SW103: Lecture 2. Magnetohydrodynamics and MHD models

Transcription

1 SW103: Lecture 2 Magnetohydrodynamics and MHD models

2 Scale sizes in the Solar Terrestrial System: or why we use MagnetoHydroDynamics Sun-Earth distance = 1 Astronomical Unit (AU) 200 R Sun 20,000 R E 1 AU = 150,000,000 km 1 R Sun = 700,000 km 1 R E = 6400 km Physics-based modeling of the solar terrestrial environment involves solving the basic equations numerically on a 3-D grid of points. A 100 x 100 x 100 grid contains 10 6 grid points Doubling the resolution increases computation by a factor of 2 4 = 16 With current computer capabilities, it s impossible to solve the heliosphere with greater than a fraction of an R Sun resolution or the magnetosphere with greater than a fraction of an R E resolution. These distances are still >> the largest plasma scale lengths. (The ion gyroradius (v th /W) and ion inertial length (c/w pi = V A /W) are typically of order or less than 100 km for thermal ions.) Thus MHD is the most viable approach of modeling solar system plasmas on the global scale.

3 Global MHD Models On the large scale, MHD works very well everywhere except in boundary layers. Boundary layers are thin surfaces or layers often separating regions of different plasma, and usually containing a current sheet. Examples: Magnetopause Shock wave Heliospheric or tail current sheet Non-MHD processes (e.g. magnetic reconnection) occur when boundary layer thicknesses approach ion scale lengths ( 1000 km). Boundary layers are usually not well resolved in MHD grids. Fortunately, processes in boundary layers important on the global scale are governed more by global boundary conditions than by the kinetic physics processes occurring in the boundary layer itself.

4 The Equations of Hydrodynamics Three variables: mass density, r, flow velocity, u, pressure, p. 1. Continuity Equation conservation of mass scalar equation for mass density, r Change in density is given by net flow of mass into or out of a volume. 2. Momentum Equation conservation of momentum or Newton s 2 nd Law vector equation for velocity, u Change in momentum density = mass x acceleration/volume = force/volume 3. Energy Equation conservation of energy The simplest assumption is adiabatic flow, no change in internal energy of the fluid scalar equation for pressure, p Can use more complicated equation if energy transport is important. Temperature, T, is got from p and r using kt = pm H /r

5 Equations of Ideal Magnetohydrodynamics (MHD) 1. Continuity Equation 2. Momentum Equation 3. Adiabatic Equation 4. Ampere s Law 5. Faraday s Law 6. Ideal Ohm s Law By substituting for j and E as indicated by the arrows, this set can be reduced to 4 equations: an equation for each of r, p, u and B. Then j, E, and T can be obtained as secondary variables from these.

6 Equations of Ideal Magnetohydrodynamics cont Substituting for j and E reduces the set to 4 equations for 4 unknowns: r, u, p, and B. 1. Continuity Equation 2. Momentum Equation 3. Adiabatic Equation 4. Magnetic Induction Equation These equations can be put into different forms to solve numerically, but the essential physics is here in this form. Then j, E, and T are obtained later if needed from

7 Plasma Approximation Displacement current Approximations n i = n e = n Plasmas remain charge neutral to a very good approximation on length scales larger than the Debye length (very small). Otherwise VERY large electric fields would result. As a result, i.e., no space charge can accumulate. We ignored the displacement current in Ampere s Law This is fine provided as Pressure In general pressure is a 3x3 tensor not a scalar. Off-diagonal elements of the tensor give rise to viscosity usually not important. Plasmas often have different velocity distributions parallel and perpendicular to the magnetic field, i.e. anisotropic temperature or pressure: p p This is usually not included in MHD, but it can be.

8 Hydromagnetic Approximation The hydromagnetic approximation assumes These are the two largest terms in the Generalized Ohm s Law Resistive term due to collisions Hall term Ambipolar term Electron inertial term For length and time scales much larger than plasma length and time scales, all these additional terms are small. Ignoring them all is known as the hydromagnetic approximation or the hydromagnetic limit. However on sub-grid scales they can be important. In particular magnetic reconnection requires at least one of these term to be non-zero.

9 Dissipation in real plasmas and numerical solutions In the ideal MHD equations there is no dissipation of energy or any heating terms the plasma is treated as adiabatic. Thus, electromagnetic energy can be converted into flow kinetic energy, or vice versa, but neither can be converted into heat. The plasma can be heated by compression but this is adiabatic and reversible. In Real Plasmas: In a real plasma, the scattering of particles in velocity space provides dissipation. These are kinetic processes not included in the MHD formulation. In Numerical Solutions: In numerical solutions, approximations caused by representing differential equations as difference equations on a finite grid introduces numerical dissipation. This can appear as diffusion, viscosity, resistivity, etc, depending on the context. In particular, numerical dissipation allows magnetic reconnection to happen in an ideal MHD calculation.

10 Magnetic Reconnection A local (very local on electron-length scale) breakdown of frozen-in-flux condition Microscale process with Macroscale consequences. Allows plasma on separate field lines to end-up on the same field line. Converts magnetic energy into thermal or flow (i.e., mechanical) energy. Changes magnetic field line connectivity or topology.

11 2D Magnetic Reconnection Occurs when antiparallel magnetic field lines break (disconnect) at an X point and reconnect with new partners Converts magnetic energy into kinetic energy Results in new topological configurations Solar flares provided the first observational evidence for the reconnection process Plays a critical role in solar flares, in CME release from the Sun, and in geomagnetic storms at Earth.

12 Stop and Think Which of the following could be the result of magnetic reconnection? 1. Two very different plasma populations (temp, composition, density, etc) at the same location. 2. Two different plasmas flowing in opposite directions along a magnetic field line. 3. Fast plasma flows 4. All of the above 5. None of the above

13 More on the momentum equation 1. Since we consider velocity at a point in space and time rather than follow the velocity of a parcel of fluid, we need to use the Lagrangian derivative that includes the convective term,, which is how velocity changes in the direction of the flow. 2. Each term in the momentum equation has dimensions of u 2 /L LHS RHS Thus the importance of each term can be judged by the relative sizes of these 4 velocities. For u << v th, V A, the terms (forces) on the RHS must nearly balance.

14 Hydromagnetic Equilibrium Momentum equation: If u = 0 (no flow) and gravity is negligible this reduces to hydromagnetic equilibrium. pressure gradients balance magnetic forces (Provided flows are slow (subsonic and sub-alfvenic) and gravity weak, i.e., u, v escape << v th, V A, this remains approximately true, ) Now the magnetic term can be rewritten gradient of magnetic pressure magnetic curvature force Thus hydromagnetic equilibrium can can be written B 2 /2m 0 is magnetic energy density. It can be thought of as magnetic pressure The magnetic curvature force can be thought of as a tension that tries to straighten curved field lines i.e., total pressure (thermal + magnetic) is constant unless field lines are curved.

15 Waves in Fluids What wave modes are present in a fluid depends on what forces are included on the RHS of the momentum equation. sound waves atmospheric gravity waves 3 modes of MHD wave: fast, shear, slow cold plasma (p = 0), Alfven and fast waves