Rescaling Transforms for Local Models of Spherical Flows

TL;DR

The paper develops a rescaling framework that maps local spherically contracting/expanding flows to Cartesian-like hydrodynamics by introducing time-dependent box metrics and rescaled variables, with a rescaled time coordinate $ au$. It derives general rescaled equations, identifies an effective heating term $ ext{Γ}$, and shows how conformal, 2D, and 1D rescalings recover known Cartesian limits under appropriate $oldsymbol{g}$-scalings, including the vanishing of $ ext{Γ}$ for certain $ ext{γ}$. It demonstrates that the incompressible small-scale limit emerges in the rescaled problem, introduces Cosserat-like anisotropic forces in the incompressible limit, and provides explicit 2D solutions (vortices) and instabilities (KHI) within the spherical-flow context, illustrating the approach's utility for studying turbulence and instabilities. The framework reveals that small-scale turbulence in contracting/expanding flows is equivalent to rescaled, decaying Kolmogorov turbulence, potentially amplified by geometric effects during collapse, and offers a path to unify and test turbulence behavior in astrophysical flows.

Abstract

Previously we developed a local model for a spherically contracting/expanding gas cloud that can be used to study turbulence and small scale instabilities in such flows. In this work we generalise the super-comoving variables used in studies of cosmological structure formation to our local spherical flow model, which make it significantly easier to derive analytical solutions and analyse the interactions of more complex flows with the background. We show that a wide class of solutions to the local spherical flow model can be obtained via a mapping from the corresponding solutions in regular Cartesian flows. The rescaling of time in the transformation results in a modification of the linear instabilities that can occur in spherical flows, causing them to have a time dependent growth rate in the physical time coordinate, and can prevent slower instabilities from operating. Finally, we show that the small scale flows in isotropic contraction/expansion can be mapped directly to Cartesian, inviscid, incompressible hydrodynamics, meaning that one expects a form of rescaled Kolmogorov-turbulence at the small scale of isotropically contracting/expanding flows.

Figures (7)

• Figure 1: Diagram showing the different transforms/spaces used this paper. a) Shows the domain, $S \subseteq \mathbb{R}^3$, in the global frame, where the size and shape of the domain changes as a result of the background flow. b) Shows the local model, $L$, with metric $\mathbf{g}$ and time-dependent coordinate chart $\varphi$ that maps the local model to $S$. c) Shows one of the rescaled local models, $L^{*}$, obtained by introducing rescaled variables analogous to the super-comoving coordinates of Shandarin80Martel98, with new metric $\boldsymbol{\gamma}$ that has determinant $1$. d) Shows the incompressible limit of c), $S^{*}$,obtained by taking the rescaled Mach number, $\mathcal{M}_0$, to zero. By composition of maps, $\Sigma = \tilde{\Sigma} \circ \tilde{f} \circ \varphi^{-1}$, we thus have a way of obtaining the incompressible limit of the original problem, a), despite the presence of compression in the background flow.
• Figure 2: Graph showing whether the background flow results in an effective heating or an effective cooling in the 2D rescaled problem. Here we are using the notation of Lynch23, with $U/\mathcal{R} = \dot{L}_1/L_1$ and $\Delta$ being the velocity divergence of the background flow. The lines are the $\Gamma=0$ line for different ratio of specific heats. Spherical flows above these lines produce effective heating, while those below cool.
• Figure 3: Maximum unstable wavelength (in terms of the box-length) for the isothermal collapse model of Shu77, for a uniform density medium, as a function of the initial Mach-number, $\mathcal{M}_0$, of the rotating band and the imbalance between the gravitational and pressure gradients in the background flow, $A$.
• Figure 4: Evolution of a rotating band on a spherical shell within the isothermal collapse of Shu77, in the global picture. The parameters for this setup are $A = 3$, $a = 1$, $R_0 = 10$, $\mathcal{M}_0 = 0.3$, $k = 5$, with $10$ boxes around spaced around the midplane, chosen so that the unstable mode occurs on a large enough scale to be visible in the plot. Time is in units of the maximum timescale. As the collapse proceeds the rotating band spins up, while the KHI grows in amplitude (potentially enough to go nonlinear).
• Figure 5: Same as Figure \ref{['KHI Shu Example']}, with $A=50$ and $\mathcal{M}_0 = 0.1$ but otherwise the same parameters as Figure \ref{['KHI Shu Example']}, resulting in no growth of the KHI on the collapse timescale.