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An overview of my PhD thesis

A PDF version of my thesis can be downloaded from Zenodo.

Chapter 1 is introductory, while chapter 13 states the main results of this thesis. The rest of the chapters are grouped into six parts:

Before describing them, here is a list of papers based on my thesis:

  1. 2502.17178 (based on chapters 2 and 6)
  2. Gopalakrishnan and Singh (2023). Mean-field dynamo due to spatio-temporal fluctuations of the turbulent kinetic energy. Journal of Fluid Mechanics, 973, A29. doi: 10.1017/jfm.2023.765. (based on chapter 3)
  3. 2502.02946 (based on chapter 4)
  4. Gopalakrishnan and Singh (2024). Small-scale dynamo with nonzero correlation time. The Astrophysical Journal, 970(1), 64. doi: 10.3847/1538-4357/ad4ee4. (based on chapter 5)
  5. Kishore and Singh (2025). The spectra of solar magnetic energy and helicity. The Astrophysical Journal, 986(2), 183. doi: 10.3847/1538-4357/add692. (based on chapter 7)
  6. 2409.14840 (based on chapter 12)

Part I: Turbulence

Chapter 2 deals with the generation of anisotropy in rotating forced turbulence. While this chapter is not directly related to convection, it will be useful in interpreting the more complicated simulations discussed in a later chapter (chapter 6). The main result of this chapter is that in solenoidally forced rotating turbulence, rotational destabilization of vortices can lead to one-dimensionalization of the flow below the forcing scale, with the velocity component along the rotational axis being larger than the other two components.

Part II: Kinematic dynamo

In general, the Lorentz force turns the evolution of the magnetic field into a nonlinear problem, which is difficult to study analytically. As a first step, one can study the kinematic limit, where the magnetic field is assumed to be so weak that the effect of the Lorentz force on the velocity field can be neglected. The statistical properties of the velocity field can then be treated as given quantities, and one is interested in the statistical properties of the magnetic field. This part of the thesis is devoted to the study of the kinematic dynamo.

Magnetic fields are often correlated at length scales much larger than that of the turbulent velocity field. Mean-field magnetohydrodynamics takes advantage of this scale-separation to make the problem analytically tractable (Moffatt 1978, Krause and Rädler 1980). Large-scale dynamo, or LSD, refers to the generation of magnetic fields ordered on the system scale by fluid motions correlated at much smaller scales.

The statistical properties of the velocity field are often treated as deterministic quantities in kinematic studies. Chapter 3 presents a simple calculation which shows that fluctuations of the turbulent kinetic energy at a particular spatial scale can induce the growth of a magnetic field at much larger scales.

The simplest treatments of the LSD are valid when the correlation time of the velocity field is zero. Chapter 4 relaxes this assumption. As a toy problem, we first examine the diffusion of a passive scalar. We then apply the same technique to the evolution equation for the averaged magnetic field. A previous attempt to solve this problem (Nicklaus and Stix 1988) turns out to have predicted a turbulent diffusivity for the magnetic field that is the same as that for a passive scalar. We find additional contributions to the turbulent diffusion of the magnetic field.

It is well known that in a turbulent fluid, intermittent magnetic fields can be generated which are typically ordered on scales comparable to or smaller than that of the velocity field (the small-scale dynamo or SSD) (Kazantsev 1968, Molchanov et al. 1985). The SSD grows on a timescale comparable to the eddy turnover time; this is much smaller than the timescale for growth of the LSD and the typical ages of astrophysical objects. Kulsrud and Anderson (1992) argue that the presence of SSD-generated magnetic fields invalidates the usual treatment of the LSD. While their conclusion has since been challenged (e.g. Subramanian 1998), magnetic fields generated by the SSD are still expected to affect the evolution of any object that contains a sufficiently turbulent plasma (i.e. where the magnetic Reynolds number, Rm\Rm, is above some critical value).

Intermittency leads to the higher moments of the magnetic field growing faster than would be expected from the growth rates of the lower moments (i.e. <Bn>\meanBr{B^n} growing faster than does <B>n\meanBr{B}^n). To study the SSD, one thus needs to consider second- or higher-order moments. In chapter 5, we use the technique described in chapter 4 to find out how the correlation time of the velocity field affects the evolution of the two-point correlation of the magnetic field.

Part III: Nonlinear dynamo

We earlier mentioned that the magnetic activity of the Sun depends on the latitude. In the CZ (convection zone), the flows themselves are anisotropic, with the vertical component being larger than the horizontal components. The rotation of the Sun is another source of anisotropy for the flows. The angle between the direction of gravity and the rotational axis is dependent on the latitude. In chapter 6, we use simulations of anisotropically forced turbulence in rotating domains (with the direction of anisotropy allowed to be different from the rotational axis) to find out how the growth rate and the saturation level of the SSD are affected. The main result of this chapter is that at high Re\Rey, the saturation level of the SSD is suppressed much more strongly by rotation than the growth rate.

Part IV: Solar magnetic fields

The preceding two parts dealt with dynamos. It is now natural to ask whether one can, using solar observations, distinguish between different dynamo mechanisms. In this context, the spatial and spectral distribution of the magnetic helicity is of interest. In chapter 7 we describe major discrepancies between results from different telescopes, suggesting that one needs to be extremely careful while interpreting current observations of photospheric magnetic fields.

Part V: Convection and plumes

Chapter 8 studies the evolution of a cool blob of fluid (a thermal) descending in a stably stratified background. The motivation here is to understand overshoot by using semi-analytic models for the evolution of a thermal. We find a scaling relation between the initial size of the thermal and the depth of its penetration. We also show that assumptions about the internal structure of the thermal have a much stronger effect on its dynamics than expected.

Supposing one understands the dynamics of isolated blobs of fluid, one then has to construct additional models for

Given the vast differences between experimentally accessible regimes and stellar regimes, three-dimensional numerical simulations are indispensable to motivate such models (or, for that matter, to guide the construction of any theory of stellar convection). One of the challenges in solving the convective conundrum is the limited parameter regime that is computationally accessible. Chapter 9 constructs a shell model for stratified convection to qualitatively study the effects of this restriction. We find indications that the convective conundrum may be explainable without appealing to rotation or magnetic fields, and that the nature of convection is sensitive to the thermal Prandtl number.

We earlier mentioned that the conventional picture of sunspot formation is starting to be questioned. One factor in favour of the conventional picture is that it readily explains why sunspots preferentially form in a certain range of active latitudes. Chapter 10 uses simulations of rotating convection to examine latitudinal effects on magnetic cycles. The goal of such a study is to understand if alternative mechanisms also allow the existence of active latitudes on the Sun. We indeed find that at certain latitudes of our setup, magnetic cycles are excited which seem to depend on the existence of an SSD.

Part VI: Helioseismology

The Sun supports a wide variety of waves. Depending on the nature of the restoring force, the corresponding modes (Leibacher and Stein 1981) are classified into various branches. The eigenfunctions of each type of mode have characteristic dependences on depth, and are thus sensitive to different layers of the Sun. Helioseismology uses observations of the amplitudes and frequencies of these modes at the solar surface to infer the properties of deeper layers (Christensen-Dalsgaard 2002).

One expects a particular mode to not be strongly dependent on the solar structure in regions where its eigenfunction is small. This is motivated by the fact that if a perturbation is added to a wave evolution equation, time-independent perturbation theory tells us that to the lowest order in the perturbation, the frequency shift of a particular mode is given by the trace of the projection of the perturbation onto the corresponding eigenmode (Sakurai 2006, p. 304). Thus, it is important to understand how the eigenfunctions of various modes depend on the radius. Chapter 11 examines a long-standing debate about the number of radial nodes of the lowest p mode. We find that the number of radial nodes of a particular p mode depends on the wavenumber. In a plane-parallel domain, the difference is due to the fact that waves of high-enough horizontal wavenumber are internally reflected before reaching the bottom of the domain, with modes which are reflected from the bottom of the domain having one node less than those which are internally reflected. However, this explanation does not work in a spherical domain, where, in the Cowling approximation, one simply finds that a p mode of a particular order has one node more when l≠0l\ne 0 than when l=0l=0.

It is difficult to confirm or rule out specific dynamo mechanisms using currently available magnetograms (chapter 7). Moreover, even when taken at face value, magnetograms only tell us about the magnetic field configuration at the surface of the Sun. Does helioseismology allow us to say anything about subsurface magnetic fields? The immediate motivation for this question is that such information could help us understand the mechanism of sunspot formation. More generally, since magnetic fields are sometimes invoked to explain the convective conundrum or the formation of the tachocline, constraining them might help confirm or rule out these explanations. In chapter 12, we examine magnetic effects on helioseismic modes. We do this by imposing prescribed magnetic fields in three-dimensional simulations of convection and measuring the responses of various modes. While we confirm the qualitative findings of previous two-dimensional forced-turbulence simulations, further work is required to make a connection with observations.

Bibliography