Deriving Coupled Temperature–Vorticity Amplitude Equations

Welcome back. In the previous lesson, we made the high-frequency phase compatible with the affine background flow by imposing

$$\dot{\zeta }=-D^T\zeta .$$

That removed the potentially large phase-transport error. We can now ask what remains once a temperature wave and a vorticity wave are placed on that transported phase. The answer is a two-dimensional amplitude system: the new velocity displaces the old temperature gradient, while the new temperature’s horizontal derivative generates new vorticity.

This is the local algebraic core of the Boussinesq amplification mechanism. In the ideal affine, uncut setting, the cancellations below are exact; later, spatial cutoffs and layer interactions will introduce residuals that must be controlled.

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The affine background and the added wave

Work with the inviscid Boussinesq equations in vorticity form:

$${\partial }_t\theta +u\cdot \nabla \theta =0,$$ $${\partial }_t\omega +u\cdot \nabla \omega ={\partial }_1\theta ,$$

with

$$\nabla \cdot u=0,\omega =\Delta \psi ,u={\nabla }^{\perp }\psi .$$

We use

$${\nabla }^{\perp }=(-{\partial }_2,{\partial }_1),J=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),$$

so that ${\nabla }^{\perp }f=J\nabla f$.

Inside the local affine core built by earlier layers, suppose the old fields have the form

$$u_{\mathrm{old}}(x,t)=D(t)x,{\theta }_{\mathrm{old}}(x,t)=G(t)\cdot x.$$

Here $D$ is trace-free, so the affine velocity is incompressible, and

$$G(t)=\nabla {\theta }_{\mathrm{old}}(t)$$

is the old, spatially constant temperature gradient. The vorticity of an affine velocity field is also spatially constant, so

$$\nabla {\omega }_{\mathrm{old}}=0.$$

Add a temperature perturbation $\vartheta$, vorticity perturbation $\varpi$, and induced velocity $v$:

$$\theta ={\theta }_{\mathrm{old}}+\vartheta ,\omega ={\omega }_{\mathrm{old}}+\varpi ,u=u_{\mathrm{old}}+v.$$

Take the shared-phase sinusoidal ansatz

$$s(x,t)=\lambda \zeta (t)\cdot x,$$ $$\vartheta (x,t)=\Theta (t)\sin s,\varpi (x,t)=\Omega (t)\cos s.$$

The fixed parameter $\lambda$ is large, $\zeta (t)$ is the transported wavevector, and $\Theta ,\Omega$ are signed amplitudes. Let

$$\kappa =|\zeta {|}^2.$$

The choice of a cosine vorticity wave gives the streamfunction

$$\psi (x,t)=-\frac{\Omega (t)}{{\lambda }^2\kappa (t)}\cos s.$$

Indeed,

$$\Delta \psi =\Omega \cos s=\varpi .$$

Differentiating $\psi$ gives the velocity induced by the wave:

$$v={\nabla }^{\perp }\psi =\frac{\Omega }{\lambda \kappa }J\zeta \sin s.$$

Thus the velocity is directed along $J\zeta$, tangent to the wavefronts.

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A compact reading of the construction’s calculation

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Subtracting the old equations: what must cancel?

Assume the old fields already satisfy the Boussinesq equations, possibly with a prescribed external forcing that we do not change when adding this new wave. Subtracting the old equations from the equations for the total fields gives the increments in the residuals.

For temperature, the new residual is

$$({\partial }_t+u_{\mathrm{old}}\cdot \nabla )\vartheta +v\cdot \nabla {\theta }_{\mathrm{old}}+v\cdot \nabla \vartheta .$$

For vorticity, it is

$$({\partial }_t+u_{\mathrm{old}}\cdot \nabla )\varpi +v\cdot \nabla {\omega }_{\mathrm{old}}+v\cdot \nabla \varpi -{\partial }_1\vartheta .$$

There are four structural simplifications.

1. Phase transport has already been arranged:

   $$({\partial }_t+Dx\cdot \nabla )s=0.$$

2. Wave self-advection vanishes. Both $\nabla \vartheta$ and $\nabla \varpi$ are parallel to $\zeta$, whereas $v$ is parallel to $J\zeta$. Since

   $$J\zeta \cdot \zeta =0,$$

   we have

   $$v\cdot \nabla \vartheta =0,v\cdot \nabla \varpi =0.$$

3. The new velocity does not advect old vorticity, because old vorticity is constant on the affine core:

   $$v\cdot \nabla {\omega }_{\mathrm{old}}=0.$$

4. The only intended cross-interactions remain:

   $$v\cdot \nabla {\theta }_{\mathrm{old}}=v\cdot G,$$

   and

   $${\partial }_1\vartheta .$$

This is why the construction uses both a common phase and an affine background. Nearly every potentially nonlinear high-frequency term has disappeared before any amplitude equation is chosen.

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Temperature residual: vorticity-induced velocity moves the old gradient

Because the phase is transported,

$$({\partial }_t+Dx\cdot \nabla )\vartheta =\dot{\Theta }\sin s.$$

The old temperature has gradient

$$\nabla {\theta }_{\mathrm{old}}=G.$$

Therefore the cross-advection term is

$$v\cdot \nabla {\theta }_{\mathrm{old}}=v\cdot G.$$

Substitute the formula for $v$:

$$v\cdot G=\left(\frac{\Omega }{\lambda |\zeta {|}^2}J\zeta \sin s\right)\cdot G.$$

Thus

$$v\cdot G=\frac{J\zeta \cdot G}{\lambda |\zeta {|}^2}\Omega \sin s.$$

The full temperature residual is consequently

$$\left[\dot{\Theta }+\frac{J\zeta \cdot G}{\lambda |\zeta {|}^2}\Omega \right]\sin s.$$

To cancel it, impose

$$\boxed{\dot{\Theta }=-\frac{J\zeta \cdot G}{\lambda |\zeta {|}^2}\Omega .}$$

This equation has a transparent interpretation. The vorticity amplitude $\Omega$ creates a velocity in the $J\zeta$ direction. That velocity samples the old temperature gradient $G$. Only the component of $G$ in the velocity direction matters, hence the projection

$$J\zeta \cdot G.$$

If the old gradient is perpendicular to $J\zeta$, the new velocity moves fluid along old isotherms and cannot change this wave’s temperature amplitude.

The prefactor also displays an important scaling fact:

$$|v|\sim \frac{|\Omega |}{\lambda |\zeta |}.$$

At higher physical frequency $\lambda |\zeta |$, a fixed vorticity amplitude induces a smaller velocity. Hence its direct ability to advect the background temperature gradient is reduced.

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Vorticity residual: the horizontal temperature derivative creates vorticity

Now turn to the vorticity equation. Phase transport gives

$$({\partial }_t+Dx\cdot \nabla )\varpi =\dot{\Omega }\cos s.$$

The Boussinesq source term is the fixed laboratory-coordinate derivative ${\partial }_1\theta$. For the wave,

$${\partial }_1\vartheta ={\partial }_1\left(\Theta \sin (\lambda \zeta \cdot x)\right)=\lambda {\zeta }_1\Theta \cos s.$$

Here ${\zeta }_1$ is the first Cartesian component of the transported wavevector. Therefore the vorticity residual is

$$\left[\dot{\Omega }-\lambda {\zeta }_1\Theta \right]\cos s.$$

Canceling it yields

$$\boxed{\dot{\Omega }=\lambda {\zeta }_1\Theta .}$$

Unlike the temperature equation, this coefficient does not involve the background gradient $G$. It comes directly from buoyancy: a temperature oscillation with a nonzero horizontal wavevector has a horizontal temperature derivative, and that derivative is a vorticity source.

This also explains why orientation matters. If

$${\zeta }_1=0,$$

the temperature wavefronts are horizontal and $\vartheta$ has no $x_1$-variation. Such a wave creates no vorticity through ${\partial }_1\vartheta$, regardless of its amplitude.

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The complete local amplitude system

Combining the phase law from the previous lesson with the two residual cancellations gives

$$\boxed{\begin{array}{rl}\dot{\zeta }& =-D^T\zeta ,\\ \dot{\Theta }& =-\frac{J\zeta \cdot G}{\lambda |\zeta {|}^2}\Omega ,\\ \dot{\Omega }& =\lambda {\zeta }_1\Theta .\end{array}}$$

It is useful to isolate the two amplitude coefficients:

$$a(t)=-\frac{J\zeta (t)\cdot G(t)}{\lambda |\zeta (t){|}^2},b(t)=\lambda {\zeta }_1(t).$$

Then

$$\dot{\Theta }=a(t)\Omega ,\dot{\Omega }=b(t)\Theta .$$

The two off-diagonal couplings have different physical origins:

Equation	Coupling mechanism	Geometric condition
$\dot{\Theta }=a\Omega$	New velocity displaces old temperature gradient	$J\zeta \cdot G\ne 0$
$\dot{\Omega }=b\Theta$	Horizontal derivative of temperature creates vorticity	${\zeta }_1\ne 0$

A useful check is the product:

$$a(t)b(t)=-\frac{(J\zeta \cdot G){\zeta }_1}{|\zeta {|}^2}.$$

The factor $\lambda$ cancels. Increasing frequency changes the relative sizes of the two individual couplings, but not this leading combined feedback strength. The orientation of $\zeta$ relative to both the fixed buoyancy direction and the old temperature gradient is what determines whether the frozen system has an amplifying or oscillatory character.

For now, do not read this as a claim of growth for every geometry. If either factor vanishes, the loop is broken. The construction’s later controlled rotations are designed precisely to manage that orientation.

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Why this is more than a linearization

The amplitude system resembles a linearized Boussinesq system, but its role here is more specific. The total fields include a nonlinear oscillatory perturbation, and the calculation does not merely discard nonlinear terms on the grounds that the perturbation is small.

Instead, the potentially troublesome nonlinear terms vanish identically:

$$v\cdot \nabla \vartheta =0,v\cdot \nabla \varpi =0.$$

They vanish because one scalar phase simultaneously controls the wave velocity direction and both wave-gradient directions. Thus, within an exactly affine background and before localization, the ansatz produces an exact local wave once the three ODEs hold.

The construction will eventually depart from this ideal setting in two ways:

• the waves must be spatially localized rather than globally affine and periodic;
• older layers make $D$ and $G$ time-dependent, and their cumulative effects must be scheduled.

Those changes produce errors, but not until after the key amplification algebra has been isolated.

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Takeaways

The phase transport equation removes the high-frequency transport residual:

$$\dot{\zeta }=-D^T\zeta .$$

The shared phase removes wave self-advection:

$$v\cdot \nabla \vartheta =v\cdot \nabla \varpi =0.$$

The two remaining Boussinesq couplings determine the amplitudes:

$$\boxed{\dot{\Theta }=-\frac{J\zeta \cdot G}{\lambda |\zeta {|}^2}\Omega ,\dot{\Omega }=\lambda {\zeta }_1\Theta .}$$

The first equation says that vorticity produces velocity, which displaces the old temperature gradient. The second says that a horizontal temperature gradient produces vorticity. Their product is independent of the base frequency $\lambda$ and depends on wave orientation relative to $G$ and the distinguished horizontal buoyancy direction.

Next, we will freeze the coefficients temporarily, diagonalize this two-by-two system, and identify the exponentially growing mode that the layer construction exploits.