From 5820761b6d9f9a449e17845a28c46459f3ad6380 Mon Sep 17 00:00:00 2001 From: Beforerr Date: Wed, 17 Jan 2024 11:57:57 -0800 Subject: [PATCH] Update dynamo theory documentation --- .../journal_club/2024-01-17_dynamo/index.qmd | 113 ++++++++++++++++-- 1 file changed, 102 insertions(+), 11 deletions(-) diff --git a/docs/blog/journal_club/2024-01-17_dynamo/index.qmd b/docs/blog/journal_club/2024-01-17_dynamo/index.qmd index 6073b94..9a1c134 100644 --- a/docs/blog/journal_club/2024-01-17_dynamo/index.qmd +++ b/docs/blog/journal_club/2024-01-17_dynamo/index.qmd @@ -47,7 +47,7 @@ If you care about magnetic fields, you might care about dynamo theory. -## Takeaway phenomenological points +## Takeaway phenomenological points {.smaller} - Many astrophysical objects have global, ordered fields - **Differential rotation**, global **symmetries** and **geometry** are important @@ -56,6 +56,9 @@ If you care about magnetic fields, you might care about dynamo theory. - Lots of “small-scale”, random fields also discovered from the 70s - These come hand in hand with global magnetism - Simultaneous development of “small-scale dynamo” theory +- Astrophysical magnetism is in a nonlinear, saturated state + - Linear theory not the whole story (or using it requires non-trivial justification) + - Multiple scale interactions expected to be important ## MHD equations @@ -67,14 +70,10 @@ Incompressible, resistive, viscous MHD $$\frac{\partial \mathbf{u}}{\partial t}+\mathbf{u} \cdot \nabla \mathbf{u}=-\nabla P+\mathbf{B} \cdot \nabla \mathbf{B}+\nu \Delta \mathbf{u}+\mathbf{f}(\mathbf{x}, t)$$ -## Lenz's Law +#### Lenz's Law Electromagnetic induction suppresses rather enhances the seed magnetic field. -## - - - ## Anti-dynamo theorem {.smaller} Observations around solar activity minimum suggest that,the large-scale solar magnetic field is axisymmetric about the rotation axis. @@ -91,19 +90,111 @@ MHD induction equation $$\begin{gathered}\frac{\partial A}{\partial t}=\underbrace{\eta\left(\nabla^2-\frac{1}{\varpi^2}\right) A}_{\text {resistive decay }}-\underbrace{\frac{\mathbf{u}_{\mathrm{p}}}{\varpi} \cdot \nabla(\varpi A)}_{\text {transport }}, \\ \frac{\partial B}{\partial t}=\underbrace{\eta\left(\nabla^2-\frac{1}{\varpi^2}\right) B+\frac{1}{\varpi} \frac{\partial(\varpi B)}{\partial r} \frac{\partial \eta}{\partial r}}_{\text {resistive decay }}-\underbrace{\varpi \mathbf{u}_{\mathrm{p}} \cdot \nabla\left(\frac{B}{\varpi}\right)}_{\text {transport }} \\ -\underbrace{B \nabla \cdot \mathbf{u}_{\mathrm{p}}}_{\text {compression }}+\underbrace{\varpi\left(\nabla \times\left(A \hat{\mathbf{e}}_\phi\right)\right) \cdot \nabla \Omega}_{\text {shearing }} .\end{gathered}$$ -## Anti-dynamo theorem +## Anti-dynamo theorem {.smaller} + +#### Cowling’s theorem + +An axisymmetric flow cannot sustain an axisymmetric magnetic field against resistive decay. + +#### Zel’dovich’s theorem + +Planar, two-dimensional motions cannot excite a dynamo. + +#### Others + +- A purely toroidal flow cannot excite a dynamo +- A magnetic field of the form $B(x, y, t)$ alone cannot be a dynamo field. + +*A minimal geometric complexity is required for dynamos to work.* + +## From toroidal to poloidal + +We have no choice but to look for some fundamentally non-axisymmetric process to provide an additional source term in MHD induction equation. + +- Turbulence and mean-field electrodynamics +- The Babcock–Leighton mechanism +- Hydrodynamical and magnetohydrodynamical instabilities (from the rotational shear layer, *tachocline*) + +## Tension: Why Is Mean-Field Electrodynamics Working? {.smaller} + +Separating the flow and magnetic field into large-scale, slowly varying “mean” component $〈U〉, 〈B〉$ and small-scale rapidly varying “turbulent” components $\boldsymbol{u}, \boldsymbol{b}$ + +$$ +\begin{gathered} +\boldsymbol{U}=\langle\mathbf{U}\rangle + \boldsymbol{u}, \\ +\boldsymbol{B}=\langle\mathbf{B}\rangle + \boldsymbol{b} . +\end{gathered} +$$ + +Occasionally interpreted as a decomposition into axisymmetric and non-axisymmetric field components in systems with a rotation axis. + +$$ +\frac{\partial\langle\mathbf{B}\rangle}{\partial t}=\nabla \times(\langle\mathbf{U}\rangle \times\langle\mathbf{B}\rangle+\xi-\eta \nabla \times\langle\mathbf{B}\rangle) +$$ +where the mean electromotive force $\xi$ is given by the average of the small-scale flow-field cross-correlation: +$$ +\xi=\left\langle\mathbf{u} \times \mathbf{b}\right\rangle +$$ + +## Mean-Field Electrodynamics + +Closure is achieved by expanding this turbulent electromotive force (emf) $\boldsymbol{\xi}$ in terms of $\langle\mathbf{B}\rangle$ and its derivatives: + +$$ +\xi_i=a_{i j}\left\langle B_j\right\rangle+b_{i j k} \frac{\partial\left\langle B_j\right\rangle}{\partial x_k}+\cdots +$$ + +This is not a ***linearization*** procedure, in that we are **not** assuming that: + +$$\left|\boldsymbol{u}\right| /|\langle\boldsymbol{U}\rangle| \ll 1$$ + +$$\left|\boldsymbol{b}\right| /|\langle\boldsymbol{B}\rangle| \ll 1$$ + +## Tension: Why Is Mean-Field Electrodynamics Working? {.smaller} + +The challenge is now to compute these tensorial quantities from known statistical properties of the turbulent flow + +#### Tractable physical regimes: -Cowling’s theorem: an axisymmetric flow cannot sustain an axisymmetric magnetic field against resistive decay. +1. The energy density of the mean magnetic field is larger than the energy density of the small-scale field; +2. The magnetic Reynolds number is low; +3. The turbulent cyclonic eddies have a lifetime shorter than their characteristic turnover time. + -## Tension: Why Is Mean-Field Electrodynamics Working? + -## Tension: The Troublesome Magnetic Helicity +## Tension(s): From Solar to Stellar Dynamos + +#### Babcock–Leighton mechanism + +![](https://media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs41116-020-00025-6/MediaObjects/41116_2020_25_Fig3_HTML.png) -## Tension: The Troublesome Solar Differential Rotation +The net effect of BMRs is taking a formerly toroidal internal magnetic field and converting a fraction of its associated flux into a net surface dipole moment. + +::: {.notes} +The larger sunspot pairs (‘‘bipolar magnetic regions’’, hereafter BMR) often emerge with a systematic tilt with respect to the E–W direction, in that on average, the leading sunspot (with respect to the direction of solar rotation) is located at a lower latitude than the trailing sunspot, the more so the higher the latitude of the emerging BMR (see, e.g., Stenflo and Kosovichev 2012; McClintock and Norton 2013). This pattern is known as ‘‘Joy’s law’’. The tilt of the magnetic axis of a BMR implies a non-zero projection along the N–S direction, which amounts to a dipole moment. The decay of BMRs and subsequent dispersal of their magnetic flux by surface flows can release a fraction of this dipole moment and contribute to the global dipole. +::: ## Tension(s): From Solar to Stellar Dynamos +#### Hydrodynamical and magnetohydrodynamical instabilities + +![tachocline](https://upload.wikimedia.org/wikipedia/commons/4/47/Tachocline.svg) + +::: {.notes} +The tachocline is the rotational shear layer uncovered by helioseismology immediately beneath the Sun’s convective envelope, providing a smooth match between the latitudinal differential rotation of the envelope, and the rigidly rotating radiative core (see, e.g., Spiegel and Zahn 1992; Brown et al. 1989; Tomczyk et al. 1995; Gough and McIntyre 1998; Charbonneau et al. 1999, and references therein). +::: + +## Tension(s): From Solar to Stellar Dynamos {.smaller} + +1. Which is the primary polarity reversal mechanism: α-effect, or Babcock–Leighton, . . . or something else? +2. How do differential rotation and meridional circulation vary with rotation rate, luminosity, and internal structure? +3. How do turbulent coefficients (α-effect, turbulent pumping, turbulent diffusion) vary with rotation rate, luminosity, and internal structure? +4. How do sunspots and BMRs form and decay in stars of varying structure (in particularly, depth of convective envelope), rotation rate and luminosity? + +Unifying dynamo framework applicable to both the sun and solar type stars of varying spectral type, luminosity, and rotation rate. + ## References