Baroclinicity-Fed Incompressible Spectrum

• Baroclinicity-fed incompressible mode spectrum is characterized by localized vorticity injection from misaligned pressure and density gradients, producing a distinct k⁻³/² energy spectrum.
• The mechanism leverages Vishniac-type instabilities along corrugated SNR interfaces, where rapid small-scale growth seeds turbulence that transfers energy to larger scales.
• Analytical and numerical studies confirm that this localized baroclinic forcing uniquely drives solenoidal turbulence, providing new insights into ISM and galactic-scale dynamics.

Baroclinicity-fed incompressible mode spectrum describes the direct excitation of solenoidal (i.e., divergence-free) turbulence in astrophysical and geophysical flows by localized baroclinic vorticity generation, specifically where pressure and density gradients are misaligned. The formal structure of this phenomenon is characterized by an explicit power-law energy spectrum for incompressible velocity fluctuations, with excitation localized in regions of strong baroclinicity—such as corrugated interfaces in supernova remnants (SNRs)—and mediated by small-scale instabilities that impart unique spectral properties at both local and galactic scales (Beattie, 9 Sep 2025).

1. Baroclinic Vorticity Generation and Mathematical Framework

In a compressible fluid, the vorticity evolution is governed by:

$$\frac{\partial \boldsymbol{\omega}}{\partial t} = \nabla \times (\mathbf{u} \times \boldsymbol{\omega}) + \frac{\nabla \rho \times \nabla P}{\rho^2} + \ldots$$

The key non-advective source is the baroclinic term:

$$\mathcal{B} = \frac{\nabla \rho \times \nabla P}{\rho^2}$$

This term is generically non-zero whenever pressure and density gradients are misaligned, which occurs naturally in multiphase media—e.g., at shock interfaces between supernova ejecta and ambient ISM. In the context of turbulent SNRs, this source term directly injects vorticity into the solenoidal (incompressible) mode spectrum, regardless of initial velocity-field irrotationality.

The analytical relation for the spectral distribution of this mechanically generated incompressible energy is derived by examining the evolution of the baroclinically generated vorticity field and its subsequent turbulent cascade.

2. Physical Mechanism: Vishniac-type Instability and Corrugated Layers

The interface between the hot SNR plasma and the cooler surrounding medium becomes highly corrugated due to a Vishniac-type instability. The growth rate for the instability scales as:

$\gamma(k) \propto k^{3/2}$

where $k$ is the wavenumber of the instability. This results in the rapid amplification of small scale perturbations along the mixing layer, creating folds and corrugations where the baroclinic vector $\mathcal{B}$ is maximized. These inhomogeneous, folded structures act as a distributed source of vorticity, directly exciting a spectrum of incompressible turbulence.

Locally, where this interface instability operates, the energy spectrum of the excited incompressible modes follows:

$P_u(k) \propto k^{-3/2}$

This spectral index is a direct reflection of the dominant folding and thin-layer instability: higher wavenumbers grow faster, thus the local turbulence injected by the baroclinic process acquires the $k^{-3/2}$ scaling.

3. Spectral Consequences and Nonlinear Transfer to Large Scales

Although the baroclinicity-fed instability pumps energy into the turbulence cascade at small scales (specifically at the cooling/mixing layer radius of SNRs), the nonlinear dynamics of incompressible turbulence in astrophysical disks are dominated by the inverse cascade process. In such systems, energy introduced at small scales is transferred nonlinearly to larger and larger scales via triadic interactions. Thus, the distinctive $k^{-3/2}$ spectrum arising from the local, baroclinically-generated, folded layer can imprint itself on much larger, galactic-scale turbulent flows.

This mechanism is structurally different from traditional turbulence cascades where large-scale solenoidal forcing is assumed; here, localized baroclinicity due to small-scale instabilities supplies the seed for the entire incompressible mode spectrum, which can extend up to kiloparsec scales in the ISM.

4. Analytical Relation and Observational Relevance

The analytical relation for the baroclinicity-fed incompressible spectrum in these systems is:

• Vorticity injection: $$\displaystyle \left[\frac{\partial \boldsymbol{\omega}}{\partial t}\right] \approx \frac{\nabla \rho \times \nabla P}{\rho^2}$$
• Instability growth rate: $\displaystyle \gamma(k) \propto k^{3/2}$
• Resulting incompressible spectrum: $\displaystyle P_u(k) \propto k^{-3/2}$

This spectral prediction matches numerically observed turbulence statistics within SNRs in high-resolution, SN-driven ISM simulations (Beattie, 9 Sep 2025), and explains how the layer-resolved injection of vorticity can propagate structure throughout a galaxy.

5. Broader Implications and Generalization

The mechanistic pathway outlined—localized baroclinic vorticity production amplified by a thin-layer instability and coupled to an inverse energy cascade—establishes a self-consistent model for sustaining large-scale, incompressible turbulence in multiphase galactic disks. This bridges distinct spectral regimes and links the microscale (mixing layer-induced vorticity) directly to macroscale ISM turbulence properties, providing a new theoretical and observational cornerstone for understanding the dynamical state of the ISM without invoking large-scale solenoidal forcing.

This framework is extendable to other astrophysical or geophysical systems where sharp interface dynamics, baroclinic misalignment, and turbulent cascades coexist, such as cold fronts in galaxy clusters, stellar wind interactions in binaries, and layered structures in planetary atmospheres.

6. Connection to Related Theoretical and Numerical Literature

Recent studies have established the baroclinic term as a dominant source of vorticity in non-isothermal, multiphase flows where rotation and shear are weak (Sordo et al., 2010). The characteristic spectral index $-3/2$ predicted for baroclinic layer-driven turbulence differs notably from classical Kolmogorov ($-5/3$) or enstrophy cascade ($-3$) scalings, reinforcing the importance of localized baroclinic injection in setting the observed spectral properties in astrophysical turbulence (Beattie, 9 Sep 2025).

In summary, the baroclinicity-fed incompressible mode spectrum arises from small-scale, interface-driven vorticity production where baroclinic mechanisms (and their associated thin-shell instabilities) serve as localized engines of turbulence that couple directly to the global solenoidal cascade through inverse nonlinear transfer, yielding a robust $k^{-3/2}$ spectrum in both simulation and analytic theory.