Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 79 / 3

Take $z$ positive upward from the release and define the pressure scale height $H_p=p_0/({\rho }_{0}g)$. Neglecting the small ambient density variation in the hydrostatic pressure calculation,Conservation of gas mass and Boyle's law giveHere $V_b(0)\simeq {\varphi }_0{V}_0$ to leading order in the dilute bubble concentration. If the thermal's total volume is $V(z)$, its bubble fraction and bubble contribution to reduced gravity areThus the volume-integrated kinematic buoyancy contribution is $g\varphi V=g{\varphi }_0{V}_0/(1-z/H_p)$, and the corresponding upward force is approximately ${\rho }_0$ times this quantity. Bubble expansion increases this total contribution even while entrainment dilutes the local concentration.

Neglect bubble rise relative to the water when the bubbles' terminal slip speed is much smaller than the thermal rise speed, their response time is short compared with $a/W$, and they remain well dispersed and retained. Small bubbles are more likely to satisfy these conditions; large bubbles can escape through the thermal. The approximations also require ${\rho }_b\ll {\rho }_0$, $\varphi \ll 1$, negligible gas dissolution and no important surface-tension correction. Pressure must remain positive and the hydrostatic expression is used only below the ocean surface.

The Boussinesq approximation retains small density differences only in the gravitational buoyancy force and uses a constant reference density in inertia and volume conservation. The entrainment coefficient $\alpha$ specifies the inward velocity of ambient fluid relative to the thermal boundary, $u_e=\alpha W$. Turbulent interface motions engulf ambient water and mix it into the coherent rising volume.

For a spherical buoyant thermal, $V=4\pi a^3/3$. To leading order in the dilute gas fraction, expansion of the gas does not dominate the total-volume balance, so fluid entrainment givesBut $dV/dt=4\pi a^{2}da/dt$ and $dz/dt=W$. Therefore, wherever the thermal is rising,This growth law for a dilute bubbly thermal is independent of stratification under the constant-$\alpha$ model. Stratification changes the speed and eventual stopping height; it does not enter this volume-per-distance relation. In real flows the entrainment coefficient itself can vary with stratification and flow structure.

For a high-Reynolds number thermal, buoyancy force scales as ${\rho }_0{g}^{\prime }{a}^3$ and inertial resistance scales as ${\rho }_0{W}^2{a}^2$. Their balance gives $W^2\propto g^{\prime }a$. It is therefore natural to write a diameter-based Froude number closureThe order-one coefficient $F$ accounts for the thermal's shape, entrainment and flow structure. The main resistance is inertial pressure or form drag and the momentum needed to entrain and accelerate surrounding water, rather than molecular skin friction. Added mass matters when the thermal accelerates, so the Froude closure is a quasi-steady approximation.

In a homogeneous ocean all of the leading buoyancy comes from the bubbles. Parts (a) and (b) giveSubstituting into the speed law yieldsFor $z\gg a_0/\alpha$, this becomesThe square root is essential both physically and dimensionally. This approximation still requires dilute bubbles and retained gas; it is not a valid prediction of infinite speed where the pressure approximation approaches zero.

For constant buoyancy frequency in the Boussinesq approximation, the ambient density isConservation of liquid mass during fluid entrainment gives $d(V\rho )/dz={\rho }_a(z)V^{\prime }(z)$, with $\rho (0)={\rho }_0$. Integrating by parts,Since $a=a_0+\alpha z$ and $V\propto a^3$,Hence the water density within the thermal isThe entrained water is denser than its new surroundings because it retains a mixture of fluid drawn from lower levels.

The bulk density is $\rho (1-\varphi )$. Dropping products of the small density contrast with $\varphi$, its net reduced gravity isThe first term is the expanding-bubble contribution and the second is the restoring effect of stratification. In the quasi-steady Froude model, ascent stops when $g^{\prime }=0$. The neutral height of a bubbly thermal in a stratified fluid therefore satisfiesThis is a fifth-degree polynomial, with leading coefficient $-{\alpha }^4/H_p$ at finite release pressure. The relevant height is the first positive zero reached while the thermal is still below the surface, dilute and within the Boussinesq range. Depending on parameters there may be no such zero before the surface, or more than one mathematical zero; a later root beyond a first stopping point is not another continuous ascent solution. The model's stopping height is a neutral-buoyancy estimate, not a claim that inertial overshoot is absent.

For a very deep release, $z/H_p\ll 1$ over the rise, so bubble expansion is negligible. The neutral-height equation reduces toThusIf the stopping thermal is also much larger than its initial size, the leading form isThe first expression does not require this extra entrainment-dominated limit.

Near the neutral height the quasi-steady speed tends to zero. Momentum can produce some overshoot and subsequent adjustment or oscillation. The water spreads horizontally as a buoyant intrusion near its equilibrium density level. Bubble slip then becomes important relative to the slowing liquid, so bubbles can separate and continue rising toward the surface, reducing the residual liquid's buoyancy. Loss of bubbles and further mixing alter the final intrusion level; the retained-bubble model no longer applies indefinitely after stopping.