Update dynamo theory documentation

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: The Troublesome Magnetic Helicity -->
 
-## Tension: Why Is Mean-Field Electrodynamics Working?
+<!-- ## Tension: The Troublesome Solar Differential Rotation -->
 
-## 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
 
 <!-- Charbonneau, P., & Sokoloff, D. (2023). Evolution of Solar and Stellar Dynamo Theory. Space Science Reviews, 219(5), 35. https://doi.org/10.1007/s11214-023-00980-0 -->