Dynamics of Mount St. Helens' 1980 pyroclastic flows, rockslide-avalanche, lahars, and blast

Journal of Volcanology and Geothermal Research, 37 (1989) 205-231 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

205

DYNAMICS OF MOUNT ST. HELENS' 1980 PYROCLASTIC FLOWS, ROCKSLIDE-AVALANCHE, LAHARS, AND BLAST

ALFRED S. M c E W E N 1 and MICHAEL C. MALIN Department of Geology, Arizona State University, Tempe, AZ 85287, U.S.A. (Received July 7, 1988; revised and accepted December 26, 1988)

Abstract McEwen, A.S. and Malin, M.C., 1989. Dynamics of Mount St. Helens' 1980 pyroclastic flows, rockslide-avalanche, lahars, and blast. J. Volcanol. Geotherm. Res., 37: 205-231. A computer model for the movement of gravity flows calculates the velocities and simulated flow paths over digital topographic models. The nonuniform flow movements are determined from initial conditions, gravitational accelerations, and resistance to motion (~r) described by the general equation ~, = ao + a lv + a2v ~, where v is velocity. Although empirical, the terms a0, al and a2 may be related to Coulomb, viscous, and turbulent resistance, respectively. The energy-line model is used by setting a0 proportional to a coefficient of friction and setting al and a~ to zero. Use of the terms ao and al results in a Bingham-like model. The models were tested against the reported velocities and distributions of the pyroclastic flows, rockslide-avalanche, lahars, and blast of the 1980 eruptions of Mount St. Helens, Washington. The energy-line model, which has been widely used for this type of effort, generally predicts velocities that are too high, resulting in flow paths that are not sufficiently responsive to the topography. Use of the a~ or a2 term usually results in better matches to the observed velocities and flow paths. An August 7th pyroclastic flow was modeled in detail for comparison with the velocities acquired from a timed sequence of photographs. Both the energy-line model and a Bingham model based on measured rheologic properties result in model velocities that are much too high. The Reynolds and Bingham numbers indicate that local turbulence was likely, which is consistent with the presence of plane-parallel and cross-bedded deposits on the steep northern flanks of Mount St. Helens. Addition of turbulent resistance to the Bingham model results in a much better match to the measured velocities. A similar model best matches the distribution of deposits from the more voluminous May 18th pyroclastic flows. Our best result for the rockslide-avalanche in the North Fork Toutle River is a Bingham model with a viscosity of 3 × 104 Pa s and a yield strength of 104 Pa. For the lahars, viscous models gives the best results, but model viscosities range from 103-104 Pa s near the flanks of the volcano and in relatively dry creeks and canyons to 10~--10~ Pa s where the lahars entered river channels and were significantly diluted. For the blast, each of the three basic types of resistance models compares reasonably well with the observed distribution and velocities, provided the resistance is small. However, the expected resistances from both boundary-layer turbulence and air drag are substantial: in order to account for the blast's travel distance of 20-30 km, some process in addition to the initial momentum and gravity must have operated, such as continued decompression far from the vent.

Introduction

The catastrophic eruption of Mount St. He-

1Present address: U.S. Geological Survey, 2255 N. Gemini Drive, Flagstaff, AZ 86001, U.S.A. 0377-0273/89/$03.50

lens (MSH) volcano, Washington, on May 18, 1980, was one of the major volcanologic events of the century. This and subsequent eruptions included four very different examples of gravity flows: pyroclastic flows, the rockslide-avalanche, lahars, and the blast (Lipman and Mul-

© 1989 Elsevier Science Publishers B.V.

50 ± 2

11+_2 62±4 ?

1791 (jun 26 jul 17)

(oct) 1792 (dec) 1794 (jan) 1795 17977 1800 (nov 2-8)

73

38+_2

1789 (jun-jul)

1813 (sep 16) (nov 18-26) 1814 (sep 10-12)

Inav 2-30) (dee)

62 287

42±1 8.86 41

fumaroles in the craters 2+2

2 +- 2

10.4 32 ± 2 ? 1.8 4.71 66.86 0.28

(apr 14 to ?) (dee) 1807 (mar 23-may 27) (jun 10-13) 1809 (jul 17-aug8) 1810 (nov 20-22) (Nov 24-28) 1812 (aug) (sep 3 30}

1 11 2

64 3 22 2 7 ? 28

14

8+2

1802 (jan 17-30)

13 33 46 2

>30

4852

>7 7

21

48

1801 (oct-nov)

? '?

39_+2

? ? ?

1784 1785 1786 (aug 4 to?} 1787 (jun 14-aug 1)

TABLE i Icontinuedl

0.14 157 0.3

Grand Brfll6 Irempart of Tremblet

Grand B~16: 3flows

Grand Brhl6

2+1 4±3 8.1 0.42 3.08 0.28 1 4

Grand Br~516

SE rift zone

Grand Br~l~ Citrons Galets. Tremblet

Grand Brfil4 Grand Brhl6

Grand Br~16: a flow of 1.5 km wide at the sea shore Grand Brfil6: many flows reach the sea S of Grand BrOil6

2

>4

2_+2 ? 2+_2 ? >1 1

4+-2

2+2 6.86

?

short activity small flows - - crater and dome fountains, lava in the crater

flows

oeeanite

many flows short eruption fountains: no apparent flow fountains fountains

huge flow of oceanite (beginning of an eruptive cycle? ) oceanite

activity (?)

one flow

one flow one flow

large flows

large eruption

fountains fountain~ large eruption

abundant Pele's hair

Pele's hairs cinders

Pele's hairs (the first days) cinders

Pele's hairs (the first days)

probable

huge: birth of Dolomieu crater

M1 on the whole island up to the north (St Denis)

forests in fire

80X 106m a {in7 days}: beginning of an eruptive cycle?

207

(Sparks, 1976; Wilson, 1980). The term "avalanche" is generally used to describe rapid movement (maximum velocity from 50 to 100 m s -1) of relatively dry (unsaturated) snow, rock, or debris (Pierson and Costa, 1987). A "rockslide-avalanche" begins its movement on discrete sliding surfaces (Mudge, 1965). A "lahar" is a debris flow or mudflow composed of volcanic detritus, usually beginning on the flank of a volcano (Mullineaux and Crandell, 1962). Downstream dilution of a lahar may transform it from a debris flow to hyperconcentrated stream flow (Pierson and Scott, 1985). "Pyroclastic surges" are low-concentration turbulent clouds analogous to turbidity currents or density currents (Fisher, 1979; Wohletz and Sheridan, 1979; Sigurdsson et al., 1987). The nature of the MSH "blast" is controversial; it is considered to be a pyroclastic surge (e.g., Moore

and Sisson, 1981; Fisher et al., 1987), a pyroclastic flow (Sparks, 1983; Walker and McBroome, 1983), or a supersonic expansion ("blast") of a multiphase mixture (Kieffer, 1981). We use the term "blast" because it is short and because the word is closely identified with the MSH event. Pre- and post-eruption DEMs of the MSH area were acquired from the National Cartographic Information Center (NCIC), U.S. Geological Survey (Elassal and Caruso, 1983; Table 1). The DEMs cover 7.5X7.5 minute quadrangles and have a spatial resolution of 30 m per picture element (pixel) and a vertical accuracy of about 3 m (except for the Hoquiam 1 ° X 2 ° quadrangle that has a spatial resolution of 90 m/pixel and a vertical accuracy of about 30 m). The Hoquiam DEM was used only for the blast models, as higher resolution DEMs are

Fig. 1. Shaded relief image of Digital Elevation Model (DEM) of Mount St. Helens area, topography after May 18, 1980. Oblique view from west with observer 30 ° above horizon; no vertical exaggeration. Width of image about 23 km,

208

not available for the northernmost reaches of the affected zone. The 7.5 × 7.5 minute DEMs were mosaicked into pre- and post-eruption altimetry images. The rockslide-avalanche caused the major topographic changes, so the preeruption topography was used to model the motions of the avalanche and blast, and post-eruption topography was used for the pyroclastic flows and lahars. Slope and slope-azimuth were computed for each pixel and stored as digital images for access by the flow-model program. The post-eruption topographic dataset is illustrated in Fig. 1.

Equations of motion Some of the theory and equations of motion for gravity flows are described in this section. The notations are defined in Table 2. We use SI units (m-kg-s) throughout this paper. In the Appendix, we describe the computer modeling program (hereafter called FLOW). FLOW computes three-dimensional flow paths over the DEM by using nonuniform models in which the flow direction and velocity are recomputed at each time step. (A uniform flow has the same velocity at all cross sections down the flow, whereas a nonuniform flow is usually either accelerating of decelerating, depending on the slope angle. ) The motions of gravity-driven flows are determined by the force of gravity and by various mechanisms of resistance to movement. Many commonly used models for flow resistance may be combined into a polynomial expression proposed for snow avalanches by Mellor (1978) and for rock avalanches and debris flows by Voight et al. ( 1983; see also Pariseau and Voight, 1979), where the stress that is resisting motion, ~, is given by: ~ =ao + a i v + a 2 v 2

(1)

where ao, a~, and a2 represent Coulomb, viscous, and turbulent resistance parameters, respectively, and v is velocity. The total stress (including gravity) is:

TAB LE 2 Definitions of notations Symbol

Definition

U4~

Coulomb shear stress Viscous shear stress Turbulent shear stress Bingham number Cohesion Drag coefficient for atmosphere Drag coefficient for ground Flow thickness in wide channel Critical or plug thickness of Bingham material in wide channel Change in flow depth with time Acceleration Acceleration of gravity Height drop of flow Yield strength in Bingham model Characteristic diameter of roughness elements Horizontal length traveled by flow Mass Radial distance from center of semicircular channel Radius of semicircular channel Critical or plug radius of Bingham material in semicircular channel Reynolds number Time Mean cross-sectional velocity Initial v Velocity of flow plug or of top center portion of flow Distance within flow measured from bottom of wide channel Azimuth of flow direction Tilt angle of top of flow in channel bend Viscosity Slope of ground Density of flow material Density of atmosphere Normal stress Shear stress Shear stress resisting movement Angle of internal friction Radius of curvature of channel bend

(11

Bi (, cg

D D,.

dD / dt dv/dt g ft k k~ L M r

R Rc

Re

t U Uql UI)

Y

fl 0

P /%

T Tr

0

Equation t I

i6,17

I 1b. l

!~, 1'7

9 ,9

l la, 12 l8 14

9,10,1 ;t ~i)

7 2 6 :~ 6

209

r=pgDsinO- (ao +alv+azv 2)

(2)

where p is flow density, g is gravitational acceleration, D is flow depth, and 0 is the surface slope in the direction of movement. The instantaneous change in momentum of a columnar element of flow is:

d(Mv) dt = MgsinO

(ao +al v+a2v2)M pD

(3)

where M is mass and t is time. If the mass of a columnar element is held constant, then the acceleration is:

dv/dt=gsinO- (ao +alv+a2v2)/(pD)

(4)

The mass of a vertical column is held constant during movement of a rigid body but not during flow of a deformable body. For example, conservation of mass in a rectangular channel requires that:

Q=vDW

(5)

where Q is the volume flux and Wis the channel width. Therefore, the flow depth changes in response to velocity changes, and velocity changes in response to channel shape. The mass of a vertical column will also change as a result of deposition or entrainment during movement. Furthermore, we use steady-state approximations and do not consider the mass flux from the vent. Three-dimensional equations of unsteady, nonuniform motion and realistic models for deposition and entrainment are needed to account for these effects, but the equations may prove analytically intractable (cf. Iverson, 1985, 1986), and such a model would result in a burdensome computing time per model run. Nevertheless, important advances in two- or three-dimensional modeling of incompressible flows, with various simplifications, have been achieved (Amsden and Harlow, 1970; Lang et al., 1979; Lang and Dent, 1987). However, we have chosen to use eq. (4) as a means of deriving empirical models of flow resistance with vand v2-dependent terms. For flows or portions of flows where the depth is observed to remain

approximately constant and the flow is approximately steady state, our model parameters may be related to physical properties such as viscosity and yield strength. In most cases, our models should be regarded as empirical. Particles transported in decelerating sediment gravity flows are deposited primarily by two different mechanisms (Fisher, 1979; Lowe, 1982). From fluidal flows (including turbulent and fluidized flows) particles tend to accumulate individually from the bed-load layers or from the suspended loads, and the deposits are well sorted and stratified. Debris flows usually deposit material en masse, when the applied shear stress drops below the yield strength, and the deposits are poorly sorted. A first-order model for debris-flow deposition is simply deposition of the entire mass when the velocity drops to zero (Johnson, 1970; Fisher, 1971). Deposition from turbulent or fluidized flows is complex, but we have introduced the term dD/ dt to estimate a linear charge in flow depth with time, from deposition and/or lateral spreading of the flow. We use nonzero values for dD/dt only for two blast models. We will proceed, below, with equations for the three resistance parameters a0, al and a2, but with the cautionary note that the parameters are physically meaningful only to the extent that the assumptions of steady-state flow and constant or linearly decreasing flow depth are accurate. The three parameters ao, al and a2 represent different resistance behaviors: Coulomb resistance, viscous resistance, and turbulent resistance, respectively.

Coulomb resistance The parameter ao represents a constant force per unit area (independent of velocity ) and can be described by a Coulomb model: rr = C + atan~

(6)

where C is cohesion, a is normal stress (Mgcos0), and O is the angle of internal friction. This model, modified by Terzaghi (1943)

210

so that effective a= a - p o r e water pressure, has been widely and successfully used to describe the deformation of soils (e.g., Lambe and Whitman, 1969; Johnson, 1984). If cohesion is ignored, eq. (6) reduces to a widely used kinematic model called by different names: the sliding block model (Pariseau and Voight, 1979), the energy-line model (Heim, 1932; Hsfi, 1975), or the grain-flow model (Bagnold, 1956; Lowe, 1976 ). The friction coefficient, tan0, may represent either sliding friction (in the sliding block model) or internal friction (in the energy-line or grain-flow models), and it can be calculated from the total vertical drop (H) divided by the total horizontal travel distance (L) of the flow. Therefore, a complete kinematic model is derived simply by measuring the geometry of the mass movement, and thus the model has been very popular; it has been applied to rock or debris avalanches (Scheidegger, 1973; Hsfi, 1975; Ui, 1983; Siebert, 1984; Ui et al., 1986; Eppler et al., 1987; Siebert et al., 1987), snow avalanches (Lied and Bakkehol, 1980; Voight, 1981 ), pyroclastic flows (Sheridan, 1979, 1980; Hoblitt, 1986; Beget and Limke, 1988), and pyroclastic surges (Malin and Sheridan, 1982; Sheridan and Malin, 1983; Armienti and Pareschi, 1987). Most natural rock types (except certain clays) have coefficients of sliding friction ranging between 0.6 and 1.0 (Jaeger and Cook, 1969), so sliding movement should be possible only over areas with an average slope of at least 30 °. Movement of a grain flow also requires a tan0 of at least 0.6 (Lowe, 1976; Middleton and Southard, 1978). A grain flow is defined by Lowe (1976) as "a sediment gravity flow in which a dispersion of cohesionless grains is maintained against gravity by grain dispersive pressure and in which the fluid interstitial to the grains is the same as the ambient fluid above the flow." Bagnold (1956) noted that tan¢ for a grain flow could be as low as 0.32, but he has more recently stated (Bagnold, 1973) that the correct value for natural sand grains is the same as that obtained from "angle of repose" exper-

iments (tan0 = 0.63 ). Lowe (1976) also pointed out that on slopes that are significantly steeper than the angle of repose, a grain flow will "accelerate, dilate, and become increasingly influenced by fluid forces." Therefore, grain flows are possible only under very restricted conditions, such as on the lee faces of sand dunes or on other angle-of-repose slopes of dry granular material such as the slopes of cinder cones. The energy-line model has been justified by analogy to grain flow (Hsfi, 1975; Sheridan, 1979; Wohletz and Sheridan, 1979; Malin and Sheridan, 1982 ), so we might expect this model to apply only to slopes of~ 30 ~ as well. Nevertheless, it has been applied largely to flows in which H/L is much less than 0.6. Bagnold ( 1954, 1956 ) showed that a grain mass can flow on very gentle slopes when immersed in interstitial mud (p= 2000 kg m :~), but it then becomes a viscous flow for which a velocity-dependent resistance model is appropriate {see below). In spite of the lack of theoretical or ex-~ perimental justification, the energy-line model might be a useful empirical model, so it is tested in this paper against the flows of MSH (described below). In general, the model gives velocity predictions that are much higher than the observed velocities. Such predictions in turn suggest that velocity-dependent (viscous or turbulent) resistance mechanisms must be important.

Viscous resistance Newtonian flow is the simplest viscous model, where:

r~=q(dv/dy)

(7)

where q is viscosity and dv/dy is the velocity gradient or shear strain rate in laminar flow. Most natural debris flows have a finite yield strength and are better characterized by a Bingham rheology where:

~,=k+q(dv/dy)

(8)

where k is the yield strength in Pascats (Pa)

211

width is twice its depth (Johnson, 1984). For an infinitely wide channel (hereafter called a "wide" channel), the velocity profile for steady, uniform flow is:

,o.

~

-~'~ d~. oo.-~ ".¢.o"•

~Ler~~i¢~

v ( y ) = ( 1 / q ) [ pgsinO(D'~-y) - k ( D - y ) ] ;

8~O0£

a

where y is height, (measured from the bottom of the channel), D is total flow depth, and D,. is the plug thickness. Eq. (10) is approximately correct for any channel that is much wider than deep, so that shearing along its sides can be ignored. The velocity of the plug is computed by substituting Re for r in eq. (9) or by substituting Dc for y in eq. (10). The plug thickness or radius, under steady, uniform flow conditions, can be related to the ground slope by:

Rc'Plug radius

b

Fig. 2. Diagrams of flow geometry and resistance mechanisms. (a) Longitudinal profile of a Bingham debris flow. (b) Cross-section of Bingham debris flow in a semi-circular channel. (c) An expanded, turbulent flow or surge and resistance mechanisms that scale with velocity squared.

and represents both cohesion and frictional strength. Due to the yield strength, a Bingham flow will have a rigid "plug" in the center and top of the channel (Fig. 2), where the gravitational stress does not exceed k. For steady, uniform channel flow (constant velocity profile as a function of time and location ), the velocity gradient for a Bingham flow has been derived by Johnson (1970, p. 497) for semicircular and infinitely wide channels. For a semicircular channel (hereafter referred to as a "circular" channel; see Fig. 2b) the steady, uniform velocity profile is: . . . . [-pgsinO(R2-r2) v(r) = t J/,Tj L '

y>_.D< (10)

k(R-r)l;

r>-Rc

(9)

where r is radial distance from the center of the channel, R is total channel radius, and R~ is the radius of the plug. Equation (9) is also approximately correct for a rectangular channel whose

Ro =2k/(pgsinO)

( 1la )

De = k/(pgsinO )

(llb)

A Bingham flow will eventually stop when 0 is low enough that D = Dc or R = R~, and this relation has been widely used to estimate k (e.g., Moore et al., 1978; Wilson and Head, 1981; Voight et al., 1983; Johnson, 1984; Eppler et al., 1987). For a nonuniform flow, eqs. (lla,b) are inappropriate. For example, when 0 is 0°, the plug thickness is infinity by these equations. The slope dependence may be eliminated by combining eqs. (10) and (11b) to give: 2(kD + ~lv) - [(2kD + 2~lv) ~ - 4kZD 2] s/z D c --

2k

(12) Rc may be similarly computed from eqs. (9) and (11a), resulting in an equation identical to (12) except that R is substituted for D. Equation ( 12 ) is used in this study for nonuniform flow models, which is equivalent to an assumption that the plug thickness is a function of velocity regardless of whether v is near the uniform (or terminal) velocity for a particular slope. The validity of this assumption needs further study, but use of eq. (12) should not result in significant errors because at high velocities Dc or Rc

212

must be very small (and have little effect on the dynamics ), and at low velocities Dc approaches D (or Rc approaches R), the necessary value when equals zero. An equation for acceleration of a Bingham flow is required for the nonuniform flow model. Pariseau and Voight (1970) presented equations for acceleration with viscous resistance plus resistance due to cohesion along a basal slip surface, but an error appears in the viscous term (the units don't match). For a Bingham flow, we must account for the effect of plug thickness on the thickness of the shear zone (assuming that the total flow depth or radius is constant). The acceleration of the plug in a wide channel is:

dVp=gsinO 2k dt p(D+D~)

2~Vp

p(DZ-D,. ~)

413a)

dt

4k p(R+R~)

4~l)p (R2-R,~ ~)

t 13b)

where vp is the plug velocity. Note that if plug acceleration is zero, eqs. (13a,b) reduce to the equations for steady, uniform plug velocity. The faster moving plug of a debris flow will move toward the flow front and, at the front, will be pushed to the sides by the body of the flow (e.g., Pierson, 1986). The pertinent velocity for describing the advance of the flow is the mean cross-sectional velocity, which for a Bingham flow in a wide channel is: l)

y~

v = (1/D)) v(y)dy + v pDc/D

( 14a )

and in a circular channel is:

v = (1/R ~) jrv(r)dr+

vpRo

Turbulent resistance The third term in eq. (1) represents three forms of resistance (Mellor, 1978): i 1 } t u r b u lent shear against the ground; (2) air resistance at the front and upper surface of the flow; and (3) frontal or plowing resistance (Fig. 2c). Frontal resistance is probably not significant when the length of the flow greatly exceeds the depth, and it is neglected in FI.OW. Turbulent shear against the ground is a complicated function of surface roughness, but many workers (e.g., Middleton and Southard, 1978; Sparks et al., 1978) have found the Chezy equation to be a useful model, where:

"6 =0.5c~pv'-'

and in a circular channel is: dvp =gsin0

flow, eq. (14a) approaches v=2vp/3 and ( 14b ) reduces to v = vp/2.

2

where Q is the drag coefficient tot the ground and can be approximated for a developing boundary layer (Schlichting, 1960) as:

c~ =O.04/ln(D/ks)

R~

In FLOW, we compute vp from the beginning velocity (Vo) and eq. (13a)or (13b), then express v (r) or v (y) (eq. 9 or 10 ) in terms of Vpprior to integration. As k approaches 0 and Newtonian

(16)

where ks is the characteristic diameter of the bed particles. For a fully developed boundary layer, Q has been approximated by Sparks et al. (1978) as:

,:~ =0.65/ [ln( D/ks )12

(1'7)

For a Bingham flow, the transition from laminar to turbulent flow depends upon two dimensionless numbers, the Reynolds number (Re) and the Bingham number (Bi) (Hedstr6m. 1952; Hampton, 1972; Middleton and Southard, 1978; Valentine and Fisher, 1986). The Reynolds number is:

Re=pvD/t 1 ( 14b )

(15)

~1~

and the Bingham number is:

Bi=kD/~lv

(19)

A plot of critical Re and Bi (when turbulence begins ) for Bingham slurries is shown in Fig. 3. The relationship shown by this plot is that when

213

I

loo

o

v

I

//"

Application to Mount St. Helens We begin our analysis of the MSH flows with a pyroclastic flow of August 7, 1980, because the velocity history of this flow is well documented and because it provides a case study to illustrate the effects of the three basic resistance parameters. This discussion is followed by analysis of the May 18th pyroclastic flows, rockslideavalanche, lahars, and blast. Input parameters for the models are listed in Table 3.

J .,,7. ; // .,,',,?""

<9

Pyroclastic flows 1 O0.....

1000 10,000 REYNOLDS NUMBER,Re

100,000

Fig. 3. Plot of critical values for onset of turbulence as a function of Reynolds and Bingham numbers. Experimental data are for pipe flow, but scales have been adjusted for wide channels. From Middleton and Southard (1978).

Bi exceeds about 1.0, the onset of turbulence occurs when: R e / B i 2 1000

(20)

Air drag along the front and upper surfaces of a flow (Perla, 1980) is: d(Mv)/dt=

-0.5pacaAv 2

(21)

where/)a is the density of the atmosphere ( ~ 1.0 kg m-3), ca is the drag coefficient for the atmosphere, and A is the area over which the air drag operates. If we ignore frontal air drag, the deceleration due to air drag on the upper surface is: dv / dt = - Ca(Pa/ fl ) V 2 / ( 2 D )

(22)

Because the deceleration is proportional to Ps/ p, air drag will not be significant for dense flows (p~ 1000-2000 kg m -a) such as pyroclastic flows, debris avalanches, and lahars, but it may be significant for low-density pyroclastic surges. According to Perla (1980), ca is likely to be within the range 0.1 to 1.

Pyroclastic flows occurred during six major eruptions in 1980 (Rowley et al., 1981, 1985). The most voluminous pyroclastic flows occurred on May 18th: a succession of flows totalling 0.12 km 3 in volume covered an area of 15.5 km 2, mostly north of the volcano. The May 18th flows were not well observed due to a heavy cloud cover. Later flows were better observed, and they were generally initiated when billowing masses of fountaining pyroclastics rose only a short distance above the inner crater. Some flows, however, resulted from collapse of Plinian eruption columns. The best observed pyroclastic flow occurred on August 7th, for which a timed sequence of photographs, all showing the flow front unobscured by ash clouds, provides the most complete velocity history ever acquired for a pyroclastic flow (Hoblitt, 1986). The path of the flow is shown in Fig. 4. (A second pyroclastic flow, east of the flow discussed here, also occurred on August 7th, but it was not well observed. ) The width and depth of the flow deposits were generally constant (Wilson and Head, 1981). Because the flow path can be described by a single flow line, the FLOWmodels are presented as twodimensional plots of velocity versus distance. Hoblitt (1986) described the movement of the August 7th flow in terms of the energy-line model. He looked at models with initial heights above the ground of 300 and 500 m (values that bracket the range of fountain height esti-

214 TABLE 3 FLOW

input parameters (see Table 2 for definition of terms; all SI units)

Model

Re,~

v,~

k

tanO

~l

(',

D

p

0 0 600 600

0.19 0.14 0 0

(~ 0 4 4

0 O 0 0.008

2 '2

0 600

0.15 0

(* 4

0 0.008

0.09 0

0 30000

4700 2250

el,,"

riD~dr

Channel' shape

[,hg.

August 7 pyroclastic flows." PY1 PY2 PY3 PY4

300 300 300 300

77 15 40 40

,b .~('

1450 1450

95 95 95 95

0 0

W W

:¢

1450

55-125 55-125

0

W

0 0

7;¢

1500

~- 170 8-170

0

\V

0 0

13 9.5

2008 2008

164- 36 164- 36

0 0

W C

1000 10

(b170 0-170 0~ 170

0.14 0.9

W W

,d

LSe

May 18 pyroclastic flows." PY5 PY6

1280 1280

40 45

Rockslide-avalanche: AV 1 AV2

700 700

50 50

0 10~

2000 2000

40 40

500 500

0 0

2000 2000 2000

164 95 95

0 600 0

0.11 0 0

a, ";i)

Lahars: LA1 LA2

Blast: BL1 BL2 BL3

() 100 ()

0 0 0.01

28 200

iOb,1 la 10c,1 lb ll)d.i h

~Ro is radial distance from vent or flow source to points where FI.OW lines begin. ha0 is initial flow direction in degrees counterclockwise from due east. 'Channel shape: W = wide, C = semi-circular.

-indicates that quantity is not relevant to model.

mates), and determined that to match the runout length, tan0 must be 0.19 or 0.23, respectively. He concluded that the first model (AH=300 m and t a n 0 = 0 . 1 9 ) gave the better match. (The initial height is converted into an initial velocity by (2gAH) 1/2 in the energy-line model, s o A H = 300 m gives Vo= 77 m s - 1. ) Hoblitt did not present the actual velocities predicted by this model, but we have done so in model PY1 (Fig. 5b). The model velocities are everywhere at least 2 times greater than the measured velocities, so apparently this is not an accurate model of the flow motion. Perhaps the problem with model PY1 is the assumption that the potential energy represented by the fountain height is converted into kinetic energy (the initial velocity) with 100% efficiency. Therefore, we tried an energy-line

model with vo lowered to 15 m s ~ (model PY2. Fig. 5c). This model results in a good match to the velocities over the first 2.5 km of movement, but the model velocities increase over the steep slopes and are again more than 2 Limes too high. The next model (PY3) is a Bingham flow model based on the work of Wilson and Head ( 1981 ), who measured the rheologic properties of the July 22nd and August 7th pyroclastic flow deposits soon after deposition. They measured channel and levee thicknesses, surface slopes, and deposit densities, and then used eq. ( 11 b ) to find yield strengths ranging from 400 to 1100 Pa, with 600 Pa as an average value at the time of emplacement. Their viscosity estimates (from penetrometer readings) range from 30 to 13,000 Pa s. For an active flow near the final

215 1.9 l.B l.V 1.0 1.5 l.Z~ 1,3 1.2 I.I

~~

~

CA]

?0.0 50.0 50.0 40.0 30.0 20.0 10.0 50.0 40. D

%. IZ

30.0 20.0

I0,0 >-I-I-4 (.3 D .J OJ

~_--/-

80.0

CD]

60.0 40.0 20.0

30.0 20.0 10.0

0.0 0.0

1.0

2.0

3.0

~.0

5.0

6,0

DISTANCE [KM]

Fig. 4. Actual path (in white) of August 7th pyroclastic flow (cf. Hoblitt, 1986). Black line outlines area of deposition from May 18th pyroclastic flows. Shaded relief base of post-eruption topography.

Fig. 5. Velocity predictions for August 7th pyroclastic flow. Elevation measurements (A) extracted from profile shown by white line in Fig. 4. Velocity predictions from models PY1, PY2, PY3, and PY4 are shown in B-E, respectively, compared to velocity observations (circles} from Hoblitt (1986). See Table 3 for model input parameters.

stages of movement, however, they estimated a viscosity of only 4 Pa s. This value seems unusually low, but a relatively low viscosity (e.g.,_< 100 Pa s) is consistent with the lowest penetrometer measurement (30 Pa s). Deposit thicknesses range from 0.5 to 1.5 m and widths range from 5 to 20 m (e.g., wide channels). The active flows may have been somewhat thicker than the deposits, but Wilson and Head ( 1981 ) argued that the reverse grading of low-density pumice occurred by flotation, so that the bulk flow could not have been greatly expanded. Therefore, we chose p-- 1450 kg m - 3 (the value recommended by Wilson and Head) and D = 2 m for model PY3 (Table 3). Although the initial velocity of the August 7th flow was not directly observed, several other M S H pyroclastic flows had initial velocities near 40 m s - 1 (Rowley et al., 1981, 1985; Hoblitt, 1986), so we se-

lected this value for Vo. The resulting model velocities (Fig. 5d) exceed the observed velocities by factors of 3 or more. Increasing ~/to 100 Pa s reduces the velocity by only 2-3 m s - 1. Reducing D to 1 m reduces the model velocity by as much as 20 m s - l , but these velocity predictions nevertheless greatly exceed the observed velocities. The Bingham model does not seem to include all of the significant resistance parameters. For the lahar and avalanche models (described below), Re/Bi is below 1000 at all model velocities, so dominantly laminar flow is likely. From the PY3 model parameters and a velocity as high as 30 m s -~ (Hoblitt, 1986), the Reynolds and Bingham numbers are 21750 and 10, respectively (eqs. 18,19 ), and Re/Bi is 2175, above the critical value of 1000 for the onset of turbulence (see Fig. 3). This result is consistent with the

216 presence of plane-parallel and cross-bedded deposits on the steep northern flank of M S H from the M a y 18th pyroclastic flows, which Rowley et al. (1985) interpreted as due to partly turbulent or transitional flow conditions. Therefore, we included a resistance parameter for a turbulent boundary layer (eqs. 15, 16) in the next FLOW model. Sparks et al. (1978) considered 1-cm roughness elements to be appropriate for most ignimbrites, giving a drag coefficient, Q, of ~ 0.008; we used this value in model PY4. The PY4 model velocities match the observed velocities to within a factor of 1/3 or less (Fig. 5e). The model flow stops about 800 m short of the observed flow termination, but this may be because the turbulence ceases at low velocities ( R e / B i is less than 1000 when v is below 20 m s - 1), whereas the turbulent resistance term operates at all velocities in FLOW. The flow path from model PY4 is indistinguishable from the observed flow path (Fig. 4), except that it is shorter. The other models ( P Y 1 - 3 ) produce straight flow lines, unlike the sinuous and channelized pyroclastic flows. Model PY4 consists of dominantly ao and a2 resistance terms, much like the most widely used models for snow avalanches (Mellor, 1978; Perla, 1980); thus Anderson and Flett (1903) may have been correct in likening the movement of pyroclastic flows to that of snow avalanches. Typical roughness elements on the steep flanks of the volcano are probably greater than 1 cm. Therefore, the turbulent resistance may have been greater than that modeled over these areas. However, there was probably no turbulence over about half of the flow distance where the velocity was below ~ 20 m s - 1. These effects compensate for each other to some degree. Although the turbulence was probably restricted to the steep slopes, most of the acceleration occurred over these slopes, so that local turbulence may have a significant overall effect on the dynamics. Some of the deviations of model PY4 from the measured velocities {Fig. 5e) may be ex-

plained in the manner discussed by Hoblitt (1986): as a pyroclastic flow travels down a steep slope or over rough ground it may ingest air, expand, and form an overriding pyroclastic surge. The surge then moves ahead of the dense basal flow, so the cloud front appears to accelerate. The surge cloud rapidly loses mass (as particles settle out of suspension) until the hot surge reaches a density less than that of the ambient atmosphere. The surge then buoyantly lifts up and the cloud front appears to decelerate until the dense basal flow regains the lead. Therefore, if model PY4 represents movement of the dense basal flow, then the observed velocities should fluctuate around the model velocities, as in Fig. 5e. Laboratory simulations ( H u p p e r t et al., 1986) have provided experimental support for Hoblitt's idea. Two FLOW models are presented in Fig. 6 to simulate the earlier, more voluminous pyroclastic flows of M a y 18th. Model PY5 uses the energy-line model with tan¢ = 0.15; the resulting flow lines are straight, unlike the sinuous patterns formed by the deposits, so the model velocities (up to 80 m s - 1) are probably too high (Fig. 6a). Model PY6 is identical to PY4 except that D is increased to 3 m and vo is 45 m s ~; the resulting flow lines are sinuous (Fig. 6b). The concentration of flow lines in a narrow central corridor extending due north from the crater and their subsequent spreading east and west on the pumice plain matches the general pattern of the flow deposits (see Plate 1 of Lipman and Mullineaux, 1981 ). The average velocities from model PY6 are 15-20 m s 1 and do not exceed 44 m s - 1, which is consistent with observed velocities (Rooth, 1980; Rowley et al., 1981; Hoblitt, 1986). Rockslide-avalanche

The rockslide-avalanche began within 20 s of the Richter magnitude 5.2 earthquake (Voight, 1981). After about 26 s and 700 m of sliding, the rock mass took on the characteristics of a debris flow with a velocity of 50 m s - ~. (These

217

Fig. 6. May 18th pyroclastic flow models PY5 (a) and PY6 (b). See Table 3 for model input parameters. Black lines outline total area of deposition of May 18th pyroclastic flows. Shaded relief base of post-eruption topography. observations define the initial position and velocity for FLOW; see Table 3. ) The volume of the initial rock mass was 2.3 km 3, but the total volume of the disaggregated deposits is ~ 2.8 km 3. The deposits cover an area of 60 km 2, extending far to the west along the North Fork Toutle River (Fig. 7), where they grade into mudflow deposits (Voight et al., 1981). The rockslideavalanche caused profound topographic changes. The thickest avalanche deposits, 245 m thick, are in front of J o h n s t o n Ridge, and deposits in the North Fork Toutle River are as much as 80 m thick. Three main slide blocks formed the avalanche (Voight et al., 1981, 1983). Block I consisted of the outer portion of the north flank of the volcano, and blocks II and III formed from retrogressive failures behind block I. The toe of block II abutted against the heel of block I, and

the slides moved as a single mass down the north flank. Block III was hidden from view by the eruption clouds. Most of block I came to rest along the base of J o h n s t o n Ridge, whereas blocks II and III were apparently more mobile. Most of the debris in the North Fork Toutle River originated from these later blocks. Pressure reduction on these hot inner blocks may have caused pore water to flash to steam, resulting in significant loss of strength. Material from both block I and "avalanche II" (block II a n d / o r block III) overtopped Johnston Ridge, but at different passes (Fisher et al., 1987). Two FLOW models for the avalanche are presented here. In the first model (AV1, Table 3), an energy-line model with tan¢ = 0.09 was utilized. Voight (1981) and Voight et al. (1983) suggested that this model could describe the average movement of the avalanche. The model is

218

Fig. 7. Avalanche FLOW models AV1 (a), AV1 with flow-line interaction model (b), AV2 (c), a n d AV2 with flow-line intern action model (d). See Table 3 for model i n p u t parameters. Black lines outline area of avalanche deposition. Shaded relief base of post-eruption topography. A p p a r e n t curvatures of lines near volcano are artifacts.

consistent with the distribution of deposits north of the volcano (Fig. 7a) and thus is consistent with the deposits from block I, but it does not reproduce the extensive westward flow down the North Fork Toutle River from avalanche II. Model velocities near the M S H cone and at Johnston Ridge compare reasonably well with the velocity estimates of Voight et al. ( 1983; see Table 4). The initial motion of at least block ! of the rockslide-avalanche was sliding, for which the energy-line model (equivalent to the sliding-block model) is appropriate (Pariseau and Voight, 1979). Each flow line is computed independently of all other flow lines in the models presented thus

far. Flow lines that cross each other at nearly the same time should have some effect on each other. We developed a model in which flow lines 'FABLE 4 Rockslide-avalanche velocities 1,()cation

Estimated velocity '' ( m s -~ )

FLOW modei velocities Im ~ ', AVI AV2

1.4 km N of M S H cone J o h n s t o n e Ridge Coldwater Creek Elk Rock

80 50-70 25-45 10-20

70 ~0

"From Voight et al. (1983).

;5 ~(~ 41~ 10-25

219

that pass within a certain (input) time and distance from each other take on the vector average velocity and direction of the two lines. This heuristic model was applied to the avalanche because (1) the flow lines appear chaotic, and (2) the interactions might help the flows turn from north to west down the North Fork Toutle River valley. An interaction model for AV1 is shown in Fig. 7b; the flow lines still make little progress to the west. The second avalanche model (AV2) is based on the rheologic properties e s t i m a t e d b y Voight et al. {1983, p. 264) for the moving debris i n t h e North Fork Toutle River valley near Elk Rock (located ~ 17 km northwest of the summit of M S H ) . They estimated a velocity of 16 m s from the superelevation around the channel curve, a flow depth of 73 m (the deposit depth), a channel slope of 1.8 °, and a density (while mobile) of 1500 kg m -~. Based on critical thicknesses and slope angles of the channel deposits at several locations, the shear strength (eq. l l b ) was calculated to range from 2X103 to 2X 104 Pa. From these data, Voight et al. (1983) computed a Bingham viscosity from 1 to 7 X 104 Pa s by assuming steady, uniform flow and that the plug velocity approximated the mean cross-sectional velocity. A viscosity calculation based on the mean cross-sectional velocity for a wide channel (eq. 14a) results in from I to 5 X 10 4 Pa s. Therefore, k= 104 Pa and ~l: 3 X 10 4 Pa s were selected as average values for use in FLOW. The resulting flow-line distribution from model AV2 (Fig. 7c, and 7d with flow-line interactions ) provides a much better match to the observed avalanche distribution. The velocities also compare well with the estimates of Voight et al. (1983; Table 4). This result is surprising because the flow depth varied greatly from 73 m along its course, and ~/and k may have varied with space and time as well. The model may work as well as it does in spite of the variations in these parameters because the tremendous inertia of the flow makes it insensitive to local variations. If we lower or raise the viscosity by'

a factor of only 1/3 (to 2X l04 Pa s or to 4X 10 4 Pa s), the runout lengths and velocities are clearly too long and high or too short and low, respectively. However, if we increase or decrease D and ~ together, or increase ~/and decrease k or vice versa, many models provide good overall matches. Therefore, FLOW can provide a reliable constraint on the viscosity only where D and k can be independently constrained (e.g., in the North Fork Toutle River valley), but it provides a useful empirical model for the overall movement.

Lahars Lahars began within minutes of the May 18th eruption and flowed down nearly all of the major drainages near the cone of Mount St. Helens (Fig. 8). The lahars formed in three ways (Janda et al., 1981): (1) from the water-saturated parts of the debris avalanche; (2) by catastrophic ejection of pyroclastics and water from the cone (pyroclastic surges evolving into lahars); and (3) by melting of debris-laden ice and snow by hot tephra. The lahar deposits range in textural characteristics from fine mudflow deposits to coarse debris-flow or avalanche deposits. Deposit depths are generally less than 10% of the depths at peak flow, as seen from mudlines and abrasion on standing trees. The most voluminous and damaging mudflow occurred in the North Fork Toutle River; it began from water-saturated portions of the avalanche. D E M s have not been acquired for most of the North Fork downstream of the avalanche deposits, so this lahar will not be considered further in this paper. Lahars to the west, south, and east of M S H will be examined (Fig. 8). Use of the energy-line model on the lahars gives obviously poor results. In order to match the observed runout distance, very low values of tan0 must be chosen ( ~ 0.05 or less), but this results in excessively high model velocities (more than 100 m s -1 } and, consequently, the flow lines are nearly straight and extend radi-

220

S. Fork

ToUter

e~

0 i

1

I

1

1

5km l

)

,

Fig. 8. Distribution of surge and lahar deposits (dotted area) near Mount St. Helens (MSH), and locations tbr velocity estimates listed in Table 5. Dashed line indicates base of MSH at 1250-m elevation. Figure boundaries correspond to boundaries of DEM used for lahar models, except for west margin. Figure simplified from Pierson (1985 }.

ally from MSH rather than following the sinuous channels. A Bingham model (Johnson, 1970, 1984) is more appropriate for lahars. Mudflow velocities may be estimated from the geometry of the deposits in two ways (cf. Pierson, 1985). First, where the flow has run up a hillside perpendicular to the flow path, assuming that all kinetic energy was converted to potential energy gives v = (2gH) 1/2 where H is the height of runup. The second method is based on the superelevation in a bend, and velocity is approximately (g~ztanfl) 1/2 where ~ is the radius of curvature of the bend and fl is the tilt angle. Both of these methods are generally thought to give minimum velocities because flow resistance is not considered. However, experimental work by Ikeya and Uehara (1982) suggests that the superelevation method results in overestimates of the actual velocities. These methods have been used to estimate MSH mudflow velocities by Fink et al. ( 1981 ) in the West Branch of Pine Creek, by Fairchild and Wigmosta ( 1983 ) in the South Fork Toutle

River, by Major and Voight (1986) on the southwest flank lahars, and by Pierson ( 1985 ) on the Pine Creek-Smith Creek-Muddy River area (see Fig. 8 and Table 5). These workers have also provided measurements of flow depth, channel width, and channel slope, so these data can be used to estimate rheologic properties of" the lahars. The South Fork Toutle River, Smith Creek, Muddy River, Ape Canyon, and East Branch Pine Creek channels are all at least 5 times wider than the flow depth, so the widechannel approximation is appropriate. For the West Branch Pine Creek, the circular-channel model seems more appropriate (Fink et al., 1981). By assuming steady, uniform flow, a Bingham rheology, and a value for yield strength (k), we can solve for the flow viscosity by using eqs. 9-11 and 14. (Not included in Table 5 are a number of velocity estimates near the base of MSH, because uniform flow is unlikely near the base of a long steep slope. ) By measuring the block and matrix densities and submerged and total heights of partly submerged blocks, k may

221 TABLE 5 Lahar viscosity calculations and model velocities Site

v (ms-')

Slope (°)

Depth (m)

Viscosities, ~ ( × 102 Pa s) Wide channels

Circular channels

Yield strength, k (Pa):

Model velocity

0

500

1000

0

500

1000

LA]

LA2

East Branch Pine Creek E1 23 8.6 E2 18 5.3 E3 21 3.7 E4 13 2.3 E5 12 2.4 MEANS:

10 15 13 14 15 13

43 76 34 40 52

42 74 33 37 48 47

41 72 31 34 45

16 28 13 15 19

16 28 12 14 18

15 27 12 13 17

8 23 19 18 14

West Branch Pine Creek Wl 31 18.0 W2 15 8.0 MEANS:

7 12 9.5

36 88

35 86

35 84

13 33

13 32 22.5

13 32

57 54

24 19

Ape Canyon A1 28

1.6

21

29

27

25

11

l0

l0

38

6

Mud~Rive~SmithCreek M1 27 1.1 M2 14 1.1 M3 7 0.6

5 5 4

1.2 2.3 1.6

0.7 1.4 0.3

0.3 0.6

0.4 0.8 0.6

0.3 0.6

SouthFork ~ u t l e R i v e r T1 33 2.3 T2 26 2.3

5 5

1.6 2.2

1.3 1.8

0.9 1.3

1.0 0.8

0.5 0.7

19 -

0.4 0.5

15 9

5 3

Velocity, slope, and depth measurements from Pierson (1985) for East Branch Pine Creek, Ape Canyon, and Muddy River/ Smith Creek; from Fink et al. (1981) for West Branch Pine Creek; and from Fairchild and Wigmosta (1983) for South Fork Toutle River. The viscosity calculations reported here for West Branch Pine Creek differ from those reported by Fink et al. ( 1981 ) by about a factor of 10 due to an error by Fink et al. (J.H. Fink, pers. commun., 1988). Sites T1 and T2 (South Fork Toutle River) are located 3 and 8 km, respectively, from the base ( 1250-m contour) of MSH. See Fig. 8 for locations of other sites. be e s t i m a t e d ( J o h n s o n , 1970). U s i n g this m e t h o d for M a y 18th flows, F i n k et al. (1981) f o u n d k ~ 400 P a in t h e W e s t B r a n c h of P i n e Creek a n d M a j o r a n d Voight (1986) f o u n d k f r o m 900 to 1200 P a for t h e s o u t h w e s t f l a n k lahars. We have c o m p u t e d q for t h r e e values of k: 0, 500, a n d 1000 P a (Table 5). T h e l a h a r s in P i n e Creek a n d Ape C a n y o n are sufficiently deep t h a t t h e choice of k has little effect on q (cf. D r a g o n i et al., 1986). As Dc a p p r o a c h e s D as the flow t h i n s or t h e g r a d i e n t flattens, k has

a n i n c r e a s i n g effect on the s o l u t i o n for ~/. A b i m o d a l d i s t r i b u t i o n of viscosity values is a p p a r e n t in T a b l e 5: l a h a r s in P i n e Creek a n d Ape C a n y o n h a d viscosities r a n g i n g f r o m 103 to 104 P a s, whereas lahars in S m i t h Creek, M u d d y River, a n d t h e S o u t h F o r k T o u t l e River has viscosities r a n g i n g f r o m 10' to 102 P a s. M a j o r a n d Voight (1986) also f o u n d ~ r a n g i n g f r o m 103 to 104 P a s for l a h a r s on t h e s o u t h w e s t flank. L a n g a n d D e n t (1987) f o u n d N e w t o n i a n viscosities of ~ 4 X 103 P a s for u p p e r reaches of the E a s t

222

Branch Pine Creek and ~ ~ 5 × 102 Pa s for lower reaches of the East Branch Pine Creek, Muddy River, and Smith Creek. Apparently, when the lahars were diluted by entering major rivers, this resulted in a 10- to 1000-fold reduction in viscosity, consistent with the behavior of clay slurries (Moore, 1965). A comparable reduction in yield strength also occurs from dilution of a debris flow (Rodine and Johnson, 1976), so the viscosity values listed in Table 5 for zero strength are probably most appropriate for the Smith Creek, M u d d y River, and South Fork Toutle River locations. These results are consistent with previous studies showing transformation of M S H lahars into hyperconcen-

trated streamflow (Pierson and Scott, 1985 ). Two examples of lahar models from FLOW are described here (see Fig. 9). A yield strength of' 500 Pa and Bingham models were used. The major lahars on the flanks of M S H were initiated by pyroclastic surges, with initial veloci ties of ~ 30 m s - ' on the south flank (Moore and Rice, 1984 ) and ~ 50 m s J on the east flank (Pierson, 1985). For FLOW, an initial velocity of 40 m s-~ was used in all directions (except to the north where the avalanche prevented lahar formation near the volcano). For the first lahar model (LA1), we chose the mean flow depth and viscosity values determined for the East Branch Pine Creek (Table

Fig. 9. Lahar models LA1 (a) and LA2 (b). See Table 3 for FLOWinput parameters. Flow lines superposed over shaded relief image of pre-eruption DEM (but FLOWmodels calculated with post-eruption DEM).

223

5 ) and a wide-channel model. The resulting flow paths (Fig. 9a) match the observed paths in the East Branch of Pine Creek, but they are too long on the south flank of M S H and they are too short in M u d d y R i v e r / S m i t h Creek. In addition, the model velocities (Table 5) are comparable (within a factor of 2) to the estimated velocities for the East Branch Pine Creek and Ape Canyon, but they are too high for the West Branch Pine Creek; they are too low in Muddy River, Smith Creek, and the South Fork Toutle River because much lower viscosities and yield strengths characterized the lahars in these drainages. The relatively short travel distance of lahars on the south flank of M S H (including the West Branch of Pine Creek) may be due to three factors (cf. Major and Voight, 1986): (1) smaller lahar volumes due to less voluminous surges and less glacial ice; (2) lower initial velocities of the surges; and (3) small channel widths (e.g. significant resistance from channel sides, which is ignored in the wide channel model ). For the second lahar model (LA2), we chose the average flow depth and viscosity values determined for the West Branch of Pine Creek (Table 5), and a circular-channel model. Note that the use of the circular channel results in calculated viscosities less than half as large as with a wide channel. The resulting flow paths (Fig. 9b) and velocities (Table 5) are good for the West Branch of Pine Creek, but the flows still travel too far on the south flank of M S H , and the model velocities are too low elsewhere. In fact, no flow line reaches the East Branch of Pine Creek. On the west flank of M S H , the flow paths from model LA2 match the observed distribution (Fig. 8), but downstream in the South Fork Toutle River the model velocities are much too low (Table 5 ). This suggests that model LA2 is about right for the initial flow on the west flank but that the rheologic properties changed when the flow was diluted by river water. Use of a lower initial viscosity results in higher velocities and less channelized flow lines on the west flank, as in model LA1, resulting in a poor

match to the observed distribution of deposits. In summary, there is no single set of FLOW parameters that can match all of the M S H lahars, because the viscosities, flow depths, and channel shapes varied with space and time. Nevertheless, the FLOW models do match the observed distributions and velocities for specific locations when the appropriate parameters are used; this in turn demonstrates that FLOW can be used to constrain the local rheologic parameters. Furthermore, the use of several FLOW models may provide a useful prediction of the range of possible effects for planning hazard mitigation. Blast The laterally directed blast was the most spectacular and destructive event at M o u n t St. Helens (Lipman and Mullineaux, 1981, and papers therein). The blast was initiated by pressure release resulting from the failure of the bulging north flank, with ejection from the walls of the north-facing slump scarps. The blast cloud rapidly spread in an arc of nearly 180 over more than 600 km ~ of mostly forested land north of the volcano. The trees were entirely removed from an inner zone, standing trees were blown to the ground in an intermediate zone, and trees were seared but left standing in an outer zone. Tree blowdown directions provide a detailed map of the flow directions of the blast front (see Plate 1 of Lipman and Mullineaux, 1981). Photographs and eyewitness accounts indicate that the blast cloud was ground hugging and had a leading edge no more than a few hundred meters high, although convection clouds rapidly billowed up to heights of several kilometers behind the flow front. The blast deposits thin from about 1 m near M S H to 1 cm near the margin, have an overall normal grading, and decrease in clast size with distance from the vent. The blast front moved rapidly, at more than 95 m s 1, but the velocity estimates range widely. If the blast destroyed the seismic sta-

224

tion 6 km north of the vent, then the velocity in this region was ~ 200 m s - 1 (Voight, 1981 ). Two observers estimated that the blast arrived at a position about 13 km north of the vent in 85 s, which gives an average velocity of 150 m s- 1 (Rosenbaum and Waitt, 1981). Observations near the North Fork Toutle River are consistent with an average velocity of 20-45 m s - ] near Elk Rock {Rosenbaum and Waitt, 1981). A more recent compilation of the blast chronology by Moore and Rice (1984), including satellite observations and photographs from M o u n t Adams, indicates a remarkably constant velocity of ~ 95 m s 1 along the entire 22km travel distance to the north of the volcano (see Fig. 10). The satellite and M o u n t Adams observations may be of fronts moving toward the northeast or northwest rather than due north, so 95 m s-1 is probably a minimum average velocity, b u t it seems unlikely that the average velocity could have exceeded about t30 m s - i . Nevertheless, the most remarkable aspect

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22.G

DISTANCE [ KM ] Fig. 10. Chronology of blast models. Elevation profile (A) extracted from pre-emption D E M as a straight line extending N 10°W from a starting point 2 km north of the vent. Time versus distance predictions shown for models BL1 (B), BL2 (C), and BL3 (D). See Table 3 for model input parameters. Circles are observations from Moore and Rice (1984, figs. 10.4 and 10.5).

of this dataset is the constancy of the velocity, in spite of the great travel distance and the rugged topography with relief of more than 0.5 km. Shown in Fig. 10a is a topographic profile extending in a straight line N 10 ° W from the vent for comparison with the Moore and Rice data and the FLOW models. (The blast traveled only about 15 km due north from the vent due to blockage by or diversion around Mount Whittier, whereas it traveled more than 20 kin northwest and northeast. ) There is disagreement over whether the blast was primarily an expanded, turbulent surge (Hoblitt et al., 1981; Moore and Sisson, 1981; Waitt, 1981, 1984; Hoblitt and Miller, 1984) or whether the ash cloud was underlain and driven by a dense basal flow of dominantly laminar motion (Sparks, 1983; Walker and McBroome, 1983; Walker and Morgan, 1984). (This has been called the "surge" versus "'flow" controversy. ) Pyroclastic flows, debris avalanches, and lahars are driven by the force of gravity, and their flow paths are strongly influenced by the topography. The forces acting on pyroclastic surges, however, are more complex. Surges are influenced by topography and gravity (e.g., Sheridan and Malin, 1983), but they may also mantle topography, like fallout tephra, due to aerodynamic processes, and compressible flow may result in significant acceleration (Kieffer, 1981; Wohletz et al., 1984). Surges are likely to have pronounced density gradients from bottom to top and front to back (Waitt, 1981; Valentine, 1987), and the density may decrease dramatically as a function of time and runou~ distance (Sparks et al., 1986). FLOW is not designed to account for these processes, but it can at least illustrate some limiting cases and per-haps provide a useful empirical model for hazards mapping. Three blast models are presented here, which represent motion resistance by each of the three terms in eq. (1). The first model (BL1) is the energy-line approach advocated by Malin and Sheridan (1982); the second model (BL2) is a

225

viscous pyroclastic flow model advocated by Walker and McBroome (1983); and the third model (BL3) uses turbulent resistance. An empirical model for surge eruptions presented by Malin and Sheridan (1982) and Sheridan and Malin (1983) is called the "energy cone" model. This model consists of an energy line rotated 360 degrees into a cone. The model's surge boundary is defined by the intersection of the cone with the ground surface (of a DEM), and the model's flow streamlines are straight lines of radii from the vent. Because the streamlines from a gravity-driven flow are diverted from straight lines of radii by topographic slopes oriented at oblique angles to the flow lines, the energy-cone model does not fully account for the effects of the topography (cf. Armienti and Pareschi, 1987). For high-velocity flows such as the blast, however, this limitation is minor. Model BL1 is an energy-line model that best matches the geographic extent of the blast (Fig. 11 a). The input parameters (Table 3) are similar to those used by Malin and Sheridan ( 1982 ), except that the initial velocity and tan¢ are both slightly lower. Note that the flow lines begin 2 km in front of the vent, approximately where the eruption fountains or jets coalesced into a ground-hugging cloud. The model velocities are too high compared with the timed observations reported by Moore and Rice (1984; Fig. 10b), but the velocities compare favorably with those in some of the earlier reports (Rosenbaum and Waitt, 1981; Voight, 1981). Also, the flow lines match the general pattern of tree blow-down directions. Therefore, this seems to be a reasonable empirical model for the blast. Model BL2 is based on the pyroclastic flow model of Walker and McBroome (1983) and Walker and Morgan (1984). Walker and MorFig. 11. Blast surge models: (a) BL1; (b) BL2; (c) BL3 (cf. Table 3 ). Black lines outline area of tree removal, blowdown, or scorching. Shaded relief base from pre-emption Hoquiam DEM quadrangle, but with modification of volcano's summit to approximate appearance at time of blast initiation.

226 gan (1984) provided a model of flow depth as a function of distance from the vent; D decreases approximately linearly with distance from an initial thickness of 28 m, which defines D and d D / d t for the model (Table 3). For the density of the active flow, we chose 1000 kg m -:~ on the basis of the estimated deposit density (Moore and Sisson, 1981) and the estimate by Walker and Morgan (1984) that the active flow was expanded ~ 30% compared with the deposit at rest. We used a yield strength of 600 Pa based on the work of Wilson and H e a d (1981), but this value for k has a small effect on the dynamics except near the terminus where the flow is very thin. Sparks (1976) estimated that partly fluidized pyroclastic flows have viscosities ranging from 5 to 100 Pa s; we chose the upper limit because this value gives the best match to the observed chronology. We used an initial velocity of 95 m s - ~, again to give the best possible match to the observed velocities. The model results (Figs. 10c and l l b ) match the observed velocities and distribution about as well as do the results of model BL1. The Reynolds and Bingham numbers for model BL2 parameters and typical values of v ( 100 m s - 1) and D ( 14 m ) are Re = 14000 and B i = 0.84, well within the turbulent range (Fig. 3). In fact, any reasonable combination of p, D, q, and k values along with a velocity of 100 m s-1 results in a prediction of turbulent flow. If the turbulence is not fully developed and continuous, then the dense pyroclastic flow model for the blast is not disallowed. However, if we add Cg= 0.008 ( ~ 1-cm roughness elements) to model BL2, the predicted travel distances are far too short, and the flow lines do not even cross J o h n s t o n Ridge, ~ 9 km from the summit of MSH. Furthermore, a more realistic size for roughness elements over a devastated forest i s ~ 1 m (Valentine, 1987), resulting in much greater turbulent resistance to movement. Therefore, accounting for the great travel distance seems to be a major problem with the pyroclastic flow model for the blast. The next model (BL3) may be thought of as

a simple surge model, but it ignores air drag, compressible flow, and other important effects. Nevertheless, this model is presented in order to illustrate that a model with turbulent resistance can match the observations as well as the first two models. The model parameters (Table 3) include an initial depth of 200 m and a density of 10 kg m -:~. The coefficient of roughness, 0.01, is appropriate for roughness elements of about 5 cm with fully developed turbulence ( eq. 17), which is probably too low for MSH. The results (Figs. 10d and 1 lc ) are about equivalent to those of models BL1 and BL2 in matching the observed velocities and distribution. Apparently, any resistance model gives about the same result provided that the resistance has a small effect on the momentum. Walker and McBroome (1983) considered it unlikely that a highly expanded pyroclastic surge could travel as far as 30 km against air resistance. The magnitude of air drag can be evaluated from equations 21 and 22. The flow density has an important effect on air drag. According to Sparks et al. (1986), the blast density decreased dramatically with increasing time and distance due to three processes: 11) decompression of the blast mixture (Kieffer. 1981 ); (2) mixing and heating of the surrounding air into the cloud; and (3) sedimentation from the flow. Kieffer (1981) suggested that the blast density decreased to less than that of the (dusty) atmosphere at the limits of the blast zone, so that the cloud lifted from the ground into the atmosphere and thus created the seared zone. Sparks et al. (1986) further argued that this buoyant uplift created the giant umbrella cloud that formed over the blast zone. in Kieflet's model, the blast's density at about half' of its total travel distance was ~ 10 kg m -:~. If we assume a dusty atmospheric density of 2 kg m :~, an atmospheric drag coefficient of 0.5 ( Perla, 1980), a flow depth of 200 m, and a velocity of 100 m s -1, the deceleration from air drag (eq. 22) is 2.5 m s -2. At that rate of deceleration tif constant), the cloud would come to a stop in only 40 s.

227 In summary, the blast must move against substantial resistance from both turbulence against the ground and air drag. There seems to be no way for the blast to travel 20 km or more simply from an initial velocity of ~ 100 m s - ' and the gravitational accelerations. Furthermore, a much higher initial velocity results in only a very small increment in travel distance because the turbulent and air resistances increase with velocity squared. There must be some additional process driving movement of the blast cloud, such as blast decompression (Kieffer, 1981 ). Ideas similar to these have been presented by Valentine (1987).

Summary and conclusions A computer model for the movement of gravity flows is presented and tested against the 1980 events at Mount St. Helens for four examples of gravity flow: pyroclastic flows, the rockslideavalanche, lahars, and the blast. The energyline model, which had been widely used for this type of effort, generally predicts velocities that are too high, resulting in flow paths that are insufficiently responsive to the topography. Velocity-dependent resistance terms (Bingham and/or turbulent models) must be applied in order to achieve accurate predictions. The August 7th pyroclastic flow was examined in detail for comparison with the velocities acquired from a timed sequence of photographs (Hoblitt, 1986). Both the energy-line model and a Bingham model based on measured rheologic properties (Wilson and Head, 1981) result in model velocities that are too high. From the Reynolds and Bingham numbers, turbulence is expected; partial or local turbulence is consistent with the presence of plane-parallel and crossbedded deposits on the steep northern flanks of the volcano (Rowley et al., 1985 ). Addition of turbulent resistance to the Bingham model results in a much better match to the measured velocities. A similar model best matches the distribution of deposits from the more voluminous pyroclastic flows of May 18th.

The rockslide-avalanche began as three slide blocks and rapidly evolved into a debris avalanche. Our best model is a Bingham flow with a viscosity of 3 X 104 Pa s and a yield strength of 104 Pa. The model matches the observed distribution and velocities surprisingly well, considering that the flow properties varied greatly with space and time, but, in contrast to the lahars, the tremendous inertia of the moving avalanche may have masked the effects of these local variations. The lahar viscosities seem to have been bimodal. Near the flanks of the volcano and in relatively dry creeks and canyons, the viscosities ranged from 103 to 104 Pa s; where the lahars entered river channels with significant water, the viscosities were reduced to 101-102 Pa s. The F L O W models match the observed distribution and estimated velocities for specific regions provided that viscosity and flow depths appropriate for the particular region are entered. For the common flow depths (5-20 m), the yield strength (in the range of 0-1000 Pa) has little effect on the mean velocities. The blast is the most difficult flow to model due to the complexities of compressible flow and rapidly changing flow density. The three basic types of resistance model each compare reasonably well with the observed distribution and velocities, provided the resistance is small and the initial velocity is high. However, the expected resistances from both boundary-layer turbulence and air drag are substantial, so movement over 20-30 km of rugged terrain must have been facilitated by some process such as continued decompression of the blast cloud (Kieffer, 1981 ).

Acknowledgements We thank W.A. Duffield and two anonymous referees for critical reviews. This research was supported by NSF (grant EAR8313091 to MCM) and by NASA (grant NGT 03-001-802 to ASM).

228

A p p e n d i x : FLOW p r o g r a m

The FLOW computer program uses the relations described under EQUATIONS OF MOTION to compute models of nonuniform flow movement over a topographic model (the D E M ) . The flow lines or paths are output in digital format into two images, with digital numbers (DNs) proportional to time (from the beginning of the model run) and velocity, corresponding to each pixel location crossed by a flow line. Each flow line in a model run begins along a circular arc defined by a source or vent location, radial distance from the source, and azimuth. This arrangement allows simulation of a variety of initial geometries, including arcuate slump scarps or radial flow from collapse of a Plinian eruption column. Each flow line is also given an initial velocity; the initial flow direction is radially outward or inward from the arc. Other input parameters define the properties of the flow (Table 2): yield strength (k), angle of internal friction (0), viscosity (q), drag coefficients for the ground and atmosphere (cg and ca), and flow density (p). Also input initially are flow depth (D) and a constant for dD/dt (m s - 1) (to estimate the decrease in D with time due to deposition or lateral spreading of the flow) and a choice of either a wide or circular channel. Currently, each model run with 10-100 flow lines requires less t h a n 1 min of computer time on a Vax 750 or Microvax-II. After the initial conditions are chosen, the movement of each flow line is determined by gravitational accelerations (in the downhill direction of each pixel) minus the resistance (parallel to the direction of motion ). The equations for time interval (At), acceleration, and velocity are all interdependent, so exact solutions are not possible and we must use some approximations. Errors from these approximations were evaluated by decreasing the time steps; for the M S H data, when the time step is set according to the pixel scale, these errors are insignificant. The new velocity at the end of

each time step must be computed explicitly (e.g., the quadratic equation for v from eq. 4 must be solved) to prevent occasional velocity oscillations in a diverging series with each time step. FLOW computes the movement for each time step by the following procedure: (1) The duration of the time step (Jt) is estimated by the pixel scale divided by the velocity of the previous time step (vo), so that the flow will travel to an adjacent pixel during At. (2) If k exceeds zero, the plug thickness is computed from eq. ( 12 ) and the use of ~.'~to approximate v. (3) Acceleration from gravity and deceleration from the resistance terms are computed for the plug or surface of the flow. For the velocitydependent resistance terms, Vp from the previous time step is used to approximate vp. If the acceleration is significant, At is adjusted accordingly. (4) To find the change in the direction of movement, components of gravitational acceleration are computed parallel (Agl) and perpendicular (Ag2) to the beginning flow direction. The new flow direction is determined by vector addition of voXAt, A~IxAt'~/2, and Ag2 X At 2/2.

(5) The new velocity of the plug, Up ( or of the top center of the flow if k = 0) is computed explicitly as a function of Vo and the acceleration terms. (6) The new cross-sectional velocity (v~ is computed as a function of Vp by eq. ( 14 ). If u is less t h a n or equal to 0, the flow line ends. (7) Three-dimensional distance traveled is computed from the average velocity, (~' + v~,) / 2, times At. (8) Two-dimensional (planimetric) travel distance is computed by correcting for the slope in the direction of movement, and used to determine the new location of the flow front on the DEM. (9) Time (since the beginning of the model ) and v are output on the time and velocity images at the new pixel locations.

229

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