Interpretation of umbrella cloud growth and morphology: implications for flow regimes of short-lived and long-lived eruptions

Abstract

New numerical and analytical modeling shows that the growth of a volcanic umbrella cloud, expressed as the increase of radius with time, proceeds through regimes, dominated by different force balances. Four regimes are identified: Regime Ia is the long-time behavior of continuously-supplied intrusions in the buoyancy-inertial regime; regime IIa is the long-time behavior of continuously-supplied, turbulent drag-dominated intrusions; regime Ib is the long-time behavior of buoyancy-inertial intrusions of constant volume; and regime IIb that of turbulent drag-dominated intrusions of constant volume. Power-law exponents for spreading time in each regime are 3/4 (Ia), 5/9 (IIa), 1/3 (Ib), and 2/9 (IIb). Both numerical modeling and observations indicate that transition periods between the regimes can be long-lasting, and during these transitions, the spreading rate does not follow a simple power law. Predictions of the new model are consistent with satellite data from seven eruptions and, together with observations of umbrella cloud structure and morphological evolution, support the existence of multiple spreading regimes.

Appendix A: Drag-dominated intrusions of constant volume

After the cessation of an eruption, the volume of fluid in the plume remains approximately constant (increasingly only slowly due to entrainment), but buoyancy forces result in continued spreading. In the absence of drag, a buoyancy-inertial spreading regime becomes established Ungarish and Zemach (2007), with a radial growth rate of t ^1/3. However, our numerical results (Fig. 10) indicate that turbulent drag has often become significant by the point at which an eruption ceases, meaning that spreading of the plume will be drag-dominated. We calculate a similarity solution to the governing equations in this regime, which exhibits a radial growth rate of r _f ∼t ^2/9. This derivation is analogous to that in (Johnson et al. 2015) for the drag-dominated spread of an intrusion supplied by a constant flux.

Fig. 10

Profiles of intrusion thickness $\mathcal {H}$ and radial velocity $\mathcal {U}$ for an intrusion of constant volume in a turbulent-drag dominated spreading regime

After the eruption has ceased, there is no longer a volume flux per radian Q feeding the intrusion, so we nondimensionalize by scaling lengths to V ^1/3, where V is the intrusion volume per radian and times to N ⁻¹, as before. At late times, the governing Eq. 2 form a dominant balance in which buoyancy spreading forces are balanced by turbulent drag. In this regime, the governing equations become (in nondimensional form)

$$ \frac{\partial h}{\partial t}+\frac{1}{r}\frac{\partial}{\partial r}\left( ru h\right)=0\qquad\text{and}\qquad \frac{h^{2}}{4}\frac{\partial h}{\partial r}=-C_{D}|u|u, $$

(A.1a,b)

respectively. We seek a similarity solution for these equations and therefore first look for scalings. Integrating (A.1a,ba) across the intrusion, we find that ${r_{f}^{2}}h\sim 1$, while from Eq. A.1a,bb the balance between driving buoyancy forces and drag results in $h^{3}/r_{f}\sim C_{D} {r_{f}^{2}}/t^{2}$. These scalings suggest that $r_{f}\sim C_{D}^{-1/9}t^{2/9}$ and that a similarity solution may exist in which

$$\begin{array}{@{}rcl@{}} h&=&\kappa C_{D}^{2/9}t^{-4/9}\mathcal{ H}(\eta),\\u&=&\kappa C_{D}^{-1/9}t^{-7/9}\mathcal{ U}(\eta),\qquad\text{and}\\\qquad r_{f}&=&\kappa C_{D}^{-1/9}t^{2/9}, \end{array} $$

(A.2)

where η = r/r _f (t) and κ is a dimensionless constant to be determined. On substitution of Eq. A.2 into the governing Eq. A.1a,b, we obtain

$$ \frac{1}{\eta}\left( \eta\mathcal{U}\mathcal{ H}\right)^{\prime}-\frac{2\eta}{9}\mathcal{ H}^{\prime}-\frac{4}{9}\mathcal{ H}=0\qquad\text{and} \qquad \frac{\mathcal{ H}^{2}}{4}\mathcal{ H}^{\prime}=-\mathcal{ U}|\mathcal{ U}|. $$

(A.3a,b)

where the prime denotes differentiation with respect to η. These are subject to boundary conditions $\mathcal {U}(1)=2/9$, representing the kinematic condition at the front, and $\mathcal {H}(1)=0$, which is the frontal Froude number condition in the drag-dominated regime. Integrating (A.3a,ba), and applying the kinematic condition, we find

$$ \eta\left( \mathcal{U} - \frac{2 \eta}{9}\right) \mathcal{H} = 0 $$

(A.4)

from which we deduce that $\mathcal U = 2 \eta /9$. From Eq. A.3a,bb, we then find

$$ \mathcal H = \left[\frac{16}{81}\left( 1-\eta^{3}\right)\right]^{1/3}. $$

(A.5)

Profiles of the thickness and velocity of the plume, $\mathcal {H}$ and $\mathcal {U}$, are illustrated in Fig. 10. Equating the total volume of the intrusion per radian (expressed as a volume of revolution) with V, we obtain

$$ \kappa^{3}{{\int}_{0}^{1}} \eta \mathcal{ H}\mathrm{d}{\eta}=1. $$

(A.6)

Evaluating (A.6) using (A.5), we find that κ = 1.62…. Thus, in dimensional variables, the long time asymptotic radius of the intrusion is r _f = 1.62(N ² V ³ t ²/C _D )^1/9.