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1 !"#$%&'()%"*#%*+,-./-*+1.2(.* *!"#$%&"'(()'*+,"-'.' Dr. D. Shaun Blmfield Astrphysics Research Grup Trinity Cllege Dublin :-#*;<="<.*-*:-><?* Mechanical examples: sund, string, water Energy transfer Restring frces: pressure, tensin, gravity Characteristics: wave speed mtin f medium directin f prpagatin Dispersin relatin really imprtant!

2 @2/A,<*:-><*B<A$<.<"#-)%"* Fr plane waves prpagating with wave vectr k = (k x, k y, k z ) and angular frequency!, [N.B. r = (x, y, z) is the psitin vectr] U = C U exp[ i( k! r " #t)] and fr prpagatin in nly the x-directin, U = C U exp[ i( kx! "t )] Cnstant phase is maintained fr a pint n the wave when, d ( kx! "t ) dt d( kx)! " dt dx phase speed dt = " k = v p :-><*C$%'A*@A<<&* Phase speed is nt the rate f infrmatin (i.e., energy) transfer Grup speed is similarly defined, but fr cnstant phase n a mdulated wave envelpe, U! exp[ i ("kx # "$t )] giving, d dt (!kx "!#t) dx dt =!#!k %!#' lim &!k ( = d# dk = v g!# $ grup speed

3 :-><*;2.A<$.2%"*B<,-)%"* Everything is cntained in dispersin relatin,! =!( k) k ften cmplex, but waves prpagate nly fr,!( k) > Dispersin relatin indicates cutffs and resnances :-#*D-E<.*+,-./-*:-><.*;2F<$<"#?* Plasma prperties: gas-like charged Fluid equatins mass cntinuity equatin f mtin energy equatin ideal gas law magnetic field (cmplicates everything) EM equatins Maxwell s equatins inductin equatin Ohm s law

4 G,'2&*HI'-)%".*!* Mass Cntinuity!" + #$ ( "v ) matter is neither created nr destryed but,!" ( ab) = ( B"! )a + a!"b!" + ( v # $ )" + "$# v Equatin f Mtin! Dv Dt = "#p + F Newtn s 2 nd law; F represents all external frces D Dt =! + v "# is the cnvective time derivative HJA-"&<&*HI'-)%"*%K*D%)%"* External frces magnetic frce: gravitatinal frce: viscus frce (ignred) J! B!g (the Lrentz frce) New Equatin f Mtin! Dv Dt = "#p + J $ B +!g Expand D/Dt,! "v "t +! ( v#$ )v = %$p + J & B +!g

5 G,'2&*HI'-)%".*!!* Ideal Gas Law p = R!T is mean atmic weight Energy Equatin L represents all energy lsses D Dt # p & % $! " ( = )L ' " is the rati f specific heats (c p /c v ; cnstant pressure t cnstant vlume), nrmally taken as 5/3 nly cnsider adiabatic case (i.e., at cnstant energy) L Apply change rule, D Expand, Dt D%&2=<&*H"<$L1*HI'-)%"* D Dt # p & % $! " ( = 1 Dp '! " Dt ) "p D!! " +1 Dt 1 #p! " #t + 1 ( v$% )p & "p #!! "! " +1 #t & "p! ( v $ %)! " +1!p + ( v "#)p $ %p &!& + ( v "#)& ) * & But,! "# frm mass cntinuity equatin, ' "t + ( v$% )# ( ) = #%$ v ' (!p + ( v"# )p + $p ( % %#" v)!p + ( v " # )p = $%p#" v

6 !"E = # $ HD*HI'-)%".*!* Ampère s Law! " B = J + 1 c 2 #E #t gradients in magnetic fields create electric currents in MHD apprx. displacement current (RHS 2 nd term) is neglected! " B = J Slenidal Cnstraint!"B n magnetic mnples Faraday s Law!B = "# $ E spatially varying electric field induces a magnetic field HD*HI'-)%".*!!* Gauss Law charge cnservatin Ohm s Law J =! ( E + v " B) generalized frm cuples EM equatins t plasma equatins thrugh v Inductin Equatin!B = " # v # B cmbinatin f Ohm s, Ampère s, and Faraday s laws diffusivity term ignred

7 :-><*M..'/A)%".* Wave amplitudes are small allws fr linearizatin f MHD equatins Basic state is a static equilibrium Eqn. f Mtin: =!"p + J # B + $ g [A] Slenidal Cnstraint:!"B [B] p = R! T Ideal Gas Law: [C] quantities X and X are the initial equilibrium state nt necessary t be static (just simplified prf) :-><*+<$#'$N-)%".* After wave initiatin, B = B + B 1 r,t v = v + v 1 r,t! =! +! 1 r,t p = p + p 1 r,t T = T + T 1 r,t X and X are the perturbed quantities X 1 and X 1 are the applied perturbatins (<< X and X quantities) Static initial cnditin: v v = v 1 ( r,t) initial quantities are time independent!x!x

8 DO;*P2"<-$2Q-)%"*3(%")"'2#1*<I"R9* Put perturbed quantities int MHD equatins and neglect prducts f small terms (i.e., X 1 Y 1 ) Mass Cntinuity:!" + #$ ( "v )!" +!" 1 + #$ (" v 1 ) + #$ (" 1 v 1 ) but with!" and drpping X 1 Y 1 terms,!" 1 + #$ (" v 1 ) DO;*P2"<-$2Q-)%"*3<I"R*%K*/%)%"9* Eqn. f Mtin:! "v "t +! ( v #$ )v = %$p + J & B +!g! "v 1 "t +! 1 "v 1 "t Neglecting X 1 Y 1 terms and substituting fr J,! "v 1 "t but,!"p + J # B + $ g, +! ( v 1 # $ )v 1 +! 1 ( v 1 # $ )v 1 =!"p! "p 1 + J # B + J # B 1 + J 1 # B + J 1 # B 1 + $ g + $ 1 g ( = #$p 1 + $ )! "v 1 "t 1 + $ 1 +! 1 g + #$p + J +! g ( = #$p 1 + $ ) 1 + $ 1 +! 1 g

9 DO;*P2"<-$2Q-)%"*3<"<$L1S2"&'()%"*<I".R9* Adiabatic Energy Eqn.:!p + ( v " # )p = $%p#" v + ( v "#) p + ( v " #)p = $%p #" v $ %p #" v !p +!p 1!p but with Inductin Eqn.: but with!b +!B 1!B and drpping X 1 Y 1 terms,!p 1 + ( v " #)p = $%p #" v 1 1!B = " # v # B + " # ( v 1 # B 1 ) = " # v 1 # B and drpping X 1 Y 1 terms,!b 1 = " # ( v 1 # B ) DO;*P2"<-$2Q-)%"*3L-.*,-TS.%,<"%2&-,*(%".#$-2"#9* Ideal Gas Law: p = R!T p + p 1 = R! T + R! 1T + R! T 1 + R! 1T 1 but p = R! T and drpping X 1 Y 1 terms, p 1 = R! 1 T + R! T 1 Slenidal Cnstraint: but with!"b,!"b!"b +!"B 1!"B 1

10 @'//-$1*%K*P2"<-$2Q<&*DO;*HI'-)%".*!" 1 + #$ (" v 1 ) [1]! "v 1 "t ( = #$p 1 + $ ) 1 + $ 1 +! 1 g [2]!p 1 + ( v " #)p = $%p #" v 1 1 [3]!B 1 = " # ( v 1 # B ) [4] p 1 = R! 1 T + R! T 1 [5]!"B 1 [6]