Space Physics ELEC-E4520 (5 cr) Teacher: Esa Kallio Assistant: Markku Alho and Riku Järvinen Aalto University School of Electrical Engineering
The 6 th week: topics Last week: Examples of waves MHD: Examples & applications Discontinuities (also shocks) This week: Waves: Sound waves MHD waves: Alfvén, fast, slow Electromagnetic (EM) waves Cold plasma waves https://www.nasa.gov/feature/goddard/2017/nasalistens-in-as-electrons-whistle-while-they-work 27.3.2018 Space physics (ELEC-E4520) 2
Material for the new topics: - Section 11 Pulsations and magnetohydrodynamic waves especially Subsections 11.4 waves in cold plasma - Section 12 Plasma waves Chapters 6 (Magnetohydrodynamics) Chapters 7 (Cold plasma waves) (Web book in preparation. A draft version can be downloaded from MyCourses) http://farside.ph.utexas.edu/teaching/plasma/plasma/index.html 27.3.2018 Space physics (ELEC-E4520) 3
Where we are now in the course? 2 1 27.3.2018 Space physics (ELEC-E4520) 4
Today you will receive some basic information about these waves: for T=0 case ~ today No oscillating B field today today (very very briefly ) Oscillating E and B fields today ~ today (in the limit c>>v A ) https://en.wikipedia.org/wiki/waves_in_plasmas 27.3.2018 Space physics (ELEC-E4520) 5
Preliminary work Linearization and linear algebra Waves Basic terminology Same basics of electromagnetic waves Waves transfers energy and information from another region without mass transfer Manifestation of collective behaviour of plasm Interacts with particles: wave particle interac anomalous resistivity) 27.3.2018 Space physics (ELEC-E4520) 6
Mathematics: Linearization and Linear algebra Linearization of an equation(s) Basic idea: if a ~ b are all SMALL values then combination of these values results, then a * b is a VERY SMALL value, which is approximated to be zero compared with a or b. For example, if a ~ b ~ 1/100, then a * b ~ 1/10000 << a and b We can then simplify (linearize) the equation: a + b + a * b + a*a + b*b = 0 a + b = 0 Linear algebra a b c d f f g h i x y z = 0 A non-trivial solution (i.e. other than x =0) obtained if det(a)=0 where Determinant: det(a), det A, A : A x = 0 When we study waves we use this knowledge (det(a)=0) to derive a dispersion relation. https://en.wikipedia.org/wiki/determinant 27.3.2018 Space physics (ELEC-E4520) 7
Waves: Basics (1/2) Question: According to the MHD theory, can there be waves? If there can be, what kind of waves there can be and which are their properties? Approach: - Analyse small variations in n, U, j, E, B (mathematically: linearize equations) - Study, can there be a solution of the form plane wave variations ~ e i(k r ωt) Terminology: angular frequency Wave length Wave vector ω l k i.e. a plane wave ω = 2 π f k 2 π / l phase speed v p v p ω k group speed v g v g dω dk (speed of information c) Note: a constant phase k r ωt = const. moves along the direction of k with a speed v = ω/k a spherical wave k v Plane of stationary phase Plane of stationary phase 27.3.2018 Space physics (ELEC-E4520) 8
Waves: Basics (2/2) Linear Polarization Plane electromagnetic (EM) wave propagating in the +x direction in a vacuum E = c B (c.f. E = vxb ) Circular Polarization Elliptical Polarization Note: equal amplitude which differ in phase by 90 Note: unequal amplitude which differ in phase by 90 27.3.2018 Space physics (ELEC-E4520) 9
ELECTROMAGNETIC (EM) WAV Many important technological applications Space filled with EM waves We will learn that plasma can reflect refract absorb EM waves 27.3.2018 Space physics (ELEC-E4520) 10
Magnetohydrodynamic (MHD) equations 27.3.2018 Space physics (ELEC-E4520) 11
http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html Warming up : Sound wave in air (Note: there are a sound wave in plasma along B) speed of sound Sound wave is longitudinal wave the displacement of the medium is parallel to the propagation of the wave. A question: So v s = γ thermal energy density mass density but what might be γ magnetic energy density mass density = γ B2 2μ 0 ρ m = B μ 0 ρ m = v A 2D: γ = 2 27.3.2018 Space physics (ELEC-E4520) 12
Warming up : Sound wave in air (contn d) 27.3.2018 Space physics (ELEC-E4520) 13
http://misclab.umeoce.maine.edu/boss/classes/sms_491_2003/week_10.htm http://dev.physicslab.org/document.aspx?doctype=3&filename=wavessound_introsound.xml Propagation of a sound wave http://wotzup.com/wp-content/uploads/2014/10/512px-comparison_us_standard_atmosphere_1962.svg.png Sound propagation in an ocean with a typical distribution of temperature and pressure. Sound refracts as the speed of sound changes, resulting in a sound channel (SOFAR channel) where sound propagates very well. Note that here and in other layered media that vary in sound speed, sound tends to get trapped in the layers where sound velocity is slowest. It is this sound channel that allows long-distance communication in the great whales. 27.3.2018 Space physics (ELEC-E4520) 14
Shock wave in air M s > 1 M s = 1 M s < 1 F/A-18 Hornet Fighter Jet M s = 0 Mach number: M = U/U wave speed Sonic Mach number: M s = U/U sound (top figure) the temperature in the low pressure regions drops condensation of the water vapor 27.3.2018 Space physics (ELEC-E4520) 15
Example of discontinuity in space plasma: Shocks Various shocks in plasma: Fast shock: standard in interplanetary space Slow shock: in the solar corona? Note: magnetic field compression Shock: Discontinuity that separates two regimes in a continuous medium Motion faster than the signal speed of the medium Only the relative speed is relevant: standing and traveling shocks follow the same rules Collisionless shock: Conventional hydrodynamic shock: momentum transport and thus propagation of information due to collisions between molecules sound speed is the signal speed Collisionless shocks: densities too low to allow for collisions. Momentum and information transport due to the plasma s collective behaviour as organized by the magnetic field 27.3.2018 Space physics (ELEC-E4520) 16
Earth s bow shock (1/2) Bow shock The solar wind moves faster than its own sound speed In the frame of the solar wind, the magnetosphere is moving at supersonic speeds. This generates a (fluid dynamics) shock wave in front of the magnetosphere called the bow shock. 27.3.2018 Space physics (ELEC-E4520) 17
Earth s bow shock (2/2) BS magnetosheath Solar wind BS = fast shock Solar wind heating 27.3.2018 Space physics (ELEC-E4520) 18
MHD WAVES Derived from MHD (i.e. single fluid) equations => low frequency and long wavelength waves ( slow and large variations ) 27.3.2018 Space physics (ELEC-E4520) 19
MHD waves (1/6) [1] [2] (Note: we assume that T = constant => v s = constant) [3] (ideal MHD: E = -VxB) background wave Linearize (X 1 Y 1 0 ) and take δ Τδt (moment. Eq.) (Note: no background flow) Alfvén speed [1] [3] 1 ρ mo = constant 27.3.2018 Space physics (ELEC-E4520) 20
MHD waves (2/6) - * dispersion relation Solution: solve det നM = 0 D, k = 0 നM θ, v p V i = 0 We get v p = and v g = dω dk 27.3.2018 Space physics (ELEC-E4520) * 21 ω k
MHD waves (3/6) : Alfvén wave Phase speed => ( ) = 0 [( )( ) - ( ) 2 ] = 0 Group speed v g = k ω = ( δω, δω, δω ) δk x δk y δk z (i.e. to where energy goes) v g = ±v A z wave normal surface: phase velocity as function of 27.3.2018 Space physics (ELEC-E4520) 22
MHD waves (4/6) [( )( ) - ( ) 2 ] = 0 => ( ) + fast MHD (Alfvén) wave - slow MHD (Alfvén) wave = 0 * low β high β * 27.3.2018 β = p Space physics (ELEC-E4520) 23 B 2 /2μ ~ o v s 2 v A 2
MHD waves (5/6): properties restoring force Fast wave: gas and B pressure are in phase (high B high p) Slow wave: they are out of phase (high B low p, low B high p) 27.3.2018 Space physics (ELEC-E4520) 24
MHD waves (6/6) Alfvén wave = shear Alfvén wave; = non-compressional (i.e. no density changes * ) Alfvén wave (k V 1 = 0) c.f. a tensed string Longitudinal sound wave propagates along the magnetic field. Note: Density and pressure changes. A fast magnetosonic wave can propagate perpendicular to the magnetic field. Note: Density, pressure and magnetic field changes. * See e.g. https://farside.ph.utexas.edu/teaching/plasma/lectures1/node65.html 27.3.2018 Space physics (ELEC-E4520) 25
Man-made EM-waves: a E-dipole antenna Electric fields produced by an electric-dipole antenna Electric and magnetic field lines produced by an electricdipole antenna http://web.mit.edu/viz/em/visualizations/coursenotes/modules/guide13.pdf The white lines and the yellow lines are the electric and the magnetic field lines 27.3.2018 Space physics (ELEC-E4520) 26
Propagation of EM wave in a conducting media Electric field generated by the oscillation of a current sheet Reflection of electromagnetic waves at conducting surface http://web.mit.edu/viz/em/visualizations/coursenotes/modules/guide13.pdf 27.3.2018 Space physics (ELEC-E4520) 27
WAVES: cold plasma model all particles move with the same velocity also EM waves (displacement current included) no pressure related waves, no instabilities 27.3.2018 Space physics (ELEC-E4520) 28
Wave dispersion relation ω = ω(k) phase speed v p v p ω k group speed v g (speed of information c) v g δω δk v g = k ω = ( δω δk x, δω δk y, δω δk z ) k =2 π / l Phase speed and group speeds are known if we know the Dispersion relation: ω = ω(k) Note: k can be complex, but wave propagates only if Re(k)>0. At cut-off frequency v p, k=0 (waves become evanescent below this frequency; reflection of a wave) Recall: Langmuir waves fast variations slow variations Example No 1: Electromagnetic wave in a cold (T=0), non-magnetized (B=0) plasma MHD large l small l propagating wave cut-off frequency (reflection) evanescent wave Note: in the wave above the phase speed is always larger than c! But the group speed is less than c and approaches 0 at k=0 27.3.2018 Space physics (ELEC-E4520) 29
Wave dispersion relation ω = ω k (contin d) phase speed v p ω k fast variations group speed (speed of information c) v g δω δk Example No 2: Electromagnetic wave in a cold (T=0) and magnetized plasma (=> electrons can gyrate around B, i.e. have cyclotron motion. New time scale: electron cyclotron frequency) (electron cyclotron) Resonance frequency (wave absorption) slow variations MHD large l small l An EM wave, so called right-hand polarized wave ( R-mode ), propagating parallel to the magnetic field Note: At resonance frequency v p = 0, k (absorption of the wave) 27.3.2018 Space physics (ELEC-E4520) 30
Propagation of EM wave in a plasma (1/2) Complete Set of Two-Fluid Equations (c.f. Extra slides 13.3) SELF STUDY MATERIAL 0 0 Same method as in MHD waves: Linearization (i.e. make a set of linear equations and use linear algebra) Cold plasma model: Analyse high frequencies Other: (1) p e = p i = 0 (note: => all electrons have the same speed and all ions have the same speed) (2) ions are static background (u i = 0) => u = u e and j =-e n e u e =-e n e u Note: displacement current (~ E/ t) can not be put to zero (as in MHD) Note: we use el. mom. eq. => mom. eq. contains the electric field E 27.3.2018 Space physics (ELEC-E4520) 31
Propagation of EM wave in a plasma (2/2) Model: Cold plasma model where ions are static background SELF STUDY MATERIAL m u t = e(e + u B) E = B t B 1 E c 2 t = * μ on o q o u Same method as in MHD waves: (1) Linearization (i.e. make a set of linear equations and use linear algebra) (2) Look for a plane wave solution നA E= 0 നA =നA (ω, k, ω ce, ω pe ) - Appleton-Hartree dispersion relation (see e.g. Koskinen&Kilpua, Eq. 7.43 or Kivelson&Russell, Eq. 12.101) (3) Solve the obtained linear equations നA E= 0 as det(a)=0 and get the dispersion relation ω = ω k 27.3.2018 * Note: there are no density variations because of linearization (ne*ue = no*ue). The only new unknown to Maxwell s Space physics (ELEC-E4520) 32 equation is Ue i.e. we need only one vector equation for U. That missing equation is the electron momentum equation.
Koskinen&Kilpua Special EM wave solutions (1/3): Propagation of EM waves in non-magnetized plasma The dispersion equation for an electromagnetic wave in a cold plasma where the background magnetic field is zero. Illustration (not to scale) of the reflection of electromagnetic waves from the ionosphere 27.3.2018 Space physics (ELEC-E4520) 33
Special EM wave solutions (2/3): Refraction and refraction of EM waves in a non-magnetized plasma Refractive index: n refractive c/v phase =c*k/ Example: non-magnetized plasma: 2 = p 2 + c 2 k 2 h => c 2 k 2 / 2 = 1 - ( p / ) 2 = n 2 => n refractive = 1 - ( p / )2 => n refractive ~ 1- n electron (cm 3 ) (9kHz/ )2 Note 1: n refractive 1 (!) Note 2: when n electron increases then n refractive decreases > p ionosphere h => ionosphere analogy water air Snell s law: n 1 *sin(j 1 ) = n 2 *sin(j 2 ) (i.e. k is same on both sides in order to have a continuous phase angle) n electron = n e2 (> n e1 ) n refractive ~ n electron = n e1 n refractive ~ 1 - n e2 (cm 3 ) (9kHz/ )2 =n r2 <n r1 k 2 j 2 k 1 j 1 1- n e1 (cm 3 ) (9kHz/ )2 =n r1 n electron 1 n refractive water => EM wave which propagates upward into higher electron density region turns back toward the Earth 27.3.2018 Space physics (ELEC-E4520) 34
Special EM wave solutions (3/3): R/L and O/X waves B o longitudinal E 1 k Parallel (k B o ) k E 1 transverse E 1 E 1 k Perpendicular (k B o ) k E 1 B o E 1 Ordinary wave (O) - linear polarization E 1 B o E 1 Extraordinary wave (X) - elliptical polarization plasma oscillation Left hand wave (L) - circular polarization Right hand wave (R) - circular polarization - Whistler 27.3.2018 Space physics (ELEC-E4520) 35
Now you have received some basic information about these waves: for T=0 case ~ today (p. 12-13) No oscillating B field today (p.29 and p. 33-34 ) today (very very briefly at p. 35) Oscillating E and B fields today (p. 22) ~ today (in the limit c>>v A ) fast MHD wave (p. 23) https://en.wikipedia.org/wiki/waves_in_plasmas 27.3.2018 Space physics (ELEC-E4520) 36
More waves Take into account propagation to arbitrary propagation angle collisions plasma pressure (warm plasma model) => warm plasma waves ion motion detailed velocity distribution distribution function (Vlasov equation) non-linear behaviour 27.3.2018 Space physics (ELEC-E4520) 37
Koskinen&Kilpua More about the property of plasma waves Clemmow-Mullaly- Allis (CMA) diagram 27.3.2018 Space physics (ELEC-E4520) 38
END OF THE THEORY FOCUSED PART 27.3.2018 Space physics (ELEC-E4520) 39