Alfvén Mode Synopsis. Fast Mode Synopsis. Sunspots Devour Acoustic Waves. Lecture 26 MHD Waves in Stratified Atmospheres - PDF

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STR 75: Solar & Stellar Magnetim ale CGEP Solar & Spae Phyi Sunpot Deour outi Wae p 4 Sunpot oraiou aborb 5% or more of the aouti energy that enter them. p 3 anel Deompoition Red NO 59 p Green NO 554 olography

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STR 75: Solar & Stellar Magnetim ale CGEP Solar & Spae Phyi Sunpot Deour outi Wae p 4 Sunpot oraiou aborb 5% or more of the aouti energy that enter them. p 3 anel Deompoition Red NO 59 p Green NO 554 olography Prof. Brad indman & Juri Toomre Leture 6 Thur 5 pr 3 eu.olorado.edu/atr75toomre Braun & Birh (8) p Bla Umbrae Pin Penumbrae Blue Plage Magnetogram Emiion Map ighfrequeny ( 4 m) aouti emiion urround the atiity. Thee are alled halo. Leture 6 MD Wae in Stratified tmophere Iothermal tmophere ith a Uniform Field Charateriti Speed Lineariation Coupled Wae Equation Fourieranel Deompoition an pi up ontribution from more than jut the unpot. The annulu enter an be ontaminated by thee halo. lfn Wae Magnetooni Wae Wea Field Limit Limit Mode Conerion 4 Remember lfn Mode Synopi Remember Fat Mode Synopi F º + V V Diperion Relation F ymptoti Diperion Relation p» Fˆ g» Fˆ Phae and Group Speed p» Vˆ g» V ˆ yˆ Phae Speed Group Speed Polariation y Ä B 5 Wea Field V V + ˆ + V + ˆ + ~outi Wae (Ga Preure) ~ Wae ( Preure and Tenion) 6 Remember Slo Mode Synopi T º V + V 4 4 T F g» Tˆ T + + p» Tˆ Wea Field V V + ˆ y ˆ + ymptoti Diperion Relation Phae and Group Speed ~lfni (Tenion / ounterating preure) Iothermal tmophere ith a Uniform Vertial Field V + ˆ + ~Duted outi (Ga Preure) 7 8 MD Equation We tart from the MD equation epreing the oneration of ma, energy, magneti flu, and momentum. We ignore ioity, thermal ondution, and other nonadiabati proee. Iothermal Baground Let the baground fluid be a planeparallel iothermal atmophere ith ontant graity. Further, onider a ontant baground magneti field of trength B, that point in the ertial diretion (antiparallel to graity). Note that thi magneti field i forefree. Continuity Equation Dr r Dt Fully Compreible ẑ B B B ˆ ontant Uniform field Energy Equation Indution Equation Momentum Equation DP Dr Dt Dt B ( B) t D ( B) B r P + gr + Dt diabati Ideal MD g / P ˆ Pe / r rˆ e T ontant Contant Denity and Preure Sale eight g g Iothermal 9 outigraity Wae ì d d d æ N ü ï h í ï ( ) ý ç ï çè î ïþ If the atmophere i iothermal, the ound peed and the denity ale height are ontant d d æ N æd h i ç ç è è In an iothermal atmophere the ame equation hold for eah eloity omponent d d æ N h + + çè Charateriti Speed Sine the atmophere i iothermal, the ound peed i a ontant. The lfn peed on the other hand arie ith height in an eponential fahion. ontant B V r B / ˆ / V e V e / r rˆ e rˆ The lfn peed i ery large high in the atmophere The lfn peed i ery mall deep in the atmophere ẑ (igh) V V» photophere Conider Linear Wae Linearie the MD equation about the baground atmophere. Bt (,) B + B(,) t Pt (, ) P( ) + P( t, ) r(,) t r () + r (,) t Wea Field (Deep) V t (, ) ( t, ) Thi ubript ill be dropped from here on. Spring 3 4 Linearied MD Equation Sine the atmophere i horiontally homogeneou and the baground magneti field i ontant, the linearied form of the MD equation are relatiely imple. Continuity Equation Energy Equation Indution Equation Momentum Equation r dr + r t P dp + r t B d B( ) + B t ( B ) B r P grˆ + t Plane Wae Sine the atmophere i horiontally homogeneou and teady, the oeffiient in the preiou et of PDE are funtion of alone. Thu, e hould ee planeae olution, t ii t ( ) r(,) r()ep P (,) t P ()ep i i t ( ) t (,) ()epi i t ( ) B (,) t B ()ep i i t ( ) For impliity, I ill drop all of the tilde from here on forard. Frequeny oriontal Waenumber Note, WLOG the horiontal aenumber i only in the diretion. 5 6 Fourier Tranformed Equation Inert the planeae funtional form (or Fourier Tranform the equation) to find the folloing Continuity Equation i r r r Energy Equation i P r r i B d B oriontal omponent Indution Equation dy i B y B oriontal yomponent i B i B Vertial omponent d Dilation i + Note, no y eloity 7 Linearied Momentum Equation Momentum Equation B æ db ir ip + ib ç çè B ædb y iry ç çè dp ir g r Tenion Preure Note that the ertial equation la any magneti fore term. oriontal omponent oriontal yomponent Vertial omponent Our ultimate goal i to ue the ontinuity, energy and indution equation to eliminate all ariable other than the eloitie. 8 3 Goal Coupled Wae Equation Derie a et of oupled ae equation from the momentum equation Eliminate all ariable other than the eloity omponent. Identify the MD ae mode (a ell a e an) 9 Vertial Momentum Equation dp i g r r The ertial momentum equation doe not ontain any magneti term. Furthermore, the energy and ontinuity equation are alo deoid of magneti term. Continuity Energy i r r r ip gr r Therefore, e an play the ame tri e ued in Leture : outi Graity Wae to ombine the preure and graity fore to reeal the buoyany fore æ i d P N ç r çè g Same a lide 33 in Leture : outi Graity Wae Vertial Equation ontinued æ i d P N ç r çè g To eliminate the preure perturbation e ue the energy equation. i P g r r Energy P i g r Combine thi ith the ertial momentum equation to find d N ( g ) g Vertial Momentum Vertial Momentum Equation d N ( g ) g Epand out the dilation in term of the to eloity d omponent, and ollet term inoling and on i + different ide of the equation. æ g i æ g çè èç g d N d d N ç + + ç Remember the definition for the buoyany frequeny æ i ç çè g d d d N ç g N g æ ç çè 3 What doe it all mean? i æ ç çè g d d d N ç Inertia Buoyany Ga Preure 4 4 oriontal Momentum Equation B æ db ir ip + ib ç çè The horiontal momentum doe poe magneti term. The dependene on the perturbed preure and the perturbed magneti field an be eliminated by uing the energy and indution equation. d i B B oriontal Indution oriontal Momentum Equation P i g r d ib ib i B B B æ db ir ip + ib ç çè i B i B P i g r Vertial Indution Energy V V i æ g ç è d d ( ) + + ç 5 6 What doe it all mean? Ga Preure y oriontal Momentum Equation ædb y B iry ç çè Sine the horiontal aeetor only ha an omponent, the y momentum equation only ontain a magneti tenion term. æ V V i g ç è d d ( ) + + ç ædb y B iry ç çè d i y B y B y Indution equation Tenion Preure Inertia 7 V + y d Tenion Inertia 8 y Three Momentum Equation æ V V i g ç è d d ( ) + + ç V + y d lfn Wae æ i ç çè g ẑ d d d N ç The y equation depend only on the y eloity y. Therefore, it deouple from the other to equation. The and equation are oupled ith eah other. ŷ Ä B ˆ 9 3 5 lfn Wae The y momentum equation deouple from the other to momentum equation. It ha imple form and deribe the hear lfn ae. d V() + y In our iothermal atmophere, the lfn peed i an eponentially inreaing funtion of height. So, the loal aelength beome larger ith height. Thi ae ha no turning point. d y It group and phae eloity are + y parallel to the field. V() It ertial propagation i independent of. d y + K () It inompreie. y 3 lfn Wae Waefuntion y d + y V () / Sine V µ e thi equation ha an analyti olution. æ y J e æ Y e W + W ç è çè here W W. W. 3 Magnetooni Wae Magnetooni Wae The and momentum equation deribe the propagation of the fat and lo magnetooni ae. d d V ( ) V i æ g + + èç d d d N i æ + ç çè g What an e ay about thee equation ithout atually oling them? The and equation are oupled. (The fat and lo magnetooni mode are oupled.) The oeffiient are nonontant. (The olution are not inuoidal.) 33 The et of equation i 4 th order! ( mode 4 th order) 34 Wea and Limit d æ d V ( ) V i g + + ç è æ i ç çè g d d d N ç In general, thee equation do not hae a imple olution beaue the fat and lo mode are oupled (and the equation i 4 th order). oeer, hen the lfn peed i either ery mall or ery large ompared to the ound peed, the to mode deouple! Deep in the atmophere V Wea Field Limit igh in the atmophere V Limit Magnetooni Wae Wea Field Limit (Deep) Wea Field Limit Fat Mode d æ d V ( ) V i g + + ç è æ i ç çè g d d d N ç When the lfn peed i mall, e an reoer the fat mode by imply negleting all term ith the lfn peed. The momentum equation implifie (but the doe not). æ V ç è d d ( ) + V + i ç g V Deep Fat Mode ( ) æ d ç è i g i æ ç çè g d d d N ç Thee to equation an be ombined together (by eliminating ) to obtain a readily reognied equation. d d æ N + + ç çè V ( ) æ d ç è i g 37 Thi i the equation for aoutigraity ae in an iothermal atmophere in the abene of a magneti field! Note, thi i a eond order ODE. 38 Wea Field Limit Slo Mode d æ d V ( ) V i g + + ç è æ i ç çè g d d d N ç When the lfn peed i mall, reoering the lo mode i triier. The lo mode i obtained by reogniing that it i hort aelength. Thu, e treat d/ a a term of order /V. (In the abene of graity the ae ould be lfni.) d æ d V ( ) V i g + + ç è d d d i æ N + ç çè g V 39 Deep Slo Mode V i d d + d d i Eamine thi lat equation in a bit more detail. If e diide by the quare of the ound peed and integrate in depth one, e dioer that the dilation i ero. The lo mode d d d i d deep in the + i atmophere i inompreie! d i + V 4 Deep Slo Mode V i d d + d i + We an no ue the inompreibility ondition to eliminate from the momentum equation. d V() + V Wea Field Waefuntion Fat mode Slo mode.. uh? The lo magnetooni ae loo lie a lfn ae in thi limit. Eept that it ha (tiny) motion along the magneti field! d i 4 4 7 ẑ Wea Field V V» Fat Wae outigraity Wae Propagate aro and along field line (doen t are about the field) V Slo Wae lfni Wae (inompreie) Propagate at the lfn peed Propagate along field line æ N () + ç çè () V Spring 43 Magnetooni Wae Limit (igh) 44 Limit Fat Mode d æ d V ( ) V i g + + ç è æ i ç çè g d d d N ç When the lfn peed i large, e an reoer the fat mode by negleting all ga preure and buoyany fore in the firt equation. V V æ ç è d d ( ) + + i ç g V igh Fat Mode d + V The fat ae in thi limit i purely a magneti ae. It poee the folloing propertie: There i a turning point (a refration point). () V() The ae propagate at the lfn peed. The propagation i NOT parallel to the magneti field. The equation depend on the tranere aenumber. V d + V 45 The eloity i ompoed of both horiontal and ertial motion. d d d N i æ + ç çè g 46 Limit Slo Mode d æ d V ( ) V i g + + ç è æ i ç çè g d d d N ç When the lfn peed i large, e an reoer the lo mode by auming that the motion i tranere to the field, i.e., i muh maller than. d d d N i ç d d æ ç çè g + V 47 igh Slo Mode d + d Thi i the ame equation one ould get for aoutigraity ae if e only onidered ertially propagating ae h. So, the lo mode in the trong field limit i a duted aouti ae. ẑ () outi ae motion are fored to align ith the field line ine the field i o trong. Fluid parel on different field line are dionneted from eah other. (The diperion relation doen t depend on ). V 48 8 Waefuntion Fat mode Slo mode.. 49 ẑ Wea Field Fat Wae Wae () Propagate at the lfn peed V V Propagate aro and along field line Slo Wae Duted aouti ae () Propagate at the ound peed Propagate along field line V» What about ere? Fat Wae outigraity Wae Propagate aro and along field line (doen t are about the field) V Slo Wae lfni Wae Propagate at the lfn peed Propagate along field line æ N () + ç çè () V Spring 5 Miedup Waefuntion Fat + Slo (outi + Tenion) Slo (Duted outi) Mode Conerion Conerion Layer 5 5 Ray plitting photophere Paul Cally 53 Paul Cally 54 9
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