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For a horizontal forced plume, the plume virtual origin lies downstream and below the physical source on the far-field vertical asymptote. The origin shift in centreline arc length is distinct from its vertical height: if $z=s-D+o(1)$ and $\hat{s}=s+s_0$, the virtual height is $z_v=-D-s_0$. Both geometric shifts scale with the jet length at fixed entrainment coefficient.

An inclined forced plume curves upward under buoyancy. In a stationary homogeneous ambient, its horizontal momentum flux is conserved and its vertical momentum flux increases. For kinematic plume fluxes, $m_x^{\prime }=0$ and $m_z^{\prime }=fq/m$, where $m=\sqrt{m_x^2+m_z^2}$ and the derivative is with respect to centreline arc length.

A buoyant plume with insufficient momentum for the corresponding pure plume balance. In the constant-entrainment top-hat model it has plume balance parameter $\Gamma >1$, unlike the momentum-excess forced plume. The equations accelerate the plume and drive the balance parameter toward one downstream. The definition depends on the chosen flux and entrainment convention; it is not merely a synonym for slow vertical speed.

Above $h_1$, plume 2 entrains middle-layer fluid rather than the original ambient. That fluid is already warm and light, so the plume's buoyancy relative to its surroundings is reduced. Its pre-interface density contrast and zero-source pure plume formula cannot simply be continued unchanged through the interface.

A consistent model treats the middle layer as a uniform new ambient, assumes a sharp interface and weak overall density differences, and carries the incident volume and momentum fluxes continuously across it. The relative buoyancy flux immediately above the interface isContinue an entraining plume model from these nonzero incident fluxes, using buoyancy measured relative to the middle layer. In that uniform layer the relative buoyancy flux is conserved, while entrainment raises the absolute buoyancy carried relative to the outside environment. This is generally a finite-flux or forced plume continuation, not a newly reset point-source similarity solution. A virtual-origin approximation can be used only after matching its fluxes appropriately.

At its matching level $h_2$, the plume density reaches the upper-layer density; it spreads and detrains, with possible finite-momentum overshoot. The model neglects mutual plume entrainment, wall heat transfer, interface thickness and detailed intrusion dynamics, and assumes each ambient layer is well mixed. In the equal-source limit $B_2^{+}=0$, the continuation is a neutrally buoyant momentum-driven flow, emphasizing why the upper interface loses its independent density meaning.

Two source scales obtained from dimensional analysis areThe jet length $L_M$ measures how far source momentum remains important relative to buoyancy-generated momentum. The source-volume length of a turbulent plume $L_Q$ measures how far imposed source volume remains important relative to entrained volume. Numerical constants and entrainment coefficient factors depend on the chosen normalization. From the equations here the useful exact volume-growth scale is ${\ell }_Q=[Q_0^3/(20{\alpha }^4{B}_0){]}^{1/5}$; the pure-similarity height whose flux equals $Q_0$ is $5{\ell }_Q/3$. A momentum crossover likewise contains a factor of order ${\alpha }^{-1/2}{L}_M$. The source radius $Q_0/\sqrt{M_0}$ is another useful geometrical combination, but it does not replace the two buoyancy/source crossover comparisons.

Integrate the differential relation from part (a) with finite source values:DefineThis plume flux-balance invariant rearranges toUse it to express $\sqrt{M}=[5B_0{Q}_0^2/(8\alpha ){]}^{1/5}[(Q/Q_0{)}^2-c{]}^{1/5}$, and substitute into $Q^{\prime }=2\alpha \sqrt{M}$. The coefficient satisfies $(2\alpha /Q_0{)}^5[5B_0{Q}_0^2/(8\alpha )]=20{\alpha }^4{B}_0/Q_0^3$, givingWith positive source fluxes, $c<1$. For $c<0$ the source has excess momentum relative to pure plume balance, so it is a momentum-dominated forced plume. For $c=0$ it is already in pure plume balance, and the exact solution is the pure similarity translated to a finite plume virtual origin. For $0<c<1$ the source has deficient momentum relative to its buoyancy and volume fluxes, a lazy plume. Here “forced” in the broad sense merely means finite source fluxes; this sign classification distinguishes its jet-like and lazy branches. At large height, finite $c{Q}_0^2/Q^2$ vanishes and every positive branch approaches pure plume balance.

Use $F=F_s>0$ and constant entrainment parameter $\alpha >0$. The printed integrals are normalized by $\pi$, so a top-hat plume model has $Q=b^{2}w$, $M=b^2{w}^2$ and $F=b^{2}w{g}^{\prime }$; the physical area is still $\pi b^2$. Keeping this normalization avoids spurious factors of $\pi$ in the flux equations.

Eliminate height between the two flux balances. The plume flux-balance invariant follows fromsoFor positive fluxes, both $M$ and $Q$ increase. In particular $Q^{\prime }\ge 2\alpha \sqrt{M_s}>0$, so $Q\to \infty$ as $z\to \infty$. Therefore $C/(\lambda Q^2)\to 0$, whatever the finite source imbalance, and $M\sim {\lambda }^{2/5}{Q}^{4/5}$. Integrating $Q^{\prime }\sim 2\alpha {\lambda }^{1/5}{Q}^{2/5}$ gives the attracting similarity solutionThis proves attraction to pure plume similarity for physical positive source fluxes. Sources with a vanishing initial volume or momentum flux enter the positive regime immediately and can be obtained by the corresponding limiting solutions; a point buoyancy source has the same similarity form with zero virtual-origin offset. The statement assumes positive buoyancy flux: a jet with $F=0$ is a different asymptotic problem.

Define the plume balance parameterPure plume balance means $\Gamma =1$ at each height, or equivalently $C=0$: momentum, buoyancy and entrainment have their similarity relation, even when the source has finite radius. A forced plume has $\Gamma <1$, meaning excess momentum relative to this balance; a lazy plume has $\Gamma >1$, meaning insufficient momentum. The invariant ensures that the sign of the imbalance is preserved while $\Gamma \to 1$. In particular, a strictly positive source remains pure at every height precisely whenFor this source the exact solution is the same similarity form with $z$ replaced by $Z=z+z_0$. Writing $a=(9\alpha F_s/10{)}^{1/3}$,Thus the radius growth of an axisymmetric pure plume is $b^{\prime }=6\alpha /5$ and the plume virtual origin lies at $-z_0$, with $z_0=5b_s/(6\alpha )$. Equivalently, in terms of top-hat source properties, pure plume balance is $g_s^{\prime }{b}_s/w_s^2=8\alpha /5$.

For the stated source-to-source geometry, the relevant horizontal distance is $L_s$. Consequently the edge criterion is $b_m=L_s$, rather than the radius at which two expanding edges first touch. Since $b(z)=b_s+6\alpha z/5$,Here $b_s=Q_s/\sqrt{M_s}={\lambda }^{-1/5}{Q}_s^{3/5}$. A positive merging height requires $L_s>b_s$; equality puts it at the source, while an already wider source satisfies the geometric condition from the start. The no-interaction premise extrapolates independent plumes even though their edges would first touch earlier, when each radius is $L_s/2$.

Use $M^{5/2}=\lambda Q^2$ with $Q=b^{2}w$ and $M=b^2{w}^2$ to get $F_s=(8\alpha /5)b{w}^3$. The individual plume properties at the specified height are thereforeThe source volume flux affects the height through the virtual origin, but it does not affect these properties at a prescribed radius in an exactly pure plume.

For the stipulated merger of two equal pure plumes, the new radius is $b_c=\sqrt{2}{b}_m$. This factor is outside $b_m$, as required by the area. Since speed and reduced gravity are held fixed, the merged fluxes are $Q_c=2Q_m$, $M_c=2M_m$ and $F_c=2F_s$. Their balance parameter isThe immediate combined plume is lazy. The same result follows from $\Gamma =5g^{\prime }b/(8\alpha w^2)$: enlarging the radius at fixed $g^{\prime },w$ increases this parameter.

At fixed $w_m,g_m^{\prime }$, the radius required for pure plume balance isThat is half the stipulated combined area $2\pi L_s^2$. This fixed-speed pure-plume merger and flux conservation comparison exposes the limitation of demanding immediate pure balance with unchanged speed and reduced gravity: reducing the area to this value would also halve all three merged fluxes. A conservative merger must instead adjust its properties and begin out of pure balance. Under the given constant-entrainment equations, the resulting lazy plume subsequently approaches the attracting similarity solution with total buoyancy flux $2F_s$.

Put $m_0=M_0/{\rho }_0$ and $f_0=F_0/{\rho }_0>0$ under the literal mass convention. Their dimensions are $[m_0]=L^4{T}^{-2}$ and $[f_0]=L^4{T}^{-3}$. Dimensional analysis givesThe jet length compares source momentum with buoyancy. The region near the source behaves as a turbulent round jet; sufficiently far away, the forced plume behaves like a pure plume. With the constant entrainment coefficient retained explicitly, substantial turning occurs over a length of order ${\ell }_J=L_J/\sqrt{E}$, where $E=2\alpha \sqrt{\pi }$. Statements involving $s\ll L_J$ or $s\gg L_J$ normally hold this dimensionless coefficient fixed.

Near the point source, $m\sim m_0$ and the Batchelor entrainment hypothesis gives $q\sim E\sqrt{m_0}s$. ThereforeIntegrating the centreline tangent givesTo leading order the centreline is a straight turbulent round jet at its source inclination. For the horizontal source, the first curvature is the horizontal forced-plume trajectoryFor a vertical source, $\cos {\theta }_0=0$, the trajectory stays vertical rather than developing this cubic transverse displacement. If $F_0=0$, there is no finite buoyancy-induced jet length and the turbulent round jet remains straight.

The dimensionless ratio $\Gamma =5F{Q}^2/(8\alpha M^{5/2})$ compares a plume to the constant-entrainment pure balance. Values one, below one and above one describe pure plume balance, a momentum-excess forced plume, and a momentum-deficient lazy plume, respectively. For fluxes normalized by pi and a top-hat profile, $\Gamma =5g^{\prime }b/(8\alpha w^2)$.