= Solution
For a horizontal source, $m_x=m_0$ throughout. Far downstream $m_z\gg m_0$, so the <kinematic plume fluxes> obey the vertical <pure plume> equations to leading order,
$$
q'=E\sqrt{m_z},\qquad m_z'=f_0q/m_z,\qquad f=f_0.
$$
Set $q=q_*f_0^{1/3}\hat s^{5/3}$ and $m_z=m_*f_0^{2/3}\hat s^{4/3}$ with $\hat s=s+s_0$. Matching exponents and coefficients gives $(5/3)q_*=E\sqrt{m_*}$ and $(4/3)m_*^2=q_*$. Consequently
$$
\boxed{m_*=(9E/20)^{2/3},\qquad q_*=\frac43(9E/20)^{4/3}},
$$
and the requested literal mass and density-weighted <buoyancy fluxes> are
$$
\boxed{Q=\rho_0q_*f_0^{1/3}\hat s^{5/3},\qquad
M_z=\rho_0m_*f_0^{2/3}\hat s^{4/3},\qquad F=F_0}
$$
to leading order. The radius is $b\sim(6\alpha/5)\hat s$ and the axial <velocity> decreases as $\hat s^{-1/3}$.
The centreline becomes nearly vertical but retains finite horizontal drift:
$$
\frac{dx}{ds}\sim\frac{m_0}{m_*f_0^{2/3}}\hat s^{-4/3},\qquad
\boxed{x_\infty-x\sim\frac{3m_0}{m_*f_0^{2/3}}\hat s^{-1/3}}.
$$
Since $dz/ds=1-O(\hat s^{-8/3})$, there is a finite positive geometric deficit $D=\lim_{s\to\infty}(s-z)$, and $z=s-D+o(1)$. The equivalent vertical <plume virtual origin> is therefore located at
$$
\boxed{(x_v,z_v)=(x_\infty,-D-s_0)}.
$$
\b[For the horizontal point-source forced plume, the virtual source is downstream and below the actual source.] The horizontal <turbulent round jet> first entrains substantial ambient fluid while rising only a little. Thus at the height where it turns upward it already has a finite radius and <volume flux>; an equivalent <pure plume> must have begun rising from below to acquire these. The turning displacement, entrained radius divided by its far-field spreading angle, and virtual vertical depth all scale as
$$
x_\infty=O(\ell_J),\qquad |z_v|=O(\ell_J),\qquad s_0=O(\ell_J),\qquad
\ell_J=\frac{L_J}{\sqrt{2\alpha\sqrt\pi}}.
$$
At fixed <entrainment coefficient> this is simply $O(L_J)$. The arclength origin $s=-s_0$ should not be interpreted as a physical height: the curved near field supplies the additional shift $D$. The numerical constants depend on the chosen <top-hat plume model> and <entrainment coefficient>; scaling does not fix them.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-345-plume.png]
{title=A horizontal forced plume and its far-field virtual origin}
The <inclined forced plume> bends from a cubic near-field trajectory toward a vertical asymptote. Dashed far-field spreading lines extrapolate to the equivalent <plume virtual origin>; the marked location is obtained from the integral model for this illustration, not a universal numerical prediction.
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