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Understanding Terms in the Forced Inviscid Boussinesq Equations

Hello, and welcome to the first lesson of this course. We will begin with the two-dimensional forced inviscid Boussinesq system: a deliberately stripped-down fluid model whose feedback loop is the starting point for the blowup constructions you want to navigate. This first module develops that model’s local wave-amplification mechanism; the second will show how infinitely many such local episodes are assembled and how the structure transfers to axisymmetric Euler flow.

For now, the objective is more basic but essential: read the equations as a dynamical system. By the end, you should be able to point to every term and say whether it transports a quantity, enforces a constraint, produces buoyancy, or represents an externally supplied input.

The system: two evolving fields and a constraint

The paper studies the forced inviscid Boussinesq equations on $R^{2}$ :

\partial_{t} θ + u \cdot \nabla θ = f_{θ},

\partial_{t} u + u \cdot \nabla u + \nabla p = θ e_{2} + f_{u},

div u = 0,

where

e_{2} = (0, 1) .

The unknowns are:

$θ (x, t)$ : a scalar temperature anomaly;
$u (x, t) = (u_{1} (x, t), u_{2} (x, t))$ : the planar velocity field;
$p (x, t)$ : pressure, more precisely a modified or kinematic pressure;
$f_{θ} (x, t)$ and $f_{u} (x, t)$ : externally imposed scalar and vector forcing terms.

Here $x = (x_{1}, x_{2})$ , with $x_{2}$ the upward coordinate.

The original paper’s short introduction is worth reading now because it fixes the physical sign conventions and, crucially, tells you exactly what its forcing terms are for.

[PDF] Blowup for the Boussinesq equations with smooth forcing

Read Section 1.1, “The Boussinesq equation,” of Blowup for the Boussinesq equations with smooth forcing. It presents the exact system used throughout this course and states the authors’ interpretation of temperature anomaly, buoyancy, pressure, and the designed external forces.

In Section 1.1, begin at the displayed system (1.1). Then read the explanatory paragraph immediately below it. Finally, read the brief paragraph beginning “The Boussinesq equation can also be written in vorticity form,” but treat the vorticity equation as a preview: we derive it in the next lesson. Focus on the distinction between the state (\theta,u), the constraint \operatorname{div}u=0, and the supplied forces (f_\theta,f_u).

A useful way to hold the system in mind is that $θ$ is both:

a scalar quantity carried around by the flow, and
a source of vertical acceleration for that same flow.

That two-way coupling is where the later amplification mechanism begins.

Transport: following a fluid parcel

Define the material derivative

D_{t} := \partial_{t} + u \cdot \nabla.

It measures the rate of change experienced by an observer moving with the fluid. If $X (t)$ is the trajectory of a parcel, defined by

\frac{d X}{d t} = u (X (t), t),

then, for any scalar field $q$ ,

\frac{d}{d t} q (X (t), t) = D_{t} q (X (t), t) .

Thus the temperature equation becomes

D_{t} θ = f_{θ} .

The term $u \cdot \nabla θ$ is advection or transport. In coordinates,

u \cdot \nabla θ = u_{1} \partial_{1} θ + u_{2} \partial_{2} θ .

It does not directly create or destroy the values of $θ$ . Instead, it rearranges them in space. If $f_{θ} = 0$ , then each moving parcel keeps its initial temperature anomaly:

\frac{d}{d t} θ (X (t), t) = 0.

This does not mean that $θ$ remains spatially simple. A velocity field can fold, compress in one direction, and stretch in another, creating very large values of $\nabla θ$ while the values of $θ$ themselves remain bounded. That distinction between scalar amplitude and scalar gradient will be central later.

The velocity equation has exactly the same transport structure:

D_{t} u + \nabla p = θ e_{2} + f_{u} .

Equivalently,

D_{t} u = - \nabla p + θ e_{2} + f_{u} .

So the fluid parcel’s acceleration is the sum of three effects:

pressure force, $- \nabla p$ ;
buoyancy force, $θ e_{2}$ ;
external vector force, $f_{u}$ .

The nonlinear term

u \cdot \nabla u

is the same advection principle applied componentwise to velocity:

u \cdot \nabla u = (u_{1} \partial_{1} u_{1} + u_{2} \partial_{2} u_{1} u_{1} \partial_{1} u_{2} + u_{2} \partial_{2} u_{2}) .

It is often called convective acceleration: a parcel can accelerate even when the velocity field is time-independent at each fixed location, simply because it moves into a region with a different velocity.

Incompressibility and the role of pressure

The final equation,

div u = \partial_{1} u_{1} + \partial_{2} u_{2} = 0,

is the incompressibility constraint.

Geometrically, the flow preserves area. A small material blob may be stretched horizontally and compressed vertically, or vice versa, but its area is unchanged. In two dimensions that is the relevant analogue of volume preservation.

Incompressibility does not mean:

that $u$ is constant in space;
that the flow cannot stretch a scalar gradient;
that the density anomaly represented by $θ$ is literally zero.

Instead, the approximation treats a reference density as constant for mass conservation and inertia, while retaining small density variations in the buoyancy term. The scalar $θ$ records the anomaly responsible for buoyant acceleration.

Pressure is not an independently transported physical substance in this model. Its principal mathematical role is to ensure that the velocity remains divergence-free. Taking divergence of the momentum equation gives, formally,

Δ p = \partial_{2} θ + div (f_{u}) - div ((u \cdot \nabla) u) .

Or, equivalently,

- Δ p = div ((u \cdot \nabla) u) - \partial_{2} θ - div (f_{u}) .

Thus pressure is found by solving an elliptic equation whose source depends on the current velocity, buoyancy, and forcing. This makes it nonlocal: changing the flow in one region can affect the pressure elsewhere.

There is also a physical bookkeeping point in the paper’s formulation. The large background hydrostatic pressure associated with gravity has already been absorbed into $p$ . What remains visibly in the equation is the relative upward force arising from a temperature or density anomaly.

For a brief physical derivation of why a Boussinesq model treats density as constant except in the buoyancy force, watch the following excerpt.

[CFD] The Boussinesq Approximation for Bouyancy Driven (Natural Convection) Flow

Watch “The Boussinesq Approximation for Bouyancy Driven (Natural Convection) Flow” from Fluid Mechanics 101 for the physical approximation behind the model. It explains why density fluctuations are neglected in the inertia and continuity terms but retained in buoyancy.

Watch the approximation, where the reference density plus a small perturbation is introduced and the selective treatment of density is explained. Then watch the reduced equations, which connects that assumption to incompressible continuity and an extra buoyancy term in momentum balance. The notation differs from the blowup paper, but the conceptual division of roles is the same.

The blowup paper uses an idealized inviscid version. In a more standard viscous and thermally diffusive Boussinesq system, one would see terms such as

ν Δ u and κ Δ θ,

with viscosity $ν$ and thermal diffusivity $κ$ . Their absence here is consequential: there is no built-in diffusive mechanism smoothing sharp velocity or temperature structures.

Buoyancy: why $θ$ matters dynamically

The term

θ e_{2} = (0, θ)

acts only in the vertical direction. In the convention of the paper, $e_{2}$ points upward and a positive $θ$ represents a lighter-than-reference fluid parcel. Hence positive $θ$ produces upward acceleration.

At first sight this coupling may seem weak: the velocity equation sees $θ$ , not $\nabla θ$ . But after taking curl, the spatial derivative $\partial_{1} θ$ becomes the source of vorticity. That is why later stages of the construction care intensely about producing large temperature gradients, especially horizontal ones.

For the present lesson, keep the causal structure at the level of the original variables:

The velocity $u$ transports the temperature anomaly $θ$ .
The resulting arrangement of $θ$ exerts a vertical buoyancy force on $u$ .
The incompressibility constraint and pressure redistribute the induced acceleration so that the velocity stays divergence-free.

This is a coupled nonlinear feedback system, not a passive-scalar equation with a prescribed background flow.

A sign check is useful. If $θ > 0$ , with no pressure gradient and no external force at a particular point, then the momentum equation says the vertical component of acceleration is positive. If $θ < 0$ , it is negative. The sign is a convention, but once $e_{2}$ has been chosen as upward, the convention is internally consistent.

What “forced” means in this paper

The two forcing terms enter different equations and should not be conflated:

Term	Equation	Direct role
$f_{θ}$	$D_{t} θ = f_{θ}$	Creates, removes, or modifies temperature anomaly along fluid paths
$f_{u}$	$D_{t} u = - \nabla p + θ e_{2} + f_{u}$	Applies an externally supplied acceleration to the fluid

They are external in the sense that the displayed Boussinesq dynamics does not determine them. Once they are specified as functions of space and time, they act as inputs to the system. They are not viscosity, not pressure, and not another name for buoyancy.

This matters especially for the theorem motivating the course. The construction does not claim that an arbitrary unforced physical fluid necessarily follows its engineered scenario. It produces carefully chosen forces that remain smooth even while the solution’s derivatives become singular at a finite terminal time. The mathematical challenge is precisely that the external inputs must not themselves hide the singularity.

The parcel-wise interpretation makes the division clean:

\frac{d}{d t} θ (X (t), t) = f_{θ} (X (t), t),

\frac{d}{d t} u (X (t), t) = - \nabla p (X (t), t) + θ (X (t), t) e_{2} + f_{u} (X (t), t) .

The first equation tells us what changes a parcel’s temperature anomaly. The second tells us what accelerates the parcel. The pressure is determined globally to preserve incompressibility, whereas $θ e_{2}$ is the internal buoyancy feedback and $f_{u}$ is supplied from outside the unforced model.

A compact reading checklist

When you encounter this system in either paper, parse it with the following checklist:

$\partial_{t}$ : change at a fixed spatial location.
$u \cdot \nabla$ : change caused by motion through a spatially varying field.
$D_{t}$ : change experienced by a moving fluid parcel.
$div u = 0$ : area preservation and the incompressibility constraint.
$\nabla p$ : constraint force that adjusts the velocity to remain divergence-free.
$θ e_{2}$ : upward buoyancy produced by a positive temperature anomaly.
$f_{θ}, f_{u}$ : externally selected scalar and momentum inputs.
No Laplacians: no viscosity and no thermal diffusion; this is the inviscid model.

Takeaways and next lesson

The forced inviscid Boussinesq system consists of a transported scalar $θ$ , an incompressible velocity $u$ , and a pressure field $p$ that enforces the divergence-free constraint. Temperature affects the flow through vertical buoyancy, while the flow transports and distorts temperature. The terms $f_{θ}$ and $f_{u}$ are separate external inputs, chosen in the blowup construction rather than generated by the Boussinesq coupling itself.

Next, we will take the curl of the momentum equation and derive the vorticity formulation

\partial_{t} ω + u \cdot \nabla ω = \partial_{1} θ + curl f_{u} .

That equation exposes the key derivative-level feedback: a horizontal temperature gradient creates vorticity.

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Deriving the 2D Boussinesq Vorticity Equation

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