Looking beyond MHD: Implications for the plasma Universe Manasvi Lingam Institute for Theory and Computation Harvard University May 21, 2018 Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 1 / 38
Overview Introduction and Motivation 1 A short introduction to MHD and “beyond MHD” models 2 The Hall effect in dynamo theory 3 Extended MHD turbulence and the solar wind 4 A brief coda on magnetogenesis 5 Conclusions 6 Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 2 / 38
Introduction and Motivation Introduction and Motivation 1 A short introduction to MHD and “beyond MHD” models 2 The Hall effect in dynamo theory 3 Extended MHD turbulence and the solar wind 4 A brief coda on magnetogenesis 5 Conclusions 6 Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 3 / 38
Introduction and Motivation Introduction The statement that 99% of the visible Universe is comprised of plasma is well-known. Astrophysical plasmas range from the tenuous intergalactic medium and interstellar medium to the interiors of stars. Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 4 / 38
Introduction and Motivation Introduction The statement that 99% of the visible Universe is comprised of plasma is well-known. Astrophysical plasmas range from the tenuous intergalactic medium and interstellar medium to the interiors of stars. Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 4 / 38
Introduction and Motivation Motivation From the previous figure, it is apparent that there exists considerable variation in plasma parameters (e.g. number density, temperature) because of the diverse array of astrophysical environments under consideration. Hence, using a one-model-fits-all approach without due caution can be dangerous since several astrophysical systems can lie outside any particular model’s domain of validity. In this talk, I will delineate a couple of simple plasma fluid models that possess a broader domain of validity compared to magnetohydrodynamics. The ensuing ramifications of using these models in the context of astrophysical mechanisms/systems will then be outlined. Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 5 / 38
A short introduction to MHD and “beyond MHD” models Introduction and Motivation 1 A short introduction to MHD and “beyond MHD” models 2 The Hall effect in dynamo theory 3 Extended MHD turbulence and the solar wind 4 A brief coda on magnetogenesis 5 Conclusions 6 Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 6 / 38
A short introduction to MHD and “beyond MHD” models Magnetohydrodynamics The most widely used plasma model in astrophysics is ideal magnetohydrodynamics (MHD). MHD is a fluid model with the following dynamical equations: ∂ρ ∂ t + ∇ · ( ρ V ) = 0 , (1) � ∂ V � ρ ∂ t + V · ∇ V = −∇ p + J × B (2) ∂ B ∂ t − ∇ × ( V × B ) = 0 , (3) where p is the total pressure and µ 0 J = ∇ × B . Here, ρ , V and B are the mass density, velocity and magnetic field respectively. Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 7 / 38
A short introduction to MHD and “beyond MHD” models Looking beyond MHD Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 8 / 38
A short introduction to MHD and “beyond MHD” models Looking beyond MHD Plasma physics is replete with fluid models that have a broader scope than MHD. In this talk, I will focus on a class of models that include ideal MHD as a limiting case. One of the chief disadvantages of ideal MHD is that it becomes inaccurate when the characteristic length scale L becomes comparable to (or smaller than) the ion and electron skin depths, denoted by d i = c /ω pi and d e = c /ω pe respectively; here, ω ps is the plasma frequency of species s . The “beyond MHD” models presented here include either or both of these length scales and are therefore valid over a broader domain compared to MHD. Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 8 / 38
A short introduction to MHD and “beyond MHD” models Hall MHD Hall MHD is the simplest of the “beyond MHD” models and has been invoked in a wide range of astrophysical environments ranging from protoplanetary discs and the solar wind to neutron star crusts and planetary magnetospheres. Hall MHD has the same governing equations as ideal MHD, except for a different Ohm’s law. The barotropic Hall MHD Ohm’s law is �� � � ∂ B V − J ∂ t − ∇ × × B = 0 . (4) ne The Hall drift, i.e. the J × B term in (4), encapsulates the fact that the electron and ion fluid velocities are not necessarily comparable in Hall MHD. Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 9 / 38
A short introduction to MHD and “beyond MHD” models Extended MHD In Hall MHD, the electrons are effectively assumed to be massless. The inclusion of finite electron inertia leads to extended MHD: ∂ρ ∂ t + ∇ · ( ρ V ) = 0 , (5) � ∂ V � � J � = −∇ p + J × B − m e ρ ∂ t + V · ∇ V e 2 J · ∇ , (6) n J × B − ∇ p e E + V × B = en � ∂ J � �� m e VJ + JV − 1 + ∂ t + ∇ · en JJ , (7) ne 2 where n is the number density, and m e ( m i ) is the electron (ion) mass. Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 10 / 38
A short introduction to MHD and “beyond MHD” models Extended MHD We can also rewrite (6) and (7): � � h + V 2 + ( ∇ × B ) × B ⋆ ∂ V ∂ t + ( ∇ × V ) × V = −∇ 2 ρ � � ( ∇ × B ) 2 d 2 − e ∇ , (8) 2 ρ 2 � ( ∇ × B ) × B ⋆ � ∂ B ⋆ ∇ × ( V × B ⋆ ) − d i ∇ × = ∂ t ρ � ( ∇ × B ) × ( ∇ × V ) � d 2 + e ∇ × , (9) ρ for the barotropic case in Alfv´ enic units, with d i and d e denoting the normalized skin depths. The dynamical variable B ⋆ is defined as � ∇ × B � B ⋆ = B + d 2 e ∇ × , (10) ρ Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 11 / 38
The Hall effect in dynamo theory Introduction and Motivation 1 A short introduction to MHD and “beyond MHD” models 2 The Hall effect in dynamo theory 3 Extended MHD turbulence and the solar wind 4 A brief coda on magnetogenesis 5 Conclusions 6 Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 12 / 38
The Hall effect in dynamo theory Magnetic fields and the dynamo mechanism [ Astrophys. J. , 829 , 51 (2016)] Magnetic fields are ubiquitous in astrophysics. They have been documented in stars, accretion discs, compact objects, galaxies and clusters of galaxies. The dynamo mechanism is widely invoked to explain the observed magnetic fields. Although dynamo theory has made considerable advances, it has typically relied upon resistive MHD (ideal MHD + resistivity) as the base physical model. Since the Hall effect has been argued to be important in several astrophysical settings, it is instructive to ask how it can affect the generation of these magnetic fields. Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 13 / 38
The Hall effect in dynamo theory Incompressible Hall MHD [ Astrophys. J. , 829 , 51 (2016)] Let us recall the equations of resistive incompressible Hall MHD. The dynamical equation for the velocity is � ∂ � � p � ∂ t + V · ∇ V = ( ∇ × B ) × B − ∇ , (11) ρ which is the same as that of ideal MHD. The Ohm’s law for the model is ∂ B ∂ t = ∇ × ( V E × B ) − η ∇ × J , (12) where J := ∇ × B , V E = V − d i J , η is the resistivity, and d i is the ion skin depth. The J × B term in the Ohm’s law of Hall MHD. Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 14 / 38
The Hall effect in dynamo theory Mean-field dynamo theory [ Astrophys. J. , 829 , 51 (2016)] Each field is decomposed into a mean-field (large-scale) component and a fluctuating (turbulent) component, i.e. X = X 0 + x for a given field x ∈ ( v , b ). In our simple model, large-scale velocity is taken to be negligible. The large-scale magnetic field evolves via ∂ B 0 = ∇ × E − d i ∇ × ( J 0 × B 0 ) − η ∇ × J 0 . (13) ∂ t The electromotive force E is E = � ( v − d i ∇ × b ) × b � , (14) and must be specified in terms of B 0 . Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 15 / 38
The Hall effect in dynamo theory Mean-field dynamo theory [ Astrophys. J. , 829 , 51 (2016)] As per the above assumptions, the electromotive force is expanded as E = α B 0 − β ∇ × B 0 + . . . , (15) and the goal of the classical dynamo theory is to compute the coefficients α and β . From (14), the EMF can be approximated as � = τ c � ∂ t v × b � + � v × ∂ t b � − d i � ( ∇ × b ) × ∂ t b � E � − d i � ( ∇ × ∂ t b ) × b � , (16) where τ c is the correlation time. Observe that the first term depends on the velocity evolution equation, whilst the last three do not. Manasvi Lingam (Harvard) Looking beyond MHD May 21, 2018 16 / 38
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