Dynamos in Stably Stratified Fluids

Abstract

While magnetic fields are thought to play a dominant role in transport processes in stellar radiative zones, present theory of radiative zone dynamos and even properties of stellar stably stratified turbulence is fragmented and incomplete. This thesis examines the fundamentals of small-scale and large-scale dynamo theory in the conditions of a stellar radiative zone: turbulence, stable stratification, and mean shear. The simultaneous operation of several effects and instabilities requires sequentially examining fluid dynamical systems of increasing complexity. A stably stratified fluid forced on a horizontal length $L$ and velocity scale $U\,$ leads to turbulence with emergent outer vertical scales and energy cascade set by the physical parameters of the fluid \{$N$, $\nu$, $\kappa$\}. These are the Brunt-V\"ais\"al\"a frequency, viscosity, and thermal diffusivity, respectively. From dimensional analysis, only three dimensionless parameters characterize the fluid: the Reynolds number $Re=UL/\nu$, the Froude number $Fr=U/NL$, and the Prandtl number $Pr=\nu/\kappa$. Understanding the scaling of the emergent outer vertical scales, the structure of the subsequent anisotropic energy cascade (see an artist's concept in Figure \ref{fig:ArtistConcept}), and possible dynamo instability is an important theoretical goal. We first develop a new theory for the anisotropic energy cascade of hydrodynamic, stably stratified turbulence in the astrophysical regime of asymptotically small Prandtl number, $Pr$. Dominant balance arguments suggest that as $Pr$ is decreased from $Pr=O(1)$ in the geophysical regime, a transition in turbulence regimes occurs when $Pr<Rb^{-1}$, where $Rb=ReFr^2$ is the buoyancy Reynolds number. This signals a shift to a regime where the vertical thermal diffusion becomes important or, equivalently, when the thermal scale becomes larger than the Ozmidov scale (the scale below which turbulence is isotropic and buoyancy effects are negligible). Critical balance arguments between linear wave and non-linear timescales provide a prediction for the scale-by-scale anisotropy of the energy cascade in the two regimes by using the appropriate asymptotic limits of the full internal gravity wave dispersion. We find that the scaling relations simply carry over from the geophysical to the astrophysical regime if the Froude number is replaced by a modified Froude number ($Fr\rightarrow Fr/(PrRb)^{1/4}\equiv Fr_M$). The qualitative effect of decreasing $Pr$ is to reduce the range of the large-scale anisotropic cascade and increase the range of the isotropic cascade, now controlled by a modified Ozmidov scale. This generally agrees with the intuition that a higher thermal diffusivity ameliorates the effects of stable stratification. Next, the possibility that the hydrodynamic stably stratified turbulence is unstable to the small-scale dynamo (SSD) is examined. An unstable SSD would amplify and sustain dynamically-important magnetic fields on length scales \textit{smaller} than characteristic hydrodynamic turbulent scales. A combination of theoretical considerations and simulations strongly suggest that the new SSD instability criterion is set by the magnetic buoyancy Reynolds number, $Rb_m\equiv PmRb_{(M)}$ (depending on $Pr$), which controls the scale separation of the resistive scale and the outer scale of the isotropic portion of the energy cascade (i.e. the Ozmidov or modified Ozmidov scale depending on $Pr$). Estimation of $Rb_m$ in stellar parameter regimes suggests that the SSD is plausibly active in stellar radiative zones despite the generally deleterious effect of stable stratification on the efficiency of dynamo action. Lastly, the possibility of a non-helical large-scale dynamo (LSD) instability in stably stratified MHD turbulence is examined in the additional presence of a mean shear in a Cartesian setting, modeling the local effect of radial differential rotation (DR). An unstable LSD would amplify and sustain dynamically-important magnetic fields on length scales \textit{larger} than the characteristic hydrodynamic turbulent scales. A combination of simulations and analytical calculations suggests that a non-helical LSD, driven by shear and magnetic fluctuations through the magnetic-shear current effect, is robust to stable stratification as long as the SSD remains unstable. The main takeaway from this thesis is that both small and large scale magnetic fields can, in principle, spontaneously emerge from the local stably stratified turbulence of a differentially rotating radiative zone. The ubiquitous horizontal shear instabilities of latitudinal DR provide stably stratified turbulence for the SSD while the radial DR provides a stable shear for operation of a LSD. These fields themselves may affect transport, but also could help kick-off further instabilities or global dynamos in the additional presence of rotation and spherical geometry, such as the Tayler-Spruit dynamo. Incorporation of these important effects to dynamo theory in stably stratified fluids is left for the future.