Configuration of magnetic field lines for fast and slow shocks. The lines are closer together for a fast shock, indicating thaFlow streamlines and velocity magnitude in the magnetosheath. These are results from a global magnetohydrodynamic simulation oESS 200CESS 200CESS 200CLecture 8Lecture 8Lecture 8The Bow Shock and The Bow Shock and The Bow Shock and MagnetosheathMagnetosheathMagnetosheath• A shock is a discontinuity separating two different regimes in acontinuous media.– Shocks form when velocities exceed the signal speed in the medium.– A shock front separates the Mach cone of a supersonic jet from the undisturbed air.• Characteristics of a shock :– The disturbance propagates faster than the signal speed. In gas the signal speed is the speed of sound, in space plasmas the signal speeds are the MHD wave speeds.– At the shock front the properties of the medium change abruptly. In a hydrodynamic shock, the pressure and density increase while in a MHD shock the plasma density and magnetic field strength increase.– Behind a shock front a transition back to the undisturbed medium must occur. Behind a gas-dynamic shock, density and pressure decrease, behind a MHD shock the plasma density and magnetic field strength decrease. If the decrease is fast a reverse shock occurs. • A shock can be thought of as a non-linear wave propagating faster than the signal speed.– Information can be transferred by a propagating disturbance.– Shocks can be from a blast wave - waves generated in the corona.– Shocks can be driven by an object moving faster than the speed of sound.• Shocks can form when an obstacle moves with respect to the unshocked gas.• Shocks can form when a gas encounters an obstacle.• The Shock’s Rest Frame– In a frame moving with the shock the gas with the larger speed is on the left and gas with a smaller speed is on the right.– At the shock front irreversible processes lead the the compression of the gas and a change in speed.– The low-entropy upstream side has high velocity.– The high-entropy downstream side has smaller velocity.• Collisionless Shock Waves– In a gas-dynamic shock collisions provide the required dissipation.– In space plasmas the shocks are collision free. Microscopic Kinetic effects provide the dissipation. The magnetic field acts as a coupling device. MHD can be used to show how the bulk parameters change across the shock. Shock FrontUpstream(low entropy)Downstream(high entropy)vuvd• Shock Conservation Laws– In both fluid dynamics and MHD conservation equations for mass, energy and momentum have the form: where Q and are the density and flux of the conserved quantity. – If the shock is steady ( ) and one-dimensional or where u and d refer to upstream and downstream and is the unit normal to the shock surface. We normally write this as a jump condition .– Conservation of Mass or . If the shock slows the plasma then the plasma density increases. – Conservation of Momentum where the first term is the rate of change of momentum and the second and third terms are the gradients of the gas and magnetic pressure in the normal direction.0=⋅∇+∂∂FtQrFr0=∂∂t1=∂∂nFn0ˆ)( =⋅− nFFdurrnˆ0][=nF0)( =∂∂nvnρ0][=nvρ0202=⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂µρBnnpnvvnn02022=⎥⎦⎤⎢⎣⎡++µρBpvn– Conservation of momentum . The subscript t refers to components that are transverse to the shock (i.e. parallel to the shock surface). – Conservation of energyThere we have used The first two terms are the flux of kinetic energy (flow energy and internal energy) while the last two terms come from the electromagnetic energy flux– Gauss Law gives – Faraday’s Law gives 00=⎥⎦⎤⎢⎣⎡−tntnBBvvrrµρ01002221=⎥⎥⎦⎤⎢⎢⎣⎡⋅−+⎟⎟⎠⎞⎜⎜⎝⎛−+µµργγρnnnBBvBvpvvrr0µBErr×0=⋅∇ BrtBE ∂∂−=×∇rr[]0=nB[]0=−tntnvBBvrr.constp =−γρ• The jump conditions are a set of 6 equations. If we want to findthe downstream quantities given the upstream quantities then there are 6 unknowns ( ,vn,,vt,p,Bn,Bt).• The solutions to these equations are not necessarily shocks. These are conservation laws and a multitude of other discontinuities can also be described by these equations.ρTypes of Discontinuities in Ideal MHDContact Discontinuity,Density jumps arbitrary, all others continuous. No plasma flow. Both sides flow together at vt.Tangential Discontinuity,Complete separation. Plasma pressure and field change arbitrarily, but pressure balanceRotational Discontinuity,Large amplitude intermediate wave, field and flow change direction but not magnitude.0≠nB0=nv0=nv0=nB()210ρµnnBv =0≠nv0≠nBTypes of Shocks in Ideal MHDShock WavesFlow crosses surface of discontinuity accompanied by compression.Parallel Shock ( along shock normal)B unchanged by shock.Perpendicular ShockP and B increase at shockOblique Shocks Fast ShockP, and B increase, B bends away from normalSlow ShockP increases, B decreases, B bends toward normal.Intermediate ShockB rotates 1800 in shock plane, density jump in anisotropic case0≠nv0=tB0=nB0,0≠≠ntBBBr•Configuration of magnetic field lines for fast and slow shocks. The lines are closer together for a fast shock, indicating that the field strength increases.• Quasi-perpendicular and quasi-parallel shocks.– Call the angle between and the normal θBn .– Quasi-perpendicular shocks have θBn> 450 and quasi-parallel have θBn< 450.–.Perpendicular shocks are sharper and more laminar.– Parallel shocks are highly turbulent.– The reason for this is that perpendicular shocks constrain the waves to the shock plane while parallel shocks allow waves to leak out along the magnetic field– In these examples of the Earth’s bow shock – N is in the normal direction, L is northward and M is azimuthal.Br• Examples of the change in plasma parameters across the bow shock– The solar wind is super-magnetosonic so the purpose of the shock is to slow the solar wind down so the flow can go around the obstacle. – The density and temperature increase.– The magnetic field (not shown) also increases.–The maximum compression at a strong shock is 4 but 2 is more typical.• Particles can be