### Brief Note on the Development of the Framework

Fluid dynamics often relies on the Navier-Stokes equations to model various flow phenomena. I have here a theoretical framework which recasts the equations of fluid mechanics in a Lagrangian formalism in a way where it can be shown that the kinetic energy of the vorticity explicitly break the scaling symmetry of the theory. This explicit symmetry breaking potentially provides a mechanism by which energy cascades and dissipation occur in turbulent flows. Additionally it demonstrates how the inclusion of vorticity fixes a scale.

This framework was developed from abstract symmetry principles and aims to address multiple problems in physics. My first goal was to understand how motion couples to the underlying geometry resulting in emergent dynamics. Fluid dynamics are fundamentally “shaped” by geometry, so I began by exploring simple fluid systems.

Some brief calculations already suggest the role of local scaling symmetries. If we define a conservative vector field to model a wave propagating in a pipe with cross sectional area

we know that it is by definition irrotational: . The mass flow field is also irrotational for constant cross sectional areas . Thus the field exhibits self-similarity at a scale (for any choice of ). However if we locally fix a scale by promoting to a varying cross sectional area, we see that the mass-flow becomes rotational This is a very basic derivation but I hope the results are clear, locally fixing a scale introduces additional dynamics, particularly rotational dynamics.

In a similar manner, by demanding that a real scalar field be invariant under local scaling transformations a rotational vector potential must be introduced resulting in a Helmholtz-decomposition. An interesting consequence is that the symmetry of the Lagrangian is explicitly broken by the kinetic term for the vector potential, again suggesting the association between vorticity and scale-fixing. Another result is that the homogeneous equations for the vector potential imply that the field is irrotational and thus conservative.

### Theoretical Background

#### Scaling Symmetry and CFT

Delving deeper into the symmetry where is the scaling factor, ties to conformal field theory (CFT) are considered. Global scaling symmetry is almost universal as it corresponds to the trivial case where is a constant, the more relevant and interesting cases are the local scaling symmetry that occurs when becomes a function of spacetime coordinates .

We recognize that the scaling transformation is a conformal dilation, indicating that the group of scalar matrices is a subset of the conformal group : This conformal dilation is a Weyl transformation, defined as the local scaling of the metric tensor: where the conformal factor generates a new metric in the same conformal class as . The field has a conformal weight , where . Identifying with , we can conclude that . Combining Equation (1) with

we have two transformation laws. In order to understand how transform, we need to formally define the theory.

### Formulation of Scalar Field Theory

To ensure that the symmetry is respected, the covariant derivative is introduced, where the field has components of pressure and velocity. In order for the derivative of to respect the symmetry we need to transform the same as so or equivalently . To satisfy Equation (3), we conclude that must transform as where . Now we have the full set of transformations, Equations (1), (2), (3) and (4)

We can then modify our Lagrangian for a free scalar field so the minimal coupling for this theory is Equation (5) has the equations of motion (EOM) where . Expanding the terms we get

Note that both Equations (5) and (6) obey the conformal symmetry we have outlined.

## Comments

This is just a sort of general overview, I will gradually edit and add to this post.

After a discussion with Eowyn, this is a correction to my post on Codidact.

Again, I want en equation of the uniform flow. Again vector potential is zero. And again we are left with:

This is the place where I was wrong in my Codidact’s post. I did not pay attention to the fact that we are using tensor notation here and just took corresponding derivatives. That was a mistake. Eowyn educated me on what this term really means.

It turns out, this is d’Alembertian. Eowyn showed me a slightly modified version of d’Alambertian which does not include the speed of light . I guess it means he is working in the coordinates where . In turn, that means that we have to be careful trying to get the governing equation of the flow.

Let me just copy the derivation from Wikipedia where no assumption of coordinates is made, but use Eowyn’s sign convention for metric :

Please correct me if I am wrong, but to the best of my limited knowledge of GR, if we neglect effects of relativity (for the case of regular flow of liquid that we observe on Earth in our every day life), the last expression reduces to:

The right hand side is, of course, a Laplacian:

Substitute Eqn. for the right hand side of Eqn. and get:

Now, plug in Eqn. into Eqn. and get the equation of motion:

From Helmholtz decomposition, it follows that and Eqn. becomes:

Note that Eqn. is true for any irrotational flow. Let’s simplify it for our case of classic uniform flow. Classic uniform flow is a parallel flow along the direction. Therefore, Eqn. can be expanded and reduced as:

Integrating Eqn., we arrive at the equation of velocity field:

Which is, indeed, the equation of the classic uniform flow.