Rankine models for time‐dependent gravity spreading of terrestrial source flows over subplanar slopes

Abstract

Geological mass flows extruding from a point source include mud, lava, and salt issued from subsurface reservoirs and ice from surface feeders. The delivery of the material may occur via a salt stock, a volcanic pipe (for magma and mud flows), or a valley glacier (for ice). All these source flows are commonly skewed by a superposed far-field velocity vector imposed by the topographic slope and thus develop plumes having a wide range of shapes. The morphological evolution of the perimeter of the plumes (in plan view) can be simulated by varying the key parameters in a simple analytical flow description on the basis of Rankine equations. Our model systematically varies the strength of the point source relative to the downslope far-field velocity of its expelled mass. The flow lines are critically controlled by the relative speed of the two rates, which can be concisely expressed by the dimensionless Rankine number (Rk, introduced in this study). For steady flows, plume widths can be expressed as a function of Rk. The viscosity of the rock, mud, or lava mass involved in the gravity flow affects Rk and thus the appearance of the plumes. For unsteady source strength, Rk becomes time dependent and the plume width varies over time. The model flow shapes suggest that the plume shapes of natural gravity flows of terrestrial surface materials (mud, lava, salt, and ice) commonly express fast initial flux of the source, followed by an exponential decline of the source strength. Flows having initially higher Rk but otherwise equal life cycles create broader plumes. Peaks in the source flux due to magmatic pulsing during the eruption cycle can explain the formation of pillow lavas. Rather than instantaneously reaching full strength before declining, some natural source flows start by swelling slowly, leading to the creation of unique plume shapes like a flying saucer.

Downbuilding salt stocks and sheets quantified in 3‐D analytical models

“…Convergent flow lines relate to a physical pressure gradient causing acceleration. We have shown in a companion study [ Weijermars et al ., ] that the basic analytical description of steady linear flow from a source can be expanded with certain time‐dependent flow parameters; thereby, our models can account for certain nonlinear progression of the particle paths. We used MATLAB code based on the algorithms explained below to track particle paths and dye certain flow segments to visualize the 3‐D growth of salt diapirs.…”

“…The principal planform shape of a thin gravity flow is controlled by the dimensionless Rankine number [ Weijermars et al ., ]: which is the ratio of the 2‐D source flow strength m s (m 2 s −1 ) and the far‐field flow rate U ∞ (m s −1 ) (due to gravity spreading on a slope) multiplied by a characteristic length scale r 0 (m). The characteristic length scale r 0 may be defined by where b is the dimensional distance between the source and the stagnation point indicated in Figure and b* is the nondimensional scaling parameter.…”

“…Gravity flows at the Earth's surface occur in a large variety of liquid, granular, and crystalline rock masses such as lava flows, volcanic mudflows, debris flows, mudslides, ice glaciers, and salt glaciers. The effects of gravity on the macroscopic flow of these materials are rather similar but may occur over different timescales, mainly due to differences in effective viscosity [ Weijermars et al ., ; Weijermars , ]. For example, low‐viscosity mud emanating from gryphons (such as those near Baku, Azerbaijan) flows rapidly downhill in the form of relatively narrow plumes as the mud cools (Figure ).…”

“…Although emplaced by similar mechanism, Iranian salt glaciers (Figure d) with relatively high effective viscosities (10 12 –10 18 Pa s) have a gravity spreading life cycle stretching over millions of years. The analytical gravity spreading model globally introduced in our companion study explains the differences in life cycle times and the variety of plume shapes observed in terrestrial gravity flows [ Weijermars et al ., ; Weijermars , ].…”

“…To analyze the emplacement kinematics of downbuilding diapirs in three dimensions, we apply a new analytical 3‐D model of gravity spreading. We use the term “gravity spreading” in a fluid, mechanical sense (for reference see Weijermars et al []). This usage may differ from that in structural geology, where attempts have been made to distinguish between gravity spreading and gravity gliding [ Ramberg , ; Schultz‐Ela , ; Brun and Fort , ; Peel , ; Weijermars and Jackson , ].…”