ESS 200CESS 200CPulsations and Magnetohydrodynamic WavesPulsations and Magnetohydrodynamic WavesLecture 17Lecture 17• Physical systems respond to perturbations by emitting waves.• In a wave, some quantity must change in an oscillatory way, i.e., it must pulsate.– Sound waves are produced by changes in pressure.– The pressure changes or perturbations move away from the source. The pressure varies periodically in space and time.– The pressure perturbation travels through the atmosphere at the local sound speed, cs= (γp/ρ)1/2where γis the ratio of the specific heat at constant pressure to the specific heat at constant volume, p is the gas pressure and r is the gas mass density.  Think about what happens to the air pressure when you clap your hands. Why do you think we hear the sound?  Sound waves propagate in all directions. In air, the fronts from a localized pressure enhancement like an explosion are spherical.• Other kinds of waves are familiar, such as radio waves, light waves, etc.• These are electromagnetic waves. An electromagnetic wave in a vacuum or in a dielectric medium can be established by varying a current in an antenna.– In an electromagnetic wave the electric field and magnetic fieldare normal to the direction of propagation.– Sound waves are polarized in the direction of propagation ( k). This means that the varying quantities (electric and magnetic fields) are changing only in directions perpendicular to k.• Waves in a plasma can combine electromagnetic and pressure perturbations. – To understand waves in plasmas, one needs to consider various properties of the plasma including density, pressure, and B.• Magnetohydrodynamic waves are found at the low frequency end of the spectrum. Their frequencies are below the ion gyrofrequency and the plasma frequency.• The first observations of ultra-low-frequency (ULF) fluctuations (periods of seconds to minutes) were made from the ground in 1861. The fluctuations were seen in suspended compass needles or equivalent.• It would be nearly 100 years before their connection with space was realized.• Early investigators grouped the waves by wave form -quasi sinusoidal - called continuous pulsations- and non-sinusoidal –called irregular pulsations. Pc-1 Pc-2 Pc-3 Pc-4 Pc-5 Pi-1 Pi-2 T(s) 0.2-5 5-10 10-45 45-150 150-600 1-40 40-150 f 0.2-5 Hz 0.1-0.2 Hz 22-100 mHz 7-22 mHz 2-7 mHz 0.025-1 Hz2-25 mHz• Oscillations may be observed in the horizontal component of B at magnetometer stations at different latitudes on the ground (see right labels on lower portion of plot).• Waves may also be observed by a spacecraft in the near-equatorial magnetosphere (see traces in upper part of plot).• Dungey (1954) suggested that MHD waves in the outer atmosphere (magnetosphere) drive the pulsating magnetic fieldobserved on the surface.• His idea was that pulsations were waves standing on field lines and reflected at the ionospheric ends.• Flux tubes “wiggle”. This makes the ionospheric end of the field line bend, i.e., produces a horizontal perturbation of B at the ground.• For a plane wave propagating in the x-direction with wavelengthλand frequency f, the oscillating quantities can be taken to be proportional to sines and cosines.– For example, the pressure in a sound wave propagating along an organ pipe might vary likep = posin(kx-ωt) where the wave frequency is f (and ω= 2πf) k = 2π/λ is called the wave number. In general it can be a vector.λis the wavelength• A convenient way of working with sines and cosines is to use complex exponentials. The next chart gives you some of the properties of complex exponentials.• Define ei(kx-ωt)= cos(kx-ωt) + i sin (kx-ωt) where (i)2= -1• Rules of use:– in any equation with real and imaginary parts such as•Re(f(x)) + i Im(f(x)) = A + iB with A and B real,– the real parts must balance and the imaginary parts must balance. Thus:•Re(f(x)) = A and Im(f(x)) = B– ei(kx-ωt)= eikxe-iωt• Differentiation and integration are as for real functions: –The x-derivative of ei(kx-ωt)is ikei(kx-ωt)– The integral over x of ei(kx-ωt)is (1/ik)ei(kx-ωt)•The phase velocity specifies how fast a feature of a monochromatic wave such as one of its peaks is moving.– This can be determined by asking (for the pressure wave example previously given) “where is sin(kx-ωt) constant?”•sin(kx-ωt) = const. if x - xo= (ω/k)(t – t0), i.e. if you follow the wave feature at velocity vph= ω/k, you remain at the same position on the wave.• The second is called the group velocity. A wave can carry information, but this requires it to be formed from a finite range of frequencies or wave numbers and we shall assume that ω= ω(k). Information is propagated at the group velocity.–vg= ∂ω(k)/∂k,• The relation ω=ω(k) is called the dispersion relation and in the general case relates the frequency to the wave vector– When the dispersion relation shows asymptotic behavior towards a certain frequency, ωres, vggoes to zero and the wave no longer propagates and all the wave energy is fed into stationary oscillations. We call this a waveresonance.• Solutions– Usually this is appnroached by assuming that the system starts in equilibrium and that perturbations are small.• Assume uniform B0, perfect conductivity with equilibrium pressure p0 and mass density 0ρEEJJuubBBpppTTTTTTrrrrrrrrr===+=+=+=000ρρρ– Continuity–Momentum– Equation of state– Differentiate the momentum equation in time, use Faraday’s law and the ideal MHD conditionwhere )(0utr⋅∇−=∂∂ρρ))((1000bBpturrr×∇×−−∇=∂∂µρρρρ∇=∇∂∂=∇20)(sCpp0)))(((()()()(2220=××∇×∇×∇×+⋅∇∇−∂∂××∇=×∇−=∂∂AAsCuCuCtuBuEtbrrrrrrrrr20)(ρµ1BACrr=0BuErrr×−=• Assume a plane wave solution • Then the operators in the MHD equations become)](exp[~ trkiuω−⋅rrr×→∇×⋅→∇⋅→∇−→∂∂kikikiitrrrω• In our linear approximation solution to the MHD equations we obtained the equation. • If we write out the components we will have a set of homogeneous equations that have a solution only if the determinant of their coefficients vanishes.• This gives (with θthe angle between k and B)which has three (or six) solutions: