1B11 Foundations of Astronomy Concepts of Magneto-HydroDynamics (MHD)

1. 1B11 Foundations of AstronomyConcepts of Magneto-HydroDynamics (MHD) Silvia Zane, Liz Puchnarewicz emp@mssl.ucl.ac.uk www.ucl.ac.uk/webct www.mssl.ucl.ac.uk/

2. 1B11 Introduction • To understand in detail the various space plasma and solar activity phenomena it is useful to recap some particular principles arising from MHD. • A plasma is a quasi-neutral gas consisting of positively and negatively charged particles (usually ions and electrons) which: • are subject to electric, magnetic and other forces, and • exhibit collective behaviour such as bulk motion, oscillations and instabilities. • Frozen-in flux approximation: a central tool to understand the behaviour of a plasma in presence of a B-field

3. 1B11 The Frozen-In Flux Approximation We start from the MHD induction equation, describing the evolution of a magnetic field in a plasma with conductivity and permeability 0 • The first term of the right hand side describes the behaviour (coupling) of the magnetic field with the plasma moving at velocity v • The second term on the right hand side represents diffusion of the magnetic field through the plasma.

4. 1B11 The magnetic Reynolds number If the scale length of the plasma is L, the gradient term is (approximately) The ratio RM between the two terms on the right hand side of the induction equation is RM= Magnetic Reynolds number • If RM<< 1 then the diffusion term dominates • If RM<< 1 then the coupling term dominates

5. 1B11 B from plasma flow • In a typical space plasma the conductivity  is very high, and the scale lengths, L, large • In the solar wind and the magnetosphere RM ~1011. Hence the diffusion term is negligible in these contests and the magnetic field convects exactly with the plasma flow. Or, the plasma particles are frozen with B and forced to move along the field lines. • This is often referred as “ideal MHD limit” or “frozen-in flux approximation”. • It is an extremely important concept since it allows us to study the evolution of the field, and particularly the topology of the field lines, by looking at the plasma flow.

6. n2 S2 n1 S1 1B11 Plasma flow from B Or course the concept can be reversed: if we know the topology of the magnetic field, then we know the plasma fluid flow. • A surface S1 in a plasma bounded by a closed contour C encloses a given amount of magnetic flux at time t1. • The surface may be subsequently deformed and/or relocated by motions of the plasma • However, under the frozen-in flux approx., the surface will enclose the same amount of magnetic flux Cat a later time t2 B-field C(t2) C(t1) Ex: If the surface is reduced in area we can infer the magnetic field strength is increased at t2.

7. n2 S2 n1 S1 1B11 Magnetic flux tube We can define a magnetic flux tube: by taking the closed loop and moving it parallel to the field it encloses. The surface, or tube S3, thus created has zero flux through it and consequently the fluid elements that form the flux tube at one moment, form the flux tube at all instants. B-field S3 Also: if two fluid elements P1 and P2 are originally linked by the field lines A and B, they will remain connected by field lines A and B whatever the individual motions V1 and V2 of the individual volumes. V P1 V P2 B B A A

8. 1B11 MHD forces The Magnetic Force FM in a MHD plasma is FM =j x B. Using the Maxwell equations: And after a bit of lengthy algebra: Rc is the local radius of curvature of the field line. It points towards the centre of curvature of the field line. Let us examine the two terms in the latter equation….

9. 1B11 MHD forces: two simple components The magnetic force can be resolved into two conceptually simple components: • A force perpendicular to the B-field which has the form of a pressure (it is the gradient of a scalar quantity B2/20), and • A force towards the instantaneous centre of curvature that depends on: i) radius of curvature and ii) field strength B. This is equivalent to a tension force acting along the field lines. Thus forcing the field lines together results in an opposing perpendicular pressure force, while trying to bend the field lines results in an opposing tension force.

10. 1B11 MHD waves This magnetic pressure and magnetic tension represent two kinds of restoring force that arise in a plasma in the presence of a magnetic field. They are associated to wave modes: waves and perturbations that propagate in the plasma. Most important ones: • Alfven waves • Magnetosonic wave modes, in which both the magnetic field strength and the plasma pressure vary.

11. 1B11 MHD waves The Alfven wave is a very important one. It is entirely due to the tension force associated with the magnetic field. • It is essentially a magnetic wave, as there is no associated compression of the plasma as in the case of sonic (pressure) waves. • It propagates preferentially along the field direction (and not across it) with speed: This wave causes magnetic field and plasma velocity perturbations which are perpendicular to the background field (and to the wave propagation vector) and thus is sometimes called transverse wave or shear wave. It is analogous to waves on a string under tension (guitar strings!)

12. 1B11 Magnetic reconnection • The concept of magnetic reconnection is key in understanding the coupling between solar wind and planetary atmospheres (auroras), as well as acceleration of particles in space and astrophysical plasmas (solar flares, etc..) • The frozen-in flux approximation (in the case of RM>>1) leads to a picture of space plasmas contained within separated regions. • For example, field and plasma from the Sun (slar winds,..) are frozen-out of the region occupied by a planetary magnetic field.

13. 1B11 Magnetic reconnection • These “separate” plasma cells are partitioned by thin current sheets, which support the change in magnetic fields across the boundary. Recall: However, exactly because of their thinness (small spatial lengths!), the magnetic Reynolds number within the current sheet may be relatively small: diffusion of the magnetic field through the plasma start to be important!

14. j  Vin E    Diffusion   B field 1B11 Magnetic reconnection If there is a strong B-field gradient and the fields on either side of the gradient are anti-parallel, then diffusion of the field at the gradient can led to a loss of total magnetic flux this situation is called magnetic annihilation. Field lines are convected into the diffusion region and merge with field lines with opposite orientation (which originally where on the other side of the gradient)

15. j  Vin  Diffusion B 1B11 Magnetic reconnection Vout ~ V Alfven The resulting “reconnected lines” are sharply bent through the current sheet. Magnetic tension forces associated with these bent lines accelerate the plasma along the current sheet and away from the diffusion region on each side. The simplest magnetic reconnection geometry. Anti-parallel field lines are separated by a thin current sheet (light grey) across which the field reverses. Due to the small scale lengths, the frozen-in flux approximation breaks down  magnetic flux diffuse from both sides. The field reconnects to form 2 hairpin like field lines, which rapidly contract away from the neutral “X”-point. Outflows jets of plasma are formed, also moving away from the “X”-point, on both sides.

16. 1B11 Magnetic reconnection Magnetic reconnection is an extremely important process: • It allows the two sides of the field gradient to be linked by the newly reconnected line. • It allows plasmas from either side to flow along the field and mix with those from other side. • Also, magnetic energy is continuously liberated in the process, causing accelerated and heated plasma flows.