Circulation and Reduced-Vorticity Coupling in Axisymmetric Euler Flow with Swirl

Good to continue with the Euler transfer. In the Boussinesq construction, the central feedback was:

$$D_t\theta =0,D_t\omega ={\partial }_x\theta ,$$

with $\theta$ a transported scalar and $\omega$ a vorticity forced by its gradient. The preceding lessons then showed how a localized high-frequency layer can repeatedly convert this coupling into growing gradients while keeping the prescribed forcing smooth.

The three-dimensional Euler result uses no external forcing. Its key observation is that axisymmetric Euler flow with swirl contains an almost identical two-dimensional subsystem when viewed on the meridional $(r,z)$-plane, particularly near a solid boundary $r=1$ and away from the symmetry axis $r=0$. This lesson derives that subsystem carefully: the conserved circulation $r{u}_{\theta }$, the reduced angular vorticity ${\omega }_{\theta }/r$, and the source term that couples them.

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1. Axisymmetry reduces a 3D problem to a meridional plane

Use cylindrical coordinates $(r,\vartheta ,z)$, where

$$r=\sqrt{x_1^2+x_2^2},z=x_3.$$

The associated orthonormal directions are $e_r$, $e_{\theta }$, and $e_z$. An axisymmetric velocity field is independent of the angle $\vartheta$, but it may still have a nonzero angular, or swirl, component:

$$u(r,z,t)=u_r(r,z,t)e_r+u_{\theta }(r,z,t)e_{\theta }+u_z(r,z,t)e_z.$$

The components $u_r$ and $u_z$ move material points in the half-plane $(r,z)$, called the meridional plane. The component $u_{\theta }$ makes those material rings rotate around the axis.

For an axisymmetric incompressible field, divergence-freeness becomes

$${\partial }_r(r{u}_r)+{\partial }_z(r{u}_z)=0.$$

Equivalently,

$${\partial }_r{u}_r+{\partial }_z{u}_z=-\frac{u_r}{r}.$$

This distinction is important. The pair $(u_r,u_z)$ is not divergence-free with respect to ordinary flat area measure $drdz$. Instead, the flux vector

$$(r{u}_r,r{u}_z)$$

is divergence-free in the meridional plane. The factor $r$ is the geometric Jacobian inherited from three-dimensional volume measure:

$$dV=rdrd\vartheta dz.$$

Define the meridional material derivative

$$D_t={\partial }_t+u_r{\partial }_r+u_z{\partial }_z.$$

Every evolution equation below is expressed along trajectories generated by this two-dimensional transport field.

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2. The transported quantity: circulation

The azimuthal component of the incompressible Euler equations is

$${\partial }_t{u}_{\theta }+u_r{\partial }_r{u}_{\theta }+u_z{\partial }_z{u}_{\theta }+\frac{u_r}{r}{u}_{\theta }=0.$$

In material notation,

$$D_t{u}_{\theta }=-\frac{u_r}{r}{u}_{\theta }.$$

At first sight, this says swirl is not conserved: radial motion changes $u_{\theta }$. But that is exactly what one expects geometrically. A particle moving radially outward travels around a larger circle, so the same angular momentum corresponds to a smaller tangential velocity.

Multiply the equation by $r$. Since

$$D_{t}r=u_r,$$

the product rule gives

$$D_t(r{u}_{\theta })=(D_{t}r)u_{\theta }+r{D}_t{u}_{\theta }=u_r{u}_{\theta }+r\left(-\frac{u_r}{r}{u}_{\theta }\right)=0.$$

Thus the natural transported scalar is

$$\boxed{\Gamma :=r{u}_{\theta },D_t\Gamma =0.}$$

The quantity $\Gamma$ is the circulation variable, or equivalently the angular momentum per unit mass of a fluid particle around the symmetry axis.

This is the first part of the Boussinesq correspondence:

$$\boxed{\Gamma =r{u}_{\theta }\text{plays the role of the transported Boussinesq scalar }\theta .}$$

There is a subtle but useful contrast with Boussinesq. There, the scalar $\theta$ is usually introduced as a density or temperature perturbation. Here, the analogue is not a passive tracer inserted into the model: it is a nonlinear geometric quantity intrinsic to Euler flow.

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3. The forced quantity: reduced angular vorticity

The vorticity is

$$\omega =\nabla \times u.$$

For axisymmetric flow, the angular component is

$${\omega }_{\theta }={\partial }_z{u}_r-{\partial }_r{u}_z$$

up to the sign convention for the cylindrical basis. The Euler paper adopts a convention consistent with its stream function formulation; the mechanism is unchanged by this sign choice.

The angular-vorticity equation is

$${\partial }_t{\omega }_{\theta }+u_r{\partial }_r{\omega }_{\theta }+u_z{\partial }_z{\omega }_{\theta }-\frac{u_r}{r}{\omega }_{\theta }=\frac{1}{r}{\partial }_z(u_{\theta }^2).$$

Equivalently,

$$D_t{\omega }_{\theta }=\frac{u_r}{r}{\omega }_{\theta }+\frac{1}{r}{\partial }_z(u_{\theta }^2).$$

There are two terms on the right:

1. $\frac{u_r}{r}{\omega }_{\theta }$ is a geometric vortex-stretching contribution.
2. $\frac{1}{r}{\partial }_z(u_{\theta }^2)$ says that axial variation in swirl produces angular vorticity.

The first term obstructs a direct Boussinesq comparison. As with the swirl equation, the correct geometric normalization removes it. Define

$$\boxed{\xi :=\frac{{\omega }_{\theta }}{r}.}$$

Using $D_{t}r=u_r$, compute

$$D_t\left(\frac{{\omega }_{\theta }}{r}\right)=\frac{1}{r}{D}_t{\omega }_{\theta }-\frac{{\omega }_{\theta }}{r^2}{D}_{t}r.$$

Substitute the angular-vorticity equation:

$$D_t\xi =\frac{1}{r}\left(\frac{u_r}{r}{\omega }_{\theta }+\frac{1}{r}{\partial }_z(u_{\theta }^2)\right)-\frac{{\omega }_{\theta }}{r^2}{u}_r.$$

The two stretching terms cancel exactly:

$$\frac{u_r{\omega }_{\theta }}{r^2}-\frac{u_r{\omega }_{\theta }}{r^2}=0.$$

Therefore

$$D_t\xi =\frac{1}{r^2}{\partial }_z(u_{\theta }^2).$$

Now write $u_{\theta }=\Gamma /r$. Since $r$ is independent of $z$,

$$u_{\theta }^2=\frac{{\Gamma }^2}{r^2},$$

and hence

$${\partial }_z(u_{\theta }^2)=\frac{1}{r^2}{\partial }_z({\Gamma }^2).$$

Thus

$$\boxed{D_t\left(\frac{{\omega }_{\theta }}{r}\right)=\frac{1}{r^4}{\partial }_z\left((r{u}_{\theta }{)}^2\right).}$$

Putting the two equations together gives the fundamental system:

$$\boxed{\begin{array}{rl}{D}_t\Gamma & =0,\\ D_t\xi & =\frac{1}{r^4}{\partial }_z({\Gamma }^2),\end{array}\Gamma =r{u}_{\theta },\xi =\frac{{\omega }_{\theta }}{r}.}$$

This is the desired circulation–reduced-vorticity coupling.

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4. Why this is Boussinesq-like, but not literally Boussinesq

The inviscid two-dimensional Boussinesq vorticity form has the schematic structure

$$D_t\theta =0,D_t\omega ={\partial }_x\theta .$$

The axisymmetric Euler pair has the corresponding structure

$$D_t\Gamma =0,D_t\xi =\frac{1}{r^4}{\partial }_z({\Gamma }^2).$$

The map is:

Boussinesq quantity	Axisymmetric Euler quantity	Role
$\theta$	$\Gamma =r{u}_{\theta }$	Transported scalar-like field
$\omega$	$\xi ={\omega }_{\theta }/r$	Vorticity-like field determining meridional velocity
${\partial }_x\theta$	$r^{-4}{\partial }_z({\Gamma }^2)$	Source for vorticity-like field
2D incompressible velocity	$(u_r,u_z)$, with weighted incompressibility	Transport field

There are three differences worth keeping separate.

The source is quadratic

The Euler source is a derivative of ${\Gamma }^2$, not of $\Gamma$. Expanding it,

$$\frac{1}{r^4}{\partial }_z({\Gamma }^2)=\frac{2\Gamma }{r^4}{\partial }_z\Gamma .$$

If the circulation has a nonzero background level $\overline{\Gamma }$, and we write

$$\Gamma =\overline{\Gamma }+\gamma ,$$

then

$${\partial }_z({\Gamma }^2)=2\overline{\Gamma }{\partial }_z\gamma +{\partial }_z({\gamma }^2).$$

For perturbations $\gamma$ small relative to $\overline{\Gamma }$, the first term provides a leading linear Boussinesq-like coupling:

$$D_t\xi \approx \frac{2\overline{\Gamma }}{r^4}{\partial }_z\gamma .$$

The quadratic remainder is then a nonlinear perturbation. This is why a positive swirl background is structurally useful in relating Euler to the Boussinesq amplification mechanism.

The coefficient varies spatially

The factor $r^{-4}$ is not constant globally. But the blowup construction is localized near the outer solid boundary $r=1$. In a small neighborhood of that boundary,

$$r\approx 1,r^{-4}\approx 1.$$

Thus the coefficient is close to a constant, and deviations from it can be treated perturbatively.

Meridional incompressibility has a geometric weight

The natural divergence-free field in $(r,z)$ variables is $(r{u}_r,r{u}_z)$, not $(u_r,u_z)$. Again, near $r=1$, the factor $r$ is close to $1$, so this geometry becomes a controlled lower-order correction to planar incompressibility.

The point is not that axisymmetric Euler is planar Boussinesq. It is that, in a shrinking region centered away from the axis, the Euler subsystem has the same leading feedback structure.

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5. The stream function and recovery of meridional velocity

To close the reduced system, one must recover $(u_r,u_z)$ from $\xi$. The Euler paper introduces a stream function $\tilde{\psi }$ such that

$$u_r=-{\partial }_z\tilde{\psi },u_z={\partial }_r\tilde{\psi }+\frac{1}{r}\tilde{\psi }.$$

This representation automatically enforces

$${\partial }_r(r{u}_r)+{\partial }_z(r{u}_z)=0.$$

Indeed,

$$r{u}_r=-r{\partial }_z\tilde{\psi },$$

while

$$r{u}_z=r{\partial }_r\tilde{\psi }+\tilde{\psi }.$$

Therefore,

$${\partial }_r(r{u}_r)+{\partial }_z(r{u}_z)=-r{\partial }_{rz}\tilde{\psi }+r{\partial }_{zr}\tilde{\psi }=0.$$

The angular vorticity satisfies the elliptic relation

$$-\left({\partial }_{rr}+\frac{1}{r}{\partial }_r+{\partial }_{zz}-\frac{1}{r^2}\right)\tilde{\psi }={\omega }_{\theta }.$$

Since ${\omega }_{\theta }=r\xi$, this becomes

$$-\left({\partial }_{rr}+\frac{1}{r}{\partial }_r+{\partial }_{zz}-\frac{1}{r^2}\right)\tilde{\psi }=r\xi .$$

Near $r=1$, the principal elliptic part is simply

$$-({\partial }_{rr}+{\partial }_{zz})\tilde{\psi }\approx \xi .$$

Likewise,

$$u_r\approx -{\partial }_z\tilde{\psi },u_z\approx {\partial }_r\tilde{\psi }.$$

So locally the meridional velocity is recovered from reduced vorticity by essentially the usual two-dimensional Biot–Savart law. The terms

$$\frac{1}{r}{\partial }_r\tilde{\psi },-\frac{1}{r^2}\tilde{\psi },$$

are the geometric corrections that the Euler paper must estimate.

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6. The Euler paper’s boundary-centred coordinates

The construction is localized near the point

$$(r,z)=(1,0),$$

on the solid radial boundary. The paper introduces local meridional coordinates

$$x=C_l(\tau {)}^{-1}z,y=(1-r)C_l(\tau {)}^{-1}.$$

Equivalently,

$$z=C_l(\tau )x,r=1-C_l(\tau )y.$$

Here $C_l(\tau )$ is a shrinking physical length scale. In the local variables, $y=0$ represents the solid boundary $r=1$.

The two coordinate choices are purposeful:

• $x$ points in the axial direction $z$, which is the direction appearing in the vorticity source ${\partial }_z({\Gamma }^2)$;
• $y$ measures inward distance from the solid boundary;
• the support is kept where $C_l(\tau )y$ is small, hence $r=1-C_l(\tau )y$ stays close to $1$.

Under this localization,

$$r^{-4}=(1-C_{l}y{)}^{-4}=1+O(C_{l}y).$$

Thus the reduced-vorticity equation has the leading form

$$D_t\xi \approx {\partial }_x({\Gamma }^2),$$

after the axial rescaling converts ${\partial }_z$ to a multiple of ${\partial }_x$. The precise rescaled equations include scaling terms and small geometric errors, but their dominant nonlinear coupling is the Boussinesq-type one derived above.

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7. The paper’s squared-circulation variable

The Euler paper uses

$$\tilde{\theta }:=(r{u}_{\theta }{)}^2={\Gamma }^2,\tilde{\omega }:=\frac{{\omega }_{\theta }}{r}=\xi .$$

This is not merely a cosmetic choice. Because $\Gamma$ is transported,

$$D_t\Gamma =0,$$

every smooth function of $\Gamma$ is transported as well. In particular,

$$D_t({\Gamma }^2)=2\Gamma D_t\Gamma =0.$$

So

$$\boxed{D_t\tilde{\theta }=0.}$$

The reduced-vorticity equation becomes especially clean:

$$\boxed{D_t\tilde{\omega }=\frac{1}{r^4}{\partial }_z\tilde{\theta }.}$$

This is exactly the form quoted in the paper.

Why square the circulation rather than use $\Gamma$ directly? Two reasons are operationally useful:

1. The physical source term naturally contains $u_{\theta }^2$, so ${\Gamma }^2$ removes unnecessary square roots and sign choices.
2. If $u_{\theta }$ is initially nonnegative, it remains nonnegative while the solution is smooth. Thus $\tilde{\theta }$ still determines the relevant swirl through

$$u_{\theta }=\frac{\sqrt{\tilde{\theta }}}{r}.$$

In the boundary-centred rescaled variables, the paper writes the source as

$$\frac{1}{r^4}{\theta }_x={\theta }_x+\frac{1-r^4}{r^4}{\theta }_x.$$

This decomposition isolates:

• the leading Boussinesq source ${\theta }_x$;
• a geometric error with coefficient $(1-r^4)/r^4$, small because $r$ remains close to $1$.

That is the precise sense in which the Euler construction inherits the Boussinesq feedback while requiring additional perturbative estimates.

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8. A compact derivation to retain

For paper navigation, the following four-line derivation is the essential one to be able to reproduce.

Start from the axisymmetric Euler component laws:

$$D_t{u}_{\theta }=-\frac{u_r}{r}{u}_{\theta },$$ $$D_t{\omega }_{\theta }=\frac{u_r}{r}{\omega }_{\theta }+\frac{1}{r}{\partial }_z(u_{\theta }^2).$$

Use $D_{t}r=u_r$. Then:

$$D_t(r{u}_{\theta })=0,$$

and

$$D_t\left(\frac{{\omega }_{\theta }}{r}\right)=\frac{1}{r^2}{\partial }_z(u_{\theta }^2)=\frac{1}{r^4}{\partial }_z\left((r{u}_{\theta }{)}^2\right).$$

Therefore, with

$$\tilde{\theta }=(r{u}_{\theta }{)}^2,\tilde{\omega }=\frac{{\omega }_{\theta }}{r},$$

one obtains

$$\boxed{\begin{array}{rl}{D}_t\tilde{\theta }& =0,\\ D_t\tilde{\omega }& =\frac{1}{r^4}{\partial }_z\tilde{\theta }.\end{array}}$$

The two divisions and multiplications by $r$ are not arbitrary normalizations. They are the exact geometric transformations that cancel the radial-expansion terms in the swirl and angular-vorticity equations.

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Takeaways

Axisymmetric Euler with swirl has a two-dimensional meridional subsystem governed by a Boussinesq-like feedback.

• The natural transported swirl quantity is the circulation

$$\Gamma =r{u}_{\theta }.$$

• The natural vorticity quantity is the reduced angular vorticity

$$\xi =\frac{{\omega }_{\theta }}{r}.$$

• They satisfy

$$D_t\Gamma =0,D_t\xi =\frac{1}{r^4}{\partial }_z({\Gamma }^2).$$

• Equivalently, the paper uses

$$\tilde{\theta }={\Gamma }^2,\tilde{\omega }=\xi ,$$

so that

$$D_t\tilde{\theta }=0,D_t\tilde{\omega }=r^{-4}{\partial }_z\tilde{\theta }.$$

• Near the boundary point $r=1$, the coefficient $r^{-4}$ and the meridional Biot–Savart law are close to their planar counterparts. The remaining cylindrical geometry is treated as a small perturbation.

Next, we will map the Boussinesq amplitude mechanism more explicitly onto this Euler system: how material coordinates, the squared-circulation variable, and spatially varying coefficients modify—but do not remove—the high-frequency amplification law.