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Is the Explicit Symmetry Breaking of Vorticity Physically Significant in Fixing a Scale?

I have been working on a theoretical framework that draws on concepts from conformal field theory (CFT) to describe fluid dynamics and could potentially be well suited for describing certain turbulent flows. In this approach, I use scalar and vector potentials to capture both conservative and non-conservative aspects of fluids, aiming to simplify complex fluid phenomena with a Maxwell-like formalism. 

Overview of the Theory

In this model, fluid flows are described using real scalar $\phi$ and vector potentials $V^\mu = (p, v)$. The key feature is the introduction of a minimal coupling term $(D_\mu \phi)(D^\mu \phi)$, where $D_\mu$ represents a covariant derivative coupling $\phi$ to $V^{\mu}$. Notably, this term is conformally invariant, meaning it respects the symmetry under conformal dilatations (Weyl transformations) $\sigma = e^{\alpha}$. The full list of transformations is:

$\phi \rightarrow \sigma \phi$

$D^{\mu}\phi \rightarrow \sigma D^{\mu}\phi$

$V^{\mu} \rightarrow V^{\mu} + \alpha^{\mu}$

$g_{\mu \nu} \rightarrow \sigma^{-2}g_{\mu \nu}$

To further characterize the fluid dynamics, I define a vorticity tensor $\Omega_{\mu \nu}$, analogous to the electromagnetic field tensor in electrodynamics. 
Instead of $E$ and $B$ fields, I define the transport field as $U = (v \cdot \nabla)v = -\nabla p - \frac{\partial v}{\partial t}$ and vorticity $W = \nabla \times v$. For the total Lagrangian for the system, $\mathcal{L}_{T}$, I included a kinetic term for the vorticity, $\frac{1}{4} \Omega_{\mu \nu} \Omega^{\mu \nu}$. However, this kinetic term explicitly breaks the conformal symmetry, leading to potentially interesting physical consequences.

$\frac{1}{4} \Omega_{\mu \nu}\Omega^{\mu \nu} \rightarrow \sigma^{-4} \frac{1}{4} \Omega_{\mu \nu}\Omega^{\mu \nu}$

Equations of Motion and Symmetry Breaking

The equations of motion derived from this framework are:

$D_{\mu}D^{\mu} \phi = 0$, which is conformally invariant, and
$\partial_\mu \Omega^{\mu \nu} = j^\nu$, where the current $j^\nu$ transforms as $j^\mu \rightarrow \sigma^{2} j^\mu$. This equation is not conformally invariant, indicating a breaking of the symmetry. $\partial_{\mu}\Omega^{\mu \nu}$ and $j^{\mu}$ do not transform the same under $\sigma$.


This explicit symmetry breaking due to the kinetic term seems to introduce a fixed scale into the theory, which in general is crucial for describing fluid phenomena that exhibit characteristic lengths, such as vortex dynamics and turbulence.

Furthermore, the conserved quantity $\frac{Q}{\rho} = \oint U \cdot dS$ acts as an analog to electric charge in electromagnetism, with units corresponding to mass flow rate (kg/s$^2$). This "charge" represents sources or sinks of advective acceleration in the fluid, similar to how electric charge induces an electromagnetic field.

I am particularly interested in understanding whether this explicit symmetry breaking is physically significant. How does this compare to explicit symmetry breaking in other physical theories, and what implications might it have for understanding complex fluid phenomena like vortex shedding and turbulence? Typically the only examples I find are in the context of quantum field theory, not classical field theory. 

Any insights or feedback from the community would be greatly appreciated!