unstable 6r gt 0 stable or lt 0 neutrally stable 6r 2 0 For the neutrally stable or marginal state the imaginary term is important If there is no imaginary part 5 0 there is a stationary pattern of motion cellular convection secondary ow If there is a non zero imaginary part the instability is oscillatory with growing amplitude oscillations Vortex Instabl Ity View Vorticity Assume the ow has small vortices The arrangement of vortices can either suppress or enhance cross ow motions In the later case this encourages instability F u 0 u9T 727 Till lTTl Fz iL1nz zo 2 Stable Unstable Tl 11H B nard Problem Start with the Boussinesq equations including thermal energy conservation a D quot quot I 07 1 iaPLgiV2L7i axi Dr p0 axi p0 3 pO 1 altT 40gt g ma g E xvzf Dt 10A1 z d fT07Fz I xg 2 T0 B nard Problem Let L7 Ouixit f ZT39xit p7 Pzpxit O Dui iiP1111 ocfT TOgivV2ui th Dr 0 3x aiuiifT39KV2fT39 at 3x Basic or mean state 2 o i 3P 1 ocf TOg 0181 P0 Bxi dz 2 Kd OTclzcz dz BC z d2 T0fz d2TO AT TTO rzd2 B nard Problem Subtract basic state from full equations to get perturbation equations Dt 3x p0 3x 1 ocfT TOg 1 0L T0g VV2ui ia p T gi VV2ui Po axi aimiifm KV2TT KV2T 31 3x aiwa T uiaiai wfuiaiKV2T39 3t 3 3xt 31 3x Linearize perturbation equations E ia p OLT39gi VV2ui ai wl KVZT 3 p0 3x a B nard Problem Scaling arguments LT39Fdw KT wr KV2T39V2T39 d2 Fd2 d Therefore the ratio of the final two terms buoyancy Viscous in the momentum equation compare as XTg xT g de 4 g VVQuI VW d2 VK Next create w equation39 from mometum Ra lt Rayleigh Number Laplacian of z direction 3V2w LV2 3717 gotV2T39 vV4w 3t 30 32 3p i a 3 3M 1 3 Divergence of vectorequation 7 77 7 gia 3t E 30 3x 3x gtO 7V2pgaaigtO LV237P O 32 DO 32 322 P0 3T39 3x 2 z a T 3iv2aip x 32 Benard Problem Combine sz BZT aVZT39 V4 aVZT39 V4 at w g0 322 g V w g H V w BC 2i wT39O a For rigid n0 slip plateboundaries u v O and u 0 gt 31 0 3xi 32 i i d2 K Nondimensionalize tainxi gtxidw gtgwT39 gtFdT39 K ai l wKV2T39 gt i VZ T39w at at 4 3V2wgavgr39vv4w gt ii VZ V2w V 1F at Pr at W w l 02ii gtwaiwT O2il 32 2 32 2 Benard Problem Nonnal mode analysis w zeikxlygtm T39 f2eikxlygta 3w A T OWe U WyH ow ViIW k2 12w K2w I V2wL2 K2w D2 K2w dz2 Updating the equations g szrew na Dz K2fWW 4 igvzjvzwz vgr Pr 3t K39V gtP D2 4202 K2W RaK2f 1 3w 1 A 1 W7TOZi7 gtWDWTOZi7 32 2 2 B nard Problem Kundu proves 039 is real 0391 O we will take that as a given This implies that the marginal state 039r O which means the onset of instability is transition to a different steady state cellular convection This is described as the principle of exchange of stabilities So examine this marginal state more closely 00 D2 K2f W D2 K22W RaKzf W D2K23WRaK2W DZ K2 Eliminate f f BC WDWD2 K22WOzi Sixth order homogeneous DE with six homogeneous BC eigenvalueproblem 2 Can solveby assuming form W i v Woeqz diWoeqz queqz qW K 2 D2 K23W RaK2W gtq2 K23 RaK2 Benard Problem D2 K23W RaK2W gtltq2 K23 RaK2 Let RaK2 RaK2ez2n71m 5 W2 K2RaK213ej2n71nl3 Janina3 2 471 71i 5 2 2 2 2 13 13 gt 12 RaK2 K2 K2R A K 13 q2 K21111Zj 1ii3 if 2 q iiq0iq1 iqik a ib a ib 12 12 13 13 qoKij 1 q1K1l jj min5 K 2 K 7 7 7 2 W alelqoz 6128 39402 a3eq z a4e 4 2 aseq z a e 4 z B nard Problem Mathematical recombination aeie be ie a cos9 al sine iar sine 51 cos 9 17 cos 9 7 sin 9 i br sin 9 17 cos 9 a 12 z39a ibcos9 al b iar z39brsin9 Acosh9Bsinh9 Aee Ae e3e Be e ABee A Be aee be 9 Using the above can rewrite W W Ae cosq02 Be coshq12 Ce coshqikz A0 sinq02 Bo sinhq12 Co sinhq1z Typically look at even vertical motion symmetric about centerline gravest lowest mode one cell and odd vertical motion antisymmetric gravest mode two cells modes separately B nard Problem Apply the boundary conditions to the even mode solution WDWD2 K22W0zi DW Aq0 sinq02 Bq1 sinhq12 qu sinhqi z o D2 K22W 14013 22 cosq02 30112 K2 2 coshq12 Cz 2 K2 2 coshq1z W o 2 14cm B coshq1 C coshq1 o DW o 2 Aq0 mm Bq1 51mg qu sinhq1 o D2 K22W o 2 14013 K2 2 cosq0 132 K2 2 coshq1 2 005 22 coshq1 o B nard Problem Writein Ax 1 form gtSlt cosqiO coshL1 coshL1 A 2 2 2 qo sin i q1 sinmil qf sinh i B o 2 2 2 2 mg Ilt2gt2 cosltq7 gt q K52 com ql K52 coshltq71gt c 2 detA O 2 K fRa de nes theline of marginal stability Ra Ramin 1708 KC 312 X 21 zzd The above approach creates rectangular top View xy plane cells Benard39 s classic result is hexagonal cells which requires a different assumptionin terms of the form of w T39 In general need to assume form W WZfxyem T39 TAZfxyeGf 2 2 Where fxy satisfies 87587 a2f O 11 K 3x 3y Benard Problem Stress Free surfaces layers of different density liquids uii w 0 O 32 3y 0 BC wT O To Oual 32 3x 3w 3x 3w g 31 32 0 w Of0rallxy at wall 0 2 02ltEE0 gtlta W0 32 0 0 2 3V 32 3x BC WD2WD2 K22WOzi O 32 0 So the problem becomes D2 K23W RaK2W D2 K22WogtD4 K2D2K4WogtD4Wo B nard Problem All even derivatives vanish on boundaries so odd mode solution W Asinrmz 112752 23 K2 Ra Gravest mode n 1 3ltn2I m22 0 T E Can also solve for rigid bottom stress free top Kc 268 Ra 1101 KC 222 Ran rt4 657 The linear theory predicts horizontal wavelength at onset of instability not m cell shape magnitude of ow High PE or direction of ow KE gm um LOW PE hear m DoubleDiffusive Instability What if the density is dependent on two opposing gradients typically due to temperature and concentration of a solute like salt 3 pon ocltf TogtBlt sogt This type of system can be unstable even With a stable density gradient if the diffusivity of the concentration is much less than thermal diffusivity Jumping to the appropriate non dimensional linear normal mode equations D2 K2f W ampD2 K2 W K D2 K22 RaK2fRs7lt2 BC fK S 0zrl K 2 Rs 4 Ra gud dTdz VK gBd4dSdz VK DoubleDiffusive Instability Simplify by eliminating to D2 K2f W D2 K22 Rs RaK2f Rs Rs KL Same as Benard convectionstress free solution S Rs Ra 1d S 3d f 657 v KS dz K dz Can get two unstable modes 39salt fingers39 and oscillating instability

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