= Solution
Use the conventional <kinematic plume fluxes>
$$
\boxed{Q=b^2w,\qquad M=b^2w^2,\qquad F=b^2wg'.}
$$
Then $\pi Q$ is the <volume flux>, $\pi\rho_r Q$ the dimensional <mass flux>, $\pi\rho_r M$ the dimensional <momentum flux>, and $\pi F$ the kinematic <buoyancy flux>. Calling $\pi Q$ a <mass flux> uses a density-normalized convention: the reference <density> has been suppressed. If density-weighted momentum and <buoyancy> fluxes are desired, multiply $M$ and $F$ by the same constant $\rho_r$ throughout. Here the conventional kinematic definitions make the <Boussinesq approximation> explicit.
For an upward plume with $Q,M>0$, the inverse relations are
$$
A=\frac{Q^2}{M},\qquad w=\frac MQ,\qquad b=\frac Q{\sqrt M},\qquad B=Ag'=\frac{FQ}{M}.
$$
The <volume conservation>, vertical <momentum conservation>, and <buoyancy> balances from part a become
$$
\boxed{\begin{aligned}
\partial_t\left(\frac{Q^2}{M}\right)+Q_z&=2\alpha\sqrt M,\\
Q_t+M_z&=\frac{FQ}{M},\\
\partial_t\left(\frac{FQ}{M}\right)+F_z&=-N^2Q.
\end{aligned}}
$$
Here the vertical <momentum conservation> equation uses $\rho_r$ in the inertia and retains $(\rho_0-\rho)gA=\rho_r B$ in the <buoyancy> force. The ambient has no imposed vertical momentum at entrainment.
To classify the resulting <partial differential equations>, work in the nonsingular primitive variables $(A,w,g')$ with $A>0$. Subtract $w$ times the volume equation from the momentum equation and $g'$ times the volume equation from the <buoyancy> equation. This gives
$$
\begin{aligned}
A_t+wA_z+Aw_z&=2\alpha w\sqrt A,\\
w_t+ww_z&=g'-\frac{2\alpha w^2}{\sqrt A},\\
g'_t+wg'_z&=-N^2w-\frac{2\alpha wg'}{\sqrt A}.
\end{aligned}
$$
The principal matrix is therefore
$$
K=\begin{pmatrix}w&A&0\\0&w&0\\0&0&w\end{pmatrix},\qquad
\det(K-\lambda I)=(w-\lambda)^3.
$$
All three <characteristic speeds> are real and equal to $w$, so the model is a <hyperbolic system> if that term means only reality of the characteristic roots. However, for $A>0$, $(K-wI)v=0$ requires $v_w=0$, leaving only two independent <eigenvectors>. Hence
$$
\boxed{\text{the top-hat system is weakly hyperbolic, not strictly or strongly hyperbolic}.}
$$
This <weak hyperbolicity of the unsteady top-hat plume> prevents a complete set of independent characteristic fields. For example, freezing $A,w$ in the homogeneous principal part gives $D_t\delta w=0$ and $D_t\delta A=-A\partial_z\delta w$. A Fourier perturbation of wave number $k$ therefore produces a term proportional to $kt$ in $\delta A$. Real characteristic roots alone do not provide uniform control of high-frequency perturbations. The flux-variable transformation above is invertible in the stated region, so changing to $(Q,M,F)$ does not remove this degeneracy.
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