ionized plasmas: effect of neutral helium in multi-fluid approach - PowerPoint PPT Presentation

Damping of Alfvén waves in solar partially ionized plasmas: effect of neutral helium in multi-fluid approach T.V. Zaqarashvili and M.L. Khodachenko Space Research Institute of Austrian Academy of Sciences, Graz, Austria

Neutral atoms in the solar atmosphere Blue solid line: ratio of neutral hydrogen and electron number densities. Green dashed line: ratio of neutral helium and electron number densities. Plasma is only weekly ionized in the photosphere, but becomes almost fully ionized in the transition region and corona. Height, km FAL93-3 model (Fontenla et al. 1993)

Neutral atoms in the solar atmosphere The ratio of neutral helium and neutral hydrogen is around 0.1 in the lower heights. But it increases quickly up to 0.22 near chromosphere/corona transition region i.e. at 2000 km. FAL93-3 model (Fontenla et al. 1993)

Neutral helium vs neutral hydrogen The ratio of neutral helium and neutral hydrogen number densities is increased in the temperature interval 10000-40000 K. FAL93-3 model (Fontenla et al. 1993)

Multi ti-fl flui uid d equations ions We consider partially ionized incompressible plasma which consists of electrons, protons, singly ionized helium, neutral hydrogen and neutral helium atoms We neglect the viscosity, the heat flux, and the heat production due to collision between particles. Then the governing equations are:     V 0 , a               V 1              a m n V V p e n E V B R ,    a a a a a a a a a     t c    1 B     E ,  c t    4     B j , c           j e n V n V n V .     e e H H He He

Multi ti-fl flui uid d equations ions For time scales longer than ion-electron collision time, the electron and ion gases can be considered as a single fluid. Then the five-fluid description can be changed by three-fluid description, where one component is the charged fluid (electron+protons+singly ionized helium) and other two components are the gases of neutral hydrogen and neutral helium gases. We use the definition of total density of charged fluid  0       H He and the total velocity of charged fluid as       V V      H H He He V .      H He The sum of momentum equations for electrons, protons and singly ionized helium is       d V   1              , w w p j B F  0 0 t H He dt c       w V V where is the relative velocity of protons and helium ions.  H He

Th Three-fl fluid uid equations ions    w V It can be shown that for the time scales longer than ion gyro period. Then we obtain the three-fluid equations as       d V 1         , p B B F  0 i 4 dt   d V     H p F , H H H dt   d V     He p F , He He He dt       B     V B .  t where                            F V V V ,         i i H He H H H He He H He He H H He H H He He He                      F V V V ,     H HeH H i HeH He H H He H H H He H                      F V V V .     He HeH He i HeH H H He He He H He He He

Multi ti-fl flui uid d linear equations ions We consider the wave propagation along unperturbed magnetic field, which is directed along the z axis. Then the linear Alfvén waves polarized in the y direction are governed by equations        u b B ( z ) ( z ) ( z ) ( z ) ( z )     y y z H He H He u u u ,       y Hy Hey t 4 ( z ) z ( z ) ( z ) ( z ) 0 0 0 0       u ( z ) ( z ) ( z ) ( z )    Hy H H HeH HeH u u u ,     y Hy Hey ( ) ( ) ( ) t z z z H H H       u ( z ) ( z ) ( z ) ( z )    Hey He H HeH HeH , u u u     y Hey Hy t ( z ) ( z ) ( z ) He He He   b u  y y B ( z ) .   z t z

Homoge mogene neous ous plasma ma We consider a homogeneous plasma and after Fourier transform derive the dispersion relation of Alfvén waves in the three-fluid plasma                4 3 a a i a ( 1 ) a ( 1 ) H He H He H H He He He H                     2 1 a a i a a 1 0 , H He H He H He H H He He where      k v k v         z A 0 z A 0 H He , , , , . a a     H He H He k v 0 0 z A H He The dispersion relation has four different roots: the two complex solutions, which correspond to Alfvén waves damped by ion-neutral collision and two purely imaginary solutions, which correspond to damped vortex solutions of neutral hydrogen and neutral helium fluids. We consider only Alfvén waves.

Upper er ch chromosp mosphe here re Chromosphere: 1995 km height above the photosphere. Zaqarashvili et al. 2011

Upper er ch chromosp mosphe here re Chromosphere: 2015 km height above the photosphere. Zaqarashvili et al. 2011

Colli llision ion frequ quencies ncies Mean ion-neutral collision frequency is (Zaqarashvili et al. 2011)   1 1 kT          2 ( n n ) .    in in i n in   m n m n m i i n n i The collision frequency is very high in the photosphere, but decreases upwards. The collision frequency between protons and neutral hydrogen atoms estimated from FAL93-3 model can be estimated as z=0: z=900: z=1900:  6 Hz 6.2 10 3 Hz Hz =8.6 10 24 in This means that the Alfvén waves with periods > 1 s can be easily considered in the single-fluid approach.

Alfvén vén wave ves s in single ngle-flui fluid d MHD HD We consider the total density        , 0 H He total velocity      u u u  0 y H Hy He Hey V ,      y 0 H He relative velocity between ions and neutral hydrogen   w u u H y Hy and relative velocity between ions and neutral helium   . w u u He y Hey Then we find that      u V w w . y y H H He He

Alfvén vén wave ves s in single ngle-flui fluid d MHD HD Consecutive subtractions of multi-fluid equations and neglect of inertial terms leads to the equations      b B       y z He HeH ( ) , w       H H H He   4 z      b B       y z H HeH w ( ) ,       He He H He   4 z           where . H He H HeH He HeH Then the sum of multi-fluid equations leads to the single-fluid equations   V b B ( z )  y y z ,    t 4 ( z ) z        b V b ( ) z     y y y c ( ) ( ) B z B z       z z   t z z B ( z ) z z where      2 B ( z ) ( z ) ( z ) ( z )               2 2 2 z He H HeH  ( z ) ( z ) ( z ) ( z )     c H He H He   4 ( z ) ( z ) ( z ) is the coefficient of Cowling diffusion.

Disp ispersi ersion on relation lation From these two equations we get           2 V V V B ( z ) ( z )     y y y   z c B ( z ) B ( z ) .         z z 2 2   t 4 ( z ) z z z V ( z ) t   A For homogeneous atmosphere we have       2 2 2 2 0 i k k V c z z A which has two complex solutions   2 2 2 k k      c z c z k V 1 i . z A 2 4 V 2 A Real part of the complex frequency gives cut-off wave number 2 V   A k .  z c