e = n 1 ( ) 3 [ m 3] = n [ m 3] n

Transcription

1 Magnetospheric Physics - Hoework Solutions, /7/4 7. Plasa definition Can a plasa be aintained at teperatures of T e K Hint: Calculate the density liit using the plasa paraeter and explain your result). Taking the square n ɛ k B T e e ) 3/ Λ nλ 3 D n / ɛ k B T e e ) n.38 3 ) 3 [ ] n 4.8 5) 3 [ 3] n. 7 [ 3] or taking nuber density as particle per c 3 n which iplies that n could be as sall as 7 or een 8 c 3. While this is true there is the additional aspect of how long such a plasa could be aintained and with a coparatiely sall alue of the plasa paraeter of for instance the collision frequency is still quite large and would lead to relatiely fast recobination leading to a loss of this plasa within a sall sall tie frae.

2 8. Moents of a distribution function A Maxwellian elocity distribution function is gien by ) 3/ f ) n exp πk B T k B T [ x + y + z z ) ]) Copute the aerage bulk) elocity of particles described by the distribution function. The kinetic energy can be split into a theral portion and a part which is caused by the bulk otion of particles. Copute the theral and the bulk kinetic energy? Deterine the alue of the bulk elocity for which the bulk kinetic energy equals the theral energy. Aerage Velocity: u n d 3 f ) Since the distribution is syetric for the x and y coponents the corresponding integrals are. For the z coponent we obtain by substituting w z z z ) 3/ u z d 3 z exp πk B T k B T x + y + z z ) ]) ) 3/ d x d y dw z w z + z ) exp πk B T k B T x + y + wz] ) ) 3/ z d x d y dw z exp πk B T k B T x + y + wz] ) Here ) 3/ πk B T d x d y dw z exp ] ) k B T x + y + wz by definition because ultiplyng the aboe equation with n is the definition of the plasa density such that u z z Another way to obtain this result is to realize that the gien distribution function is a Maxwell distribution transfored into the frae oing with z. Since the aerage elocity of a Maxwell distribtion is the aerage elocity of the transfored distribution ust be z. Total kinetic energy: E kin,tot d 3 ) x + y + z f ) With the transforation w z z z this becoes: E kin,tot ) 3/ n πk B T d x d y dw z x + y + w z + z ) ) exp k B T x + y + wz] )

3 ) 3/ n πk B T + z n + n z ) d x d y dw z x + y + wz exp k B T x + y + wz] ) ) 3/ d x d y dw z w z exp πk B T k B T x + y + wz] ) ) 3/ d x d y dw z exp k B T x + y + wz] ) πk B T Here the first ter is the kinetic energy in restfrae of the Maxwellian or the theral kinetic energy, the second ter is because of anti-syetry in w z note the integral has the bounds w z ) and the last ter represents the bulk kinetic energy. Because of the noralization of the integral the bulk kinetic energy is siply E bulk n z ρu z The theral kinetic energy is E th ) 3/ n πk B T 3 ) 3/ n πk B T d x d y dw z x + y + w z d x d y dw z x exp ) exp k B T k B T x + y + wz] ) [ x + y + w z] ) because the x ter, the y ter, and the wz ter hae identical contributions to the energy. Substituting x k B T/ ṽ x, and the sae for y and w z yields E th 3 ) 3/ n k B T π 3 nk BT 3 nk BT ) 3/ π π ) 3/ dṽ x dṽ y d w z ṽx exp [ ] ) ṽx + ṽy + w z dṽ x ṽ x exp ṽ ) x dṽ x ṽx exp ṽ x ) dṽ y d w z exp [ ] ) ṽy + w z d r π r exp ) r With dṽ y d w z exp [ ṽy + w z] ) dx exp ) x π d r π r exp ) r [ exp )] r π )

4 The theral energy becoes E th 3 nk BT ) / dṽ x ṽx exp ) ṽ x π We can use integration by parts and the expression in ) or alternatiely we can use the integral expressions K K + + exp a u ) π du a u exp a u ) π du a 3 to obtain the theral energy density as E th 3 nk BT Bulk kinetic energy equal to the theral energy: nu z 3 nk BT or ) 3kB T u z

5 9. MHD equations Assue a scalar pressure, L, Q E, and Q ρ. Consider a function hρ p) ρ a p b and deterine a and b such that the resulting equation for h assues a total deriatie, i.e., h/ + u h. For γ 5/3 this becoes the equation for an entropy function because entropy is consered for adiabatic changes. Hint: Use the continuity and the pressure equation to eliinate the tie deriaties of ρ and p). The total deriatie of hρ p) ρ a p b is dh dt ) ρ a p b + u ρ a p b) bρ a p b [ p + u p ] [ ] ρ + aρ a p b + u ρ Using the continuity and the pressure equations we obtain dh dt ρ p + u ρ ρ u + u p γp u γbρ a p b u aρ a p b u [γb + a] ρ a p b u such that a/b γ. Choosing b yields a γ and d p/ρ γ ) dt

6 . Noralization a) Following the exaple presented in class, deterine typical alues for the electric field, E, and the pressure p fro the MHD equations in ters of, ρ, B. b) Using the noralization procedure, derie the coefficients of the inertial ter and of the Hall ter in generalized Ohs s law. Show that these coefficients are c/ω pe ) /L and c/ ω pi ) respectiely. c) What are the alues of these coefficients for the plasa paraeters fro proble 4 and R E? a) Fro class we know the noralization for j B µ. We re-write the oentu equation ρu + ρuu) p + j B ) for all ariables in ters of ρ ρ ρ, u û, B B B, etc where the indicates noralised alues. The oentu equation becoes Diision by B µ yields ρ t ρû t ρ µ t B + ρ ρûû) p p + B µ ĵ B ρû t + ρ µ B ρûû) p µ p + B ĵ B Now we set the coefficients to unity. The second ter yields B / µ ρ ) which is the typical Alfen speed. The first ter gie just the identity /t and the pressure ter yields p B /µ. Doing the sae for Oh s law E + u B ] e j + uj + ju) e n M eρ p e + i j B + ηj 3) eρ yields E B b) The coefficient of the inertial ter in generalized Ohs s law becoes e j e B e n B t e n B µ c ɛ e e n ) λe L c ω pe L e µ e n L

7 and for the Hall ter where we used the definitions and the condition: ɛ µ c. j B B µ i n ) / en B en µ en µ / i µ e n ) / ) i / e ɛ c e e n λ i c ) iɛ / e n ) / c ω pi eletron plasa frequency : ne e ) / ω pe ɛ e ion plasa frequency : ne e ) / ω pi ɛ i electron inertial length : λ e c/ω pe ion inertial length : λ i c/ω pi c) Nuerical alues for the coefficients using n c 3 and : λe ω pe ñ / s s ) e / ω pi ωpe ω pe.3 3 s 836 i λ e c/ω pe k λ i c/ω pi k.3 3 ) ) λ i The results deonstrate that the electron inertial ters play a role only for structures on the few k scale and are inute on the R E scale. Ione inertial effects becoe iportant on a few hundred k scale but are rather sall R E scale structure.