Inclined forced plume

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.

For zero source volume flux, constant Batchelor entrainment, and a horizontal source, the near-source trajectory is cubic: $z\sim \alpha \sqrt{\pi }{f}_0{x}^3/(3m_0^{3/2})$. Far from the source the centreline approaches a vertical asymptote, with its horizontal distance from that asymptote proportional to $z^{-1/3}$.