Anelastic MHD Equations : Theory

Transcription

1 Anelasti MHD Equations : Theory Abhishek Kr. Srivastava Armagh Observatory Northern Ireland

2 Different zones of the Sun Comlex magneti field is the key behind all tyes of solar ativity and formation of different solar strutures.

3 Solar Convetion Resonses on Photoshere Magneti flux emergene as seen by MDI Magnetogram Suer-granular motion at solar surfae. Diameter ~ 1300 km, life time ~ 8-15 min, flow veloities ~ 1- km/s

4 Simulation of Convetion Proess and Magneti Field Evolution Azimuthal Magneti Field in a rotating onvetive sherial shell.blue and red reresent resetively the negative and ositive magneti field olarities. (Credit : A. S. BRUN, Colorado University)

5 Solar Magneti Field Evolution and Reversal Solar magneti field generates due to the dynamo ation in the ore of the Sun. The magneti field olarity reversal takes lae in 11 years.

6 Global Variability of Sun s Dynamis Large sale streamers in white light Photoshere s magneti field Fe XIV 5303 orona (Credit : LASCO/SOHO)

7 Solar Corona in X-Rays Key behind atmosheri ativity, heating, and the formation of variety of strutures is its omlex magneti field. Flaring loos Coronal hole Small sale X-ray BP (Credit : X-Ray Solar Telesoe, Hinode)

8 Ative Region Dynamis Comlex loo system. Cooler loo tos. Muh hotter ores, loser to loo foot oints. In Fe XV 84 EUV line, 40 EIS- SLOT, Hinode.

9 The Key behind the dynamis and heating of the different art of solar atmoshere is its omlex magneti field evolution and its link with the lasma fluid, whih generates the different tye of MHD waves. The different tye of MHD waves deosited their mehanial energy by dissiation roesses to heat the solar atmoshere and aelerate the solar wind. Magneti field reonnetion is another major roess whih may results the small-sale flaring events (nano and miro-flares), as well as also resonsible for the small-sale transient and exlosive events, e.g., jets, siules, blinkers, bright oints et. Hene, magneti reonnetion is the another mehanism whih is resonsible for atmosheri heating and dynamis.

10 Fluid + Magneti field Magneto-Fluid (e.g., solar lasma) resents with different roerties in different solar zones Convetion zone fluid Photosheri fluid Coronal fluid

11 Basi Magnetohydrodynami Equations

12 Continuity Equation Rate of hange of mass density Divergene of mass flux ρ t + ( ρ v) 0 Inomressible flow ( ρ v) 0

13 Continuity equation shows the onservation of mass inside the flux tube. This imlies that there must be a soure and a sink of the fluid mass for the onsidered region. Under the assumtion of inomressibility, the general mass onservation equation simlifies into volume onservation equation.

14 Momentum Equation Convetive aeleration Pressure gradient Lorentz magneti fore ρ v + ρ 1 t µ ( v ) v P +ρg + ( B) B + Π Unsteady aeleration of fluid artiles Gravitational fore Visous stress tensor

15 This equation shows the onservation of momentum. Pressure Gradient is a surfae fore and arises from the normal stresses. Visous stress tensor have 9 omonents. The divergene of visous stress tensor simlifies into in the ase of inomressible fluids. This is known as kinemati visosity. η v The divergene of visous stress tensor simlifies into 4 in the ase of omressible fluids. η v + η ( v) 3

16 Indution Equation B E j B / µ j σ( E + v B) t Faraday s Law Current Density Generalized Ohm s Law Solving three equations simultaneously to eliminate E and j Indution Equation B t ( v B) ( η B) Advetion term Diffusion term

17 * The Reynolds number is defined as L v / η. R m * The magneti Reynolds Number is a (dimensionless) measure of the degree of ouling between the lasma flow in a region and the ambient magneti field. * The Magneti Reynolds Number is very high in the ase of solar orona where the lasma frozen-in onet holds. This means the diffusive term is aroximately negligible and urrent density is essentially arallel to the magneti field lines. * If the sunsot radius is 10,000 km, and magneti diffusivity is 1000 sq. meter er seond, then magneti field of sun-sot will diffuse away in 3000 years!!. (Diffusion time Square of the length/magneti Diffusivity

18 Energy Equation γ ρ γ 1 D Dt γ ρ L Where, Energy loss or gain by thermal ondution Energy loss through radiation L α j.( κ T ) + χρ T ( H o + + H σ ν ) Heating funtion due to small sale reonnetions Joule heating term due to large sale urrents Visous heating term

19 Equation of State Equation of state defines the relation between density, temerature, and ressure. It may be in several forms deending uon the hysial onditions. For inomressible fluid, it will be ρ onstant However, for the erfet gas, it beomes P ρ R T R is universal gas onstant. The ressure is roortional to the density and temerature both. The temerature omes from the energy equation or taken as a onstant under isothermal assumtion.

20 Three MHD modes 1. Alfvén waves : transverse wave, roagates also arallel to B, hase veloity is the Alfvén veloity, magneti field erturbation found.. Slow MHD wave : longitudinal sound wave, roagates arallel to B, hase veloity is the adiabati sound veloity, found also in a omressible & non-onduting fluid. 3. Fast MHD wave :longitudinal wave, roagates erendiular to B, known also as the omressional Alfven wave or the fast Alfven wave, and sometimes as the magnetoaousti wave, assoiated B and E found.

21 Anelasti MHD Aroximation to Model Convetion Zone MHD Tubes Low Mah number flows are onsidered uto the multile density and temerature sale heights. Low Mah number flows imlies that the loal Alfveni seed will be less than the sound seed. Aliation regime lies between the inomressible and fully omressible magneto-onvetion. The density erturbation only omes in the buoyany fore term. However, bakground density and other thermodynamial variables hanges with the height/deth. Hene, the non-trivial deth stratifiation an be Inororate in the simulation with filtering fast magnetoaousti waves and onsidering quasi-omressible onditions.

22 Mixing Length Theory (MLT) of Convetion (The motivation for the Anelasti aroximation) The weakly unstable onfigurations suffies to arry the stars luminosity, and onvetion fluxes are highly suffiient to transort the heat. Large sale onvetions by the order of loal ressure sale height In the hydrostati equilibrium and low Mah number flows, the ontribution of aousti wave is negligible. This is one of the examles of ommon hysial situation both in MLT and anelasti aroximation. MLT Theory guides at every ste in the derivation of Anelasti MHD equations, to established the bakground onfiguration and erturbations.

23 Shwarzshild Criterion for Stable Convetion and Bakground Conditions in Anelasti Aroximation In adiabati onvetion, the bubble's density hange whih is the funtion of ressure and entroy, deends only on hange in ressure beause the ideal adiabati ondition inside the bubble restrits the entroy variation with height z. The differene in the bubble s density hange and the hange in the bubble s viinity is given as (Shwarzshild, 1906, 1958) ρ ' ρ ρ s ' d d z z ρ s d d s z z...(1) The bubbles will be lighter and float uward stably only when ρ s d d s z z ρδ d s d z z 0...( ) 1.The onvetion zone is in equilibrium at lowest order under adiabati (Isentroi) and hydrostati onditions.. The onvetion rovides first order erturbations in the thermodynamial arameters and the arameters related to fluid motions.

24 Using first law of thermodynamis ρt ds ρ dt δ d...(3), and hydrostati equilibrium onditions d dz ρ g...( 4), the adiabati temerature gradient in lowest order equilibrium ondition will be T z ad δ g...(5) Conlusions ome that hydrostati equilibrium and strit isentroiity must be the bakground onditions.

25 Determination of Perturbations and Saling The rising bubble exhange off the exess of entroy with environment over the mixing length l. So, the amount of heat given off er gram of bubble is T l d s d z d T l d z ad d T d Z l ( T )...(6) Where mixing length relates with ressure sale height H d z / d (ln ) by l α H. The alha is onstant equal to unity. The heat flux sent off by bubble an be obtained by multilying heat amount to the average bubble seed and loal density: F onv 1 ρ v l ( T )... (7)

26 Tyial bubble s seed an be estimated equating its kineti energy to the average work done in its fall or rise to the reresentative distane l/ by the fore of buoyany ρ v δ ρ g l T 8 T... (8) (Kim et al., 1996) Solving equations (7) and (8) under MLT aroximations will result the two erturbed arameters : H d s d z H T T d lnt ln d ad + d lnt...(9), ln d s ρ s v ( T ( T / ρ) / ) ρ γ α T ρ γ α T g H...(10),

27 where, α ( ln ρ / ln T ) T is the isothermal dimensionless omressibility. The Mah number is known as : M V / s...(11). For an ideal gas, γ 5 / 3; α δ 1...( 1) T P These may be deth deendent in the ase of ionized lasma gas (Lantz, 199)

28 On solving the above set of equations, will result : M U 1 ( ) 1/ U ( F onv / ρ) s 1/3...(13) where, α δ αt U1 4 γ 1/ ~ O (1)...(14 a ), and 1/3 α δ r ( 1) α αt γ 1/3 U1 ~ O (1)...(14b ). δ γ

29 ********************************* Derivations************************************** 1)...( 4 4 / 4 / E T T H g H T T l g V M T T T s γ α δ α α γ α δ α γ δ E1 an be obtained by using equation 8, 9, 10, and 11.

30 On multilying and simlifying equations 7 and 8, we get 1 ρv l ( T ) ρv F onv δ ρ g l T > v δ g l T 1/3 ( F onv / ρ) 1/3 > v α δ g H T 1/3 ( F onv / ρ) 1/3 > α δ ρh v T H 1/3 ( F onv / ρ) 1/3 >

31 > ρ ρ α δ 1/ 3 1/ 3 ) / ( onv F T v > ρ δ α 1/3 1/3 * ) / ( onv F r v > ρ δ α δ 1/3 1/ 3 * ) / ( onv F r v > ρ δ α γ α 1/3 1/3 * ) / ( 1) ( onv T v F r v

32 Beause, δ v r*. α T Hene, M α δ r * 1/3 ( F onv / ρ) s 1/3 ( 1) ααt γ δ γ 1/3 ( F onv / ρ) s 1/3 ( E) Taking root of E1 and equating with E will give, M 1/ 3 1/ ( Fonv / ρ) U1 ( ) U...( E3), s,whih is equation 13 with the oeffiients given by 14 a, b. ***********************************************************************************

33 1. > Provides the sale for all the erturbations. Its magnitude governs the basi validity of MLT. These equations holds only if the dearture from equilibrium will be small.. Its value is 0.1 at the hotoshere, while at the bottom of onvetion zone it is These estimations are based on standard solar model that itself make use of MLT theory.

34 Sales The erturbations and veloities vary very small from bakground equilibrium. Hene, the naturally stable stratifiation should be a good zero order aroximation to the thermodynamis Heat transort is due to the rise and fall of adiabati bubbles over the mixing length. This is of the order of loal ressure sale height. Thermodynami flutuations due to onvetion are equated by the differene of Its vertial gradient and its adiabati gradient, times the mixing length. The thermodynami to bakground ratio sales varies as a square of Mah number.

35 Conlusions Mixing Length Theory (MLT) is a bakbone behind the anelasti aroximation. The anelasti aroximation is very useful to model the buoyant flux tube dynamis in the onvetion zone of sun and magnetially ative stars. This aroximation is based on the quasi-omressibility ondition, hene allow the large time stes of simulation and simlify the omutational stuff. This aroximation is suessful in Its goal to model the onvention zone dynamis (Fan et al., 1999).

36 What will ome in the next talk? The anelasti form of the MHD equations to model solar onvetion and the omutational tehnique (ANMHD 3-D MHD simulation ode) to solve them!!! Aliations of ANMHD simulation ode!!!

37 Aims of the Projet ANMHD Hinode is roviding the high resolution observations first time to study the dynamis of solar atmoshere. The magneti field is the key for all tye of solar ativity and its strutures. The ouled motion of lasma and magneti field generates all the atmosheri dynamis. Hene, ANMHD simulation ode will be helful for my grou to model the flux tube whih omes out in the lower atmoshere from onvetion zone, e.g., in the ative region loos, over the sunsots, flare related sites et. This tye of modelling with the reently develoed faility of Magneti footrinting of solar oronal magneti field with MDI data (Dr. R. C. Malean), and EUV observational analysis will enlighten the related siene. This ode may be also useful to study the magneti field evolution of magnetially ative stars, e.g., AD Leo Flaring stars.

38 Referenes and Aknowledgment 1. Kim,Y.-C., Fox, P.A., Demarque, P., Sofia, S., 1996, AJ, 461, Lantz, S.R., Fan, Y., 1999, AJ, 11, Shwarzshild, M., 1958, Struture and evolution of the star, 47, NY : Dover. 4. Y. Fan, E.G., Zweibel, M.G. Linton, G.H. Fisher, 1999, 51, Solar Magnetohydrodynamis, by E.R. Priest, 198, Sringer. 6. Lantz, S.R., 199, Ph.D. Thesis, Cornell University. 7. Shwarzshild, M., K., 1906, Gottingen Nahr., 435. We are thankful to G.H. Fisher and grou from UCL Berkeley for sharing the ode ANMHD, whih is develoed under the Solar MURI rojet. Solar MURI is a ollaborative rojet studying magneti erutions on the Sun and their effets on the Earth's sae environment. ("MURI stands for Multidisilinary University Researh Initiative, a researh rogram funded by DoD.)

39 Thank You Very Muh We shall ontinue it in seond art..