Mapping Boussinesq Amplitude Mechanisms to Euler Amplitude Laws

Good to continue. In the previous lesson we derived the exact axisymmetric Euler subsystem

$$D_t\tilde{\theta }=0,D_t\tilde{\omega }=\frac{1}{r^4}{\partial }_z\tilde{\theta },$$

where

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

This already resembles the Boussinesq pair

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

The task now is more precise: translate the high-frequency amplitude mechanism from Boussinesq into Euler. We will see that the same feedback survives, but it is expressed through a moving material coordinate system and is perturbed by cylindrical geometry. The key distinction is that material coordinates remove transport from the scalar equation, but they do not make the dynamics trivial: the deformation of the flow map transfers the growth into physical gradients.

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1. The local Boussinesq dictionary

Near the Euler blowup point $(r,z)=(1,0)$, introduce boundary-centred coordinates

$$x=\frac{z}{\ell (t)},y=\frac{1-r}{\ell (t)},$$

where $\ell (t)$ is a small physical length scale. Thus

$$z=\ell (t)x,r=1-\ell (t)y.$$

The axial coordinate $z$ becomes the Boussinesq tangential coordinate $x$, and inward distance from the boundary becomes the Boussinesq normal coordinate $y$. The basic correspondence is therefore

Boussinesq	Axisymmetric Euler near $r=1$
$\theta$	$\tilde{\theta }=(r{u}_{\theta }{)}^2$
$\omega$	$\tilde{\omega }={\omega }_{\theta }/r$
${\partial }_x\theta$	$r^{-4}{\partial }_z\tilde{\theta }$
planar velocity from $-\Delta \psi =\omega$	meridional velocity from a cylindrical elliptic equation
flat incompressibility	weighted incompressibility

At the level of the two evolution equations, the mapping is immediate:

$$D_t\theta ={\partial }_t\theta +u\cdot \nabla \theta =0$$

corresponds to

$$D_t\tilde{\theta }={\partial }_t\tilde{\theta }+u_r{\partial }_r\tilde{\theta }+u_z{\partial }_z\tilde{\theta }=0,$$

while the Boussinesq vorticity forcing

$${\partial }_x\theta$$

corresponds to

$$\frac{1}{r^4}{\partial }_z\tilde{\theta }.$$

The local identification of directions is

$${\partial }_x^{\mathrm{Boussinesq}}⟷{\partial }_z^{\mathrm{Euler}}.$$

So, in a small region where $r$ remains close to $1$,

$$r^{-4}{\partial }_z\tilde{\theta }\approx {\partial }_z\tilde{\theta },$$

and the leading forcing is the Boussinesq forcing in renamed coordinates.

That is only the first part of the transfer. The more consequential part concerns what happens to a high-frequency wave under the flow.

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2. What the Boussinesq amplitude mechanism was really tracking

Recall the local high-frequency picture from Module 1. Take a wave perturbation with phase

$$\varphi (t,x)=q(t)\cdot x,$$

and write, schematically,

$${\theta }_{\mathrm{wave}}\sim A(t)e^{i\varphi (t,x)},{\omega }_{\mathrm{wave}}\sim B(t)e^{i\varphi (t,x)}.$$

Here $q(t)$ is the physical wavevector. The scalar and vorticity amplitudes are $A$ and $B$.

For a flat two-dimensional Biot–Savart law, the velocity induced by a vorticity wave has leading Fourier form

$$V\sim i\frac{q^{\perp }}{|q{|}^2}B.$$

Thus, at frequency $|q|$,

$$|V|\sim \frac{|B|}{|q|},|\nabla V|\sim |B|.$$

A vorticity wave can therefore produce an order-one strain despite having velocity amplitude smaller by a factor of $|q{|}^{-1}$.

The other key fact was the shared-phase cancellation. Since

$$\nabla e^{iq\cdot x}=iq{e}^{iq\cdot x},$$

and the leading velocity is perpendicular to $q$,

$$q^{\perp }\cdot q=0,$$

a wave does not advect another field with the same phase at leading order. This lets one isolate the amplitude dynamics rather than being overwhelmed by high-frequency self-transport.

The growth mechanism is not merely

$$\dot{B}\sim |q|A.$$

That relation alone would produce vorticity from a scalar gradient, but not a closed feedback loop. The loop arises when the wave velocity displaces a nonconstant background scalar. If the background has scalar gradient

$$G=\nabla \overline{\theta },$$

then the wave velocity changes the scalar-wave amplitude at leading order through

$$\dot{A}\sim -V\cdot G.$$

Meanwhile, the vorticity equation has the leading forcing

$$\dot{B}\sim i{q}_{x}A.$$

Suppressing lower-order background-vorticity and transport terms, the frozen-coefficient system has the form

$$\frac{d}{dt}\left(\begin{array}{c}A\\ B\end{array}\right)=\left(\begin{array}{cc}0& -i\frac{q^{\perp }\cdot G}{|q{|}^2}\\ i{q}_x& 0\end{array}\right)\left(\begin{array}{c}A\\ B\end{array}\right).$$

After choosing compatible sine and cosine phases, this becomes a real $2\times 2$ system. If the signs and geometry are arranged so that the product of the off-diagonal coefficients is positive, it has one exponentially growing mode.

The point to retain is:

A scalar perturbation produces vorticity through its gradient; the vorticity produces a velocity; that velocity moves the scalar relative to its background gradient; this regenerates the scalar perturbation.

The Euler construction has the same leading loop.

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3. The exact Euler system in material coordinates

Before writing an Euler amplitude system, it is helpful to separate Eulerian position from material label.

Let

$$X(t,a)=(R(t,a),Z(t,a))$$

be the meridional trajectory of the fluid particle labelled by $a$. It solves

$$\frac{d}{dt}X(t,a)=(u_r,u_z)(t,X(t,a)).$$

For any Eulerian scalar $f(t,r,z)$, define its pullback to labels by

$$F(t,a)=f(t,X(t,a)).$$

The chain rule gives

$$\frac{d}{dt}F(t,a)=(D_{t}f)(t,X(t,a)).$$

Apply this to the squared-circulation scalar $\tilde{\theta }$. Since

$$D_t\tilde{\theta }=0,$$

we obtain the exact Lagrangian conservation law

$$\tilde{\Theta }(t,a):=\tilde{\theta }(t,X(t,a))=\tilde{\theta }(0,X(0,a)).$$

With the usual convention $X(0,a)=a$,

$$\boxed{\tilde{\Theta }(t,a)={\tilde{\theta }}_0(a).}$$

Thus the Euler scalar is genuinely frozen into the flow. This is not an approximation.

The reduced vorticity pullback

$$\tilde{\Omega }(t,a):=\tilde{\omega }(t,X(t,a))$$

satisfies

$$\frac{d}{dt}\tilde{\Omega }(t,a)=\frac{1}{R(t,a{)}^4}{\partial }_z\tilde{\theta }(t,X(t,a)).$$

The right side still contains an Eulerian derivative. To express it using labels, differentiate the identity

$$\tilde{\theta }(t,X(t,a))=\tilde{\Theta }(a).$$

Let

$$F(t,a)=D_{a}X(t,a)$$

be the deformation gradient. Then

$${\nabla }_{r,z}\tilde{\theta }(t,X(t,a))=F(t,a{)}^{-T}{\nabla }_a\tilde{\Theta }(a).$$

Therefore

$$\boxed{\frac{d}{dt}\tilde{\Omega }(t,a)=\frac{1}{R(t,a{)}^4}{e}_z^{T}F(t,a{)}^{-T}{\nabla }_a{\tilde{\theta }}_0(a).}$$

Equivalently,

$$\frac{d}{dt}\tilde{\Omega }(t,a)=\frac{1}{R(t,a{)}^4}\left(F(t,a{)}^{-1}{e}_z\right)\cdot {\nabla }_a{\tilde{\theta }}_0(a).$$

This is the conceptual Euler amplitude law in material variables.

It says that the source of reduced vorticity comes from three ingredients:

1. the fixed initial circulation gradient ${\nabla }_a{\tilde{\theta }}_0$;
2. the inverse deformation $F^{-1}$, which converts a label-space gradient into a physical gradient;
3. the cylindrical coefficient $R^{-4}$.

The transported scalar itself does not grow along an individual trajectory. Its physical gradient can grow because the flow compresses material distances.

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4. Wavevectors: stationary in labels, moving in physical space

Suppose the initial squared-circulation perturbation has a label-space phase

$$k_0\cdot a.$$

In material coordinates, this phase is fixed:

$${\tilde{\Theta }}_{\mathrm{wave}}(a)\sim A(a)e^{i{k}_0\cdot a}.$$

But physical observers do not differentiate with respect to $a$; they differentiate with respect to $(r,z)$. The physical wavevector is therefore

$$\boxed{q(t,a)=F(t,a{)}^{-T}{k}_0.}$$

This is exactly the Lagrangian version of the wavevector transport law derived earlier for an affine Boussinesq background.

Indeed, differentiating $F$ along a trajectory gives

$$\frac{d}{dt}F=(\nabla u)(t,X)F.$$

Consequently,

$$\frac{d}{dt}{F}^{-T}=-(\nabla u{)}^T(t,X)F^{-T},$$

and so

$$\boxed{\frac{d}{dt}q=-(\nabla u{)}^{T}q.}$$

For an affine flow $u(x)=Mx$, this reduces to the earlier formula

$$\dot{q}=-M^{T}q.$$

Thus there is no conceptual conflict between the two descriptions:

• in material coordinates, the phase $k_0\cdot a$ is unchanged;
• in physical coordinates, the wavevector $q=F^{-T}{k}_0$ evolves.

The high-frequency cascade is therefore a statement about deformation. A material sheet carrying initially moderate oscillations may be compressed in the axial direction, causing its physical gradient to become large.

For the Euler source term, what matters is specifically the axial component

$$q_z=e_z\cdot q.$$

A WKB-scale scalar wave satisfies

$${\partial }_z{\tilde{\theta }}_{\mathrm{wave}}\sim i{q}_{z}A{e}^{i\varphi }.$$

Hence, along a trajectory,

$$\frac{d}{dt}{\tilde{\Omega }}_{\mathrm{wave}}\sim \frac{i{q}_z}{R^4}A{e}^{i\varphi }.$$

This is the Euler analogue of the Boussinesq contribution

$$\dot{B}\sim i{q}_{x}A.$$

The replacement is simply

$$\boxed{q_x^{\mathrm{Boussinesq}}⟷R^{-4}{q}_z^{\mathrm{Euler}}.}$$

Near $R=1$, the factor $R^{-4}$ is close to $1$, and the two laws agree at leading order.

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5. Recovering the coupled amplitude feedback in Euler

The exact Lagrangian equation may make it look as if $\tilde{\theta }$ is completely passive and only $\tilde{\omega }$ evolves. That would miss the feedback.

The feedback appears when we study a high-frequency perturbation around a lower-frequency background:

$$\tilde{\theta }=\overline{\theta }+\vartheta ,\tilde{\omega }=\overline{\omega }+\varpi ,u=\overline{u}+v.$$

Linearizing the Euler subsystem gives

$$({\partial }_t+\overline{u}\cdot \nabla )\vartheta +v\cdot \nabla \overline{\theta }=0,$$ $$({\partial }_t+\overline{u}\cdot \nabla )\varpi +v\cdot \nabla \overline{\omega }=\frac{1}{r^4}{\partial }_z\vartheta .$$

The velocity $v$ is obtained nonlocally from $\varpi$ through the meridional Biot–Savart law.

Now freeze the slowly varying background near one packet centre. Let

$$G=\nabla \overline{\theta },H=\nabla \overline{\omega },$$

and let $q$ be the local physical wavevector. At principal high-frequency order, the local Biot–Savart relation is the same as in the plane:

$$V\approx i\frac{q^{\perp }}{|q{|}^2}B.$$

The amplitudes then satisfy the schematic Euler system

$$\dot{A}=-V\cdot G,$$ $$\dot{B}=\frac{i{q}_z}{r^4}A-V\cdot H.$$

The central two-way part is

$$\boxed{\dot{A}\approx -i\frac{q^{\perp }\cdot G}{|q{|}^2}B,\dot{B}\approx \frac{i{q}_z}{r^4}A.}$$

Compare it directly with Boussinesq:

$$\boxed{\dot{A}\approx -i\frac{q^{\perp }\cdot G}{|q{|}^2}B,\dot{B}\approx i{q}_{x}A.}$$

The transfer is therefore:

Mechanism	Boussinesq	Euler near $r=1$
Scalar-wave regeneration	$-V\cdot \nabla \overline{\theta }$	$-V\cdot \nabla \overline{\tilde{\theta }}$
Vorticity-wave creation	$i{q}_{x}A$	$i{r}^{-4}{q}_{z}A$
Wave velocity	$i q^\perp B/	q
Frequency evolution	$\dot{q}=-(\nabla \overline{u}{)}^{T}q$	same deformation law
Growing eigenmode	from compatible off-diagonal signs	same, after controlling coefficient variation

In a real phase convention, write this schematically as

$$\frac{d}{dt}\left(\begin{array}{c}A\\ B\end{array}\right)\approx \left(\begin{array}{cc}0& a(t)\\ b(t)& 0\end{array}\right)\left(\begin{array}{c}A\\ B\end{array}\right),$$

where

$$a(t)\sim \frac{q^{\perp }\cdot G}{|q{|}^2},b(t)\sim \frac{q_z}{r^4}.$$

If

$$a(t)b(t)>0$$

in the relevant phase convention, the instantaneous eigenvalues are approximately

$${\lambda }_{\pm }(t)\approx \pm \sqrt{a(t)b(t)}.$$

The growing eigenvector is the Euler counterpart of the Boussinesq growing mode. The construction chooses packet orientations, signs, and backgrounds so that this positive feedback persists for the required time.

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6. Material coordinates do not eliminate the feedback

There are two equivalent but differently useful descriptions.

Eulerian perturbation viewpoint

Here one keeps a fixed background flow $\overline{u}$. The scalar perturbation satisfies

$$({\partial }_t+\overline{u}\cdot \nabla )\vartheta =-v\cdot \nabla \overline{\theta }.$$

The right side visibly shows the feedback: the velocity induced by vorticity moves the scalar wave across a background scalar gradient.

Fully Lagrangian viewpoint

Here $\tilde{\theta }\circ X$ is exactly constant:

$$\tilde{\Theta }(t,a)={\tilde{\theta }}_0(a).$$

But the physical gradient is

$${\nabla }_{r,z}\tilde{\theta }=F^{-T}{\nabla }_a{\tilde{\theta }}_0.$$

The feedback is now hidden in the fact that $F$ depends on the velocity, and the velocity depends nonlocally on reduced vorticity.

These are the same mechanism seen from two coordinate systems. The Eulerian picture makes the amplitude loop transparent; the Lagrangian picture makes transport and gradient growth transparent.

A useful way to phrase the difference is:

In material labels, circulation structure is preserved. In physical space, the same preserved structure is squeezed, tilted, and stretched by the flow.

The reduced-vorticity source detects the resulting physical axial gradient.

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7. Cylindrical geometry: what changes and why it is perturbative

The Boussinesq-to-Euler transfer is not an identity on a whole cylinder. Four geometric differences must be controlled.

7.1 The forcing coefficient $r^{-4}$

The Euler forcing is

$$\frac{1}{r^4}{\partial }_z\tilde{\theta }.$$

In the localized coordinates,

$$r=1-\ell y,$$

so

$$\frac{1}{r^4}=(1-\ell y{)}^{-4}.$$

For a packet supported where $\ell y$ is small,

$$\frac{1}{r^4}=1+O(\ell y).$$

This makes the Euler vorticity source

$$\frac{1}{r^4}{\partial }_z\tilde{\theta }={\partial }_z\tilde{\theta }+\left(\frac{1-r^4}{r^4}\right){\partial }_z\tilde{\theta }.$$

The first term is the Boussinesq source. The second is a geometric error. It is small because the construction keeps the physical support close to $r=1$, not because high frequency automatically makes it small.

Along an individual path $R(t,a)$, the coefficient also changes in time:

$$\frac{d}{dt}\left(R^{-4}\right)=-4R^{-5}{u}_r.$$

Thus a frozen-coefficient amplitude matrix is only valid over an interval on which radial motion is controlled. This is one reason the support-control estimates are structural rather than cosmetic.

7.2 The elliptic law is not exactly planar

In physical meridional variables, the stream function satisfies

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

Near $r=1$, its principal part is

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

After boundary-centred rescaling, the paper obtains an equation of the form

$$-({\partial }_{xx}+{\partial }_{yy})\psi +\frac{\ell }{r}{\partial }_y\psi +\frac{{\ell }^2}{r^2}\psi =r\omega .$$

The leading elliptic operator is planar. The terms involving $\ell$ and ${\ell }^2$ are lower order in the localized regime.

This is what justifies using the planar principal symbol

$$V\approx i\frac{q^{\perp }}{|q{|}^2}B$$

in the Euler amplitude calculation. It is an approximation with an error estimate behind it, not a claim that the cylindrical Biot–Savart law is literally the two-dimensional one.

7.3 Incompressibility carries a radial weight

For the meridional flow,

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

Thus a meridional particle map is not exactly area-preserving with respect to $drdz$. Rather, it preserves the cylindrical volume measure $rdrdz$, inherited from three-dimensional volume.

At the level of the meridional deformation gradient $F$, this gives the weighted Jacobian relation

$$R(t,a)\det F(t,a)=R(0,a).$$

Near $r=1$,

$$R(t,a)\approx R(0,a)\approx 1,$$

so

$$\det F(t,a)\approx 1.$$

The local deformation is therefore approximately planar area-preserving, which is the setting of the Boussinesq wave analysis. But the approximation is supported by localization away from the symmetry axis; it would fail as $r$ approached zero.

7.4 The shared-phase cancellation becomes a principal-order cancellation

For planar Boussinesq waves, phase-velocity orthogonality gives an exact leading cancellation:

$$q^{\perp }\cdot q=0.$$

In Euler near $r=1$, the principal Biot–Savart symbol still has this form, so the same cancellation controls the leading high-frequency interaction.

However, the full cylindrical velocity recovery has variable coefficients and lower-order terms. Consequently, the cancellation is not a globally exact identity for arbitrary packets. The construction must estimate the residual produced by those geometric corrections.

This distinction is worth remembering:

• flat principal part: phase orthogonality prevents leading self-advection;
• cylindrical remainder: produces errors that are small only because the packet is sharply localized near $r=1$.

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8. Material coordinates versus dynamic rescaling

One final distinction will prevent a common confusion when reading the Euler paper.

In the original physical variables,

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

In full material coordinates,

$$\frac{d}{dt}\tilde{\Theta }=0.$$

But the paper also introduces dynamically rescaled variables. A rescaled scalar, denoted there by $\theta$, is multiplied by a time-dependent normalization factor and evaluated at a shrinking spatial scale. Its equation takes the form

$${\theta }_t+\left(c_{l}x+u\cdot \nabla \right)\theta =c_{\theta }\theta .$$

The term

$$c_{\theta }\theta$$

does not mean that the physical squared circulation is no longer transported. It records the chosen renormalization. Dynamic rescaling is a moving coordinate frame designed so that a concentrating profile can look approximately stationary.

It is useful to keep the three levels separate:

Description	Scalar behaviour
Physical Eulerian variables	$\tilde{\theta }$ is transported
Lagrangian labels	$\tilde{\theta }\circ X$ is constant
Dynamically rescaled variables	the rescaled $\theta$ has scaling and dilation terms

The amplitude mechanism is physical. Dynamic rescaling does not create it; it provides a coordinate system in which sustained growth can be analysed as stability near an approximate steady profile.

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Takeaways

The Boussinesq amplitude mechanism transfers to axisymmetric Euler with swirl through the identifications

$$\theta ⟷\tilde{\theta }=(r{u}_{\theta }{)}^2,\omega ⟷\tilde{\omega }=\frac{{\omega }_{\theta }}{r},{\partial }_x⟷{\partial }_z.$$

The exact Euler reduced system is

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

In material coordinates, the scalar is frozen:

$$\tilde{\theta }(t,X(t,a))={\tilde{\theta }}_0(a),$$

but its physical gradient evolves through the inverse deformation gradient:

$$\nabla \tilde{\theta }(t,X)=(D_{a}X{)}^{-T}{\nabla }_a{\tilde{\theta }}_0.$$

Thus the physical wavevector is

$$q=(D_{a}X{)}^{-T}{k}_0,$$

and satisfies

$$\dot{q}=-(\nabla u{)}^{T}q.$$

At frozen leading order, the Euler amplitude system is the Boussinesq system with the replacement

$$q_x⟼r^{-4}{q}_z.$$

Near the boundary $r=1$, this is a small perturbation of the planar law. The extra work in the Euler argument is to control:

• variation of $r^{-4}$ along and across packets;
• cylindrical corrections to the elliptic velocity recovery;
• weighted, rather than flat, meridional incompressibility;
• residual self-interactions beyond the planar principal symbol.

Next, we will step back and compare this predominantly low-to-high amplification mechanism with convex integration, where high-frequency corrections are instead used to cancel a low-frequency Reynolds-stress defect.