= Solution
Use a <two-sided top-hat line plume> with full width $w(z)$, upward speed $U(z)$, and <reduced gravity> $b(z)$ uniform across the plume. Fluxes are measured per unit span along the line source: the <volume flux> is $V=wU$, the kinematic <momentum flux> is $M=wU^2$, and the <buoyancy flux> is $B=wUb$. If the inward edge speed is $eU$, where $e>0$ is the <entrainment coefficient>, the two exposed edges give
$$
\frac{dV}{dz}=2eU,\qquad \frac{dM}{dz}=wb=\frac BU,\qquad \frac{dB}{dz}=0.
$$
These are the <volume conservation>, <momentum conservation>, and <buoyancy flux> balances for a <Boussinesq approximation> plume in an unstratified ambient. Entrained ambient fluid supplies neither vertical momentum nor reference <buoyancy>.
A <pure plume> has no persistent source length scale. Since a line-source $B$ has dimensions $L^3T^{-3}$, <dimensional analysis> gives constant $U\propto B^{1/3}$, $w\propto z$, and $V\propto z$. Substituting constant $U$ into $M=UV$ and the momentum balance determines the coefficients:
$$
2eU^2=\frac BU,\qquad \boxed{U=(2e)^{-1/3}B^{1/3},\quad w=2ez,\quad V=(2e)^{2/3}B^{1/3}z.}
$$
Thus $\lambda=(2e)^{2/3}$ and $\mu=2e$ when width is full width and velocity and <buoyancy> have top-hat profiles. If width means half-width, $\mu=e$ instead. Other prescribed profile shapes change these numerical factors; the linear growth laws remain the same. The point-source $B$ in Question 1 has different dimensions, so its $z^{5/3}$ law must not be used here.
Let $C(x,z,t)$ be local contaminant <concentration> and let $c(z,t)=\int C(x,z,t)\,dx$ be <horizontally integrated plume concentration>. Its conserved amount per unit span is $\int_0^\infty c\,dz=V_c$. The dilute contaminant is a <passive scalar>; the maintained plume is unaffected by its impulsive release. A one-dimensional effective transport closure takes the integrated scalar flux to be
$$
J=a c-K(z)c_z,\qquad a=\alpha B^{1/3},\qquad K(z)=Dz,\qquad D=\beta B^{1/3}.
$$
The constant advective speed and the <eddy diffusivity> scaling $K\sim Uw\propto B^{1/3}z$ follow from the line-plume scales. The <entrainment coefficient> alone does not determine the scalar dispersion coefficient: this Fickian closure for the integrated variable is an additional modelling assumption. <Mass conservation> $c_t+J_z=0$ then gives the displayed transport equation in the PDF. With this effective closure and advective speed chosen as $U$, $\alpha=(2e)^{-1/3}$; if $K=\kappa Uw$, then $\beta=\kappa(2e)^{2/3}$, with independent dimensionless mixing coefficient $\kappa$.
There is a second possible closure convention. If one instead applies local diffusive flux to <horizontally averaged plume concentration> $\overline C=c/w$, its integrated diffusive flux is $-Kw\overline C_z=-K(c_z-c/z)$. For $K=Dz$, the total scalar flux is then $(U+D)c-Dzc_z$. The same printed equation is obtained with $a=U+D$, rather than $a=U$. Thus the printed $\alpha$ and $\beta$ should be regarded as effective coefficients unless the averaging and closure conventions are specified; no equality between $\alpha$ and the velocity prefactor is universal.
Assume $a,D>0$, $z>0$, an initial impulse $V_c\delta_0$ at the origin, no further contaminant input, and zero endpoint scalar flux for $t>0$. For a <similarity solution>, put
$$
r=\frac aD=\frac\alpha\beta,\qquad \eta=\frac z{Dt},\qquad c=\frac{V_c}{Dt}f(\eta).
$$
Substituting into the <advection-diffusion equation> gives
$$
\eta f''+(1-r+\eta)f'+f=0,\qquad [\eta f'+(\eta-r)f]'=0.
$$
Decay at infinity makes the integrated constant zero. Hence $f'/f=r/\eta-1$, and normalization with the <gamma function> gives $f(\eta)=\eta^re^{-\eta}/\Gamma(1+r)$. The resulting <gamma impulse solution for linearly increasing diffusivity> is
$$
\boxed{c(z,t)=\frac{V_c}{\beta B^{1/3}t\,\Gamma(1+\alpha/\beta)}\left(\frac z{\beta B^{1/3}t}\right)^{\alpha/\beta}\exp\left(-\frac z{\beta B^{1/3}t}\right).}
$$
Its <integral> over $z>0$ is $V_c$. Its flux is $J=(z/t)c$, which vanishes at both endpoints for $t>0$. The scale $Dt$ shrinks to zero as $t\downarrow0$, so the normalized solution has <weak convergence of probability measures> to the required unit source impulse. The mass at the boundary is a full unit impulse on the half-line, not half of a whole-line impulse.
For the integrated quantity, $\partial_z\log c=r/z-1/(Dt)$, and consequently
$$
\boxed{z_{\max,c}=rDt=\alpha B^{1/3}t.}
$$
This verifies the printed location for <horizontally integrated plume concentration>. It is a maximum because the derivative changes from positive to negative there. The normalized profile is a <gamma distribution> with shape $1+r$: its <expected value> is $(a+D)t$ and its <variance> is $(a+D)Dt$, so neither the mean position nor the spread should be mistaken for the modal position.
The PDF changes from “integral” to “averaged” concentration in its final request. A genuine <horizontally averaged plume concentration> divides by the growing width $w=\mu z$, and is therefore
$$
\boxed{\overline C(z,t)=\frac{V_c}{\mu(Dt)^2\Gamma(1+r)}\left(\frac z{Dt}\right)^{r-1}e^{-z/(Dt)}.}
$$
It is normalized by $\int_0^\infty w\overline C\,dz=V_c$, not by $\int\overline C\,dz$. When $a>D$, its interior maximum is
$$
\boxed{z_{\max,\overline C}=(a-D)t=(\alpha-\beta)B^{1/3}t.}
$$
When $a=D$ it decreases from a finite boundary supremum; when $0<a<D$ it is singular at the ideal point source and has no positive interior maximum. A finite source regularizes that singularity. Thus the final printed maximum is correct for the integrated variable $c$, or for an “average” using a fixed reference width, but is not generally the maximum of the local mean $c/w(z)$. Both quantities have been given rather than silently identifying them. If $D=0$, the smooth similarity formula is replaced by the advected impulse $V_c\delta(z-at)$; positive <eddy diffusivity> is essential to the gamma profile.
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