Journal of Volcanology and Geothermal Research 139 (2005) 89–102 www.elsevier.com/locate/jvolgeores

Evaluating Titan2D mass-flow model using the 1963 avalanches, ,

M.F. Sheridana,*, A.J. Stintona, A. Patrab, E.B. Pitmanc, A. Bauerb, C.C. Nichitac

aDepartment of Geology, 876 Natural Science Complex, University at Buffalo, Buffalo NY, 14260, USA bDepartment of Mechanical and Aerospace Engineering, University at Buffalo, Buffalo NY, 14260, USA cDepartment of Mathematics, University at Buffalo, Buffalo NY, 14260, USA Accepted 29 June 2004

Abstract

The Titan2D geophysical mass-flow model is evaluated by comparing its simulation results and those obtained from another flow model, FLOW3D, with published data on the 1963 Little Tahoma Peak avalanches on Mount Rainier, Washington. The avalanches, totaling approximately 10106 m3 of broken lava blocks and other debris, traveled 6.8 km horizontally and fell 1.8 km vertically (H/L=0.246). Velocities calculated from runup range from 24 to 42 m/s and may have been as high as 130 m/s while the avalanches passed over . Titan2D is a code for an incompressible Coulomb continuum; it is a depth-averaged, dshallow-waterT, granular-flow model. The conservation equations for mass and momentum are solved with a Coulomb-type friction term at the basal interface. The governing equations are solved on multiple processors using a parallel, adaptive mesh, Godunov scheme. Adaptive gridding dynamically concentrates computing power in regions of special interest; mesh refinement and coarsening key on the perimeter of the moving avalanche. The model flow initiates as a pile defined as an ellipsoid by a height (z) and an elliptical base defined by radii in the x and y planes. Flow parameters are the internal friction angle and bed friction angle. Results from the model are similar in terms of velocity history, lateral spreading, location of runup areas, and final distribution of the Little Tahoma Peak deposit. The avalanches passed over the Emmons Glacier along their upper flow paths, but lower in the valley they traversed stream gravels and glacial outwash deposits. This presents difficulty in assigning an appropriate bed friction angle for the entire deposit. Incorporation of variable bed friction angles into the model using GIS will help to resolve this issue. D 2004 Elsevier B.V. All rights reserved.

Keywords: GIS; TIN; mass-flow model; Mount Rainier; avalanche; adaptive gridding

1. Introduction

* Corresponding author. Tel.: +1 716 645 6800x3984; fax: +1 Debris avalanches and flows are frequently asso- 716 645 3999. ciated with volcanic activity or collapse of over- E-mail address: [email protected] (M.F. Sheridan). steepened slopes due to water saturation or prolonged

0377-0273/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jvolgeores.2004.06.011 90 M.F. Sheridan et al. / Journal of Volcanology and Geothermal Research 139 (2005) 89–102 periods of erosion. They pose a significant threat to majority of the deposit fills an area of 1.3 km2 that the population living on and around volcanoes. lies between the terminal moraine and the terminus Between 1900 and 1985, approximately 76,000 of Emmons Glacier, where the deposit has a people have been killed by debris avalanches, debris maximum thickness of 30 m. The thickness variation flow and pyroclastic flows related to volcanic activity within the deposit was determined using several (Tilling, 1989). This number includes the estimated cross-sections from fig. 10 in Crandell and Fahne- 29,000 killed by pyroclastic flows at St. Pierre on stock (1965). These sections were originally sur- Martinique in 1902 and the 23,000 killed by lahars veyed for a study of the White River geomorphology from the eruption of Nevado del Ruiz, Columbia in by Fahnestock (1963) just prior to the occurrence of 1985 (Tanguy et al., 1998). the avalanches. As global population grows, pressure increases to Crandell and Fahnestock (1965) identified seven develop available land. This has resulted in an different avalanche units based on surface features, increase in the numbers living on or close to active textural and color variations seen in field mapping, volcanoes. For this very reason, it is necessary to and aerial photographs. This avalanche deposit is develop accurate and usable prediction models, so that similar in appearance to deposits at other volcanoes the impact of a potential hazardous event can be such as Mount St. Helens, though on a much smaller correctly determined and appropriate actions taken. A scale. Large blocks up to 184050 m rest on and variety of models exist for simulating various types of are partially buried by a matrix of grayish-red sand- geophysical mass flows at volcanoes, such as sized material of the same composition. The deposit FLOW3D (Kover, 1995), LaharZ (Iverson et al., surface has several curvilinear ridges and troughs 1998) and DAN (Hungr, 1995), all of which have demarking lateral and distal margins of the various their advantages and disadvantages. avalanche units. Of the seven avalanche units identi- This study compares simulations using Titan2D, a fied, Unit 3 is presumed to be the largest and the new geophysical mass-flow model developed at the furthest traveled. University at Buffalo (Pitman et al., 2003; Patra et al., During movement, at least one of the avalanches submitted for publication), with an earlier model, ran up the lower west-facing slope of Goat Island FLOW3D. The 1963 Little Tahoma Peak avalanches Mountain to a maximum height of 90 m. Ava- on Mount Rainier, Washington were selected to lanche Unit 3 also ran up the north-facing slope of validate the models on the basis of the wealth of , opposite the terminal mor- published data available on the dynamics and features aine, to a height of 43 m. This indicates that Unit of these avalanches and their deposits. 3 was deflected by the terminal moraine through the gap between it and the valley wall incised by the White River. Unit 3 continued to flow another 2. The 1963 Little Tahoma Peak avalanches 600 m downstream past the moraine, coming to rest about 1.6 km upstream from the White River Little Tahoma Peak is located on the eastern flank Campground. of Mount Rainier volcano (Fig. 1). The steep north Velocities calculated from the runup heights give face rises some 600 m above the Emmons Glacier. values of 42 m/s at Goat Island Mountain and 24 m/s On December 6th 1963, and possibly over a period at the terminal moraine. These are assumed to be of several weeks afterwards (Norris, 1994), a series minimum velocities at the two locations. Crandell and of seven avalanches descended from the north-facing Fahnestock (1965) determined a velocity of 134 m/s slope. After impacting the Emmons Glacier at the for the units at the point they left Emmons Glacier and base of the peak, the avalanches proceeded to flow became airborne, hitting the ground some 600 m over the glacier and down the White River Valley for down valley. It is at this point that they presumed that a distance of 6.8 km while descending approximately the avalanches trapped air that enabled them to travel 1900 m. An estimated 10106 m3 of brecciated 2800 m beyond the glacier’s terminus. According to andesitic lava flows and other debris covers 5.1 km2 seismic records from the time, the largest avalanche of the White River Valley and Emmons Glacier. The created a signal that was recorded for approximately M.F. Sheridan et al. / Journal of Volcanology and Geothermal Research 139 (2005) 89–102 91

Fig. 1. Location of Little Tahoma Peak avalanche deposits. Red outline indicates mapped extent of area over which the 1963 avalanches passed as mapped by Crandell and Fahnestock (1965); superimposed on USGS aerial photographs acquired in 1994. Insert shows location with respect to Mount Rainier National Park.

300 s on the Longmire seismic station (LON) hazards. These include velocity, basal friction, seismograph (Norris, 1994). areal distribution and deposit thickness and the volume of material at the source region. These will be discussed in detail in the following section. 3. Avalanche dynamics Table 1 compares several parameters between the work of Crandell and Fahnestock (1965) and the There are several parameters important in results from the flow models FLOW3D and modeling debris avalanches and assessing their Titan2D. 92 M.F. Sheridan et al. / Journal of Volcanology and Geothermal Research 139 (2005) 89–102

Table 1 where h is the angle between downslope direction and Comparison of published data on the Little Tahoma Peak a normal to the curved path. Hence, magnitude of the avalanches and the results of simulations done with the FLOW3D and TITAN2D models velocity is given by: Published FLOW3D TITAN2D v2 ¼ rgsinacosh ð4Þ data Run out length (km) 6.8a 6.8 6.8 The actual path of curvature is not always Fall height (km) 1.9a 1.9 1.9 circular, being a function of the velocity and the 3 6a 6 Volume (m )1010 N/A 110 surface slope vectors; that is, r is not constant, but (total) (single) Maximum 30a N/A 3 (single) it does have a finite value at any location in space, thickness (m) (total of 7) from which the velocity can be calculated. While Maximum 134a 82 75 this is quite a simple relationship, it is also quite velocity (m/s) rigorous in that friction is not ignored—friction a Maximum run up 90 201 60 changes velocity and hence the radius of curvature. height (m) Bed friction angle – 8.58 128 However, approximations do have to be made for Internal friction – 0.01 338 the values of r, a and h. McSaveney (1978) used (velocity) this simple relationship to calculate a velocity of 12 Time (s) V300b 172 52 m/s for the Sherman Glacier rock avalanche as it Data for the TITAN2D model correspond to the simulated flow moved over and ice during the latter stages of shown in Figs. 3 and 4. a flow. This relationship was also applied to the Little Data from Crandell and Fahnestock (1965). Tahoma Peak avalanches, approximately 1 km from b Data from Norris (1994). the base of the peak, where the avalanches are curving down the slope of the glacier. Here, a 3.1. Velocity velocity of 57 m/s was calculated. This velocity may be too low for the avalanches at this point, as The velocity of a moving body of material can be the calculation was done without taking into account calculated under several circumstances in which the any extra momentum generated by free fall from the trace of the flow path and the underlying topography face of Little Tahoma Peak. is known. Three such circumstances are: (1) move- ment along a curved path across sloping topography; 3.1.2. Runup obstacles aligned perpendicular to flow (2) runup onto obstacles aligned perpendicular to the A second method for determining velocity is to use direction of flow; and (3) superelevation along the the height to which flows runup on obstacles aligned outside of bends in a confined channel. perpendicular to the direction of flow using the simple relationship: 3.1.1. Movement along a curved path According to basic physics, a particle of mass m pffiffiffiffiffiffiffiffi and velocity v moves in a circular path of radius r v ¼ 2gh ð5Þ when subjected to a radial force Fr according to the relationship: where h is the runup height and g is the

2 acceleration due to gravity. The Little Tahoma Peak Fr ¼ mv =r ð1Þ avalanches ran up Goat Island Mountain (see Fig. 1) If the force producing the curvature of path is a in two places, on the west-facing slopes to a height gravity force Fg, acting down slope (slope=a), then: of about 90 m, and on the north-facing slope to a height of 43 m. In these two areas, the minimum Fg ¼ mgsina ð2Þ velocity required to run up to these heights is 43 and this has a component perpendicular to the path and 29 m/s, respectively, as determined by Crandell that is equal to Fr. Thus: and Fahnestock (1965). These are considered mini- mum velocities because Eq. (6) does not account for 2 Fr ¼ mv =r ¼ mgsinacosh ð3Þ friction. M.F. Sheridan et al. / Journal of Volcanology and Geothermal Research 139 (2005) 89–102 93

3.1.3. Superelevation sections resurveyed by Crandell and Fahnestock As material flows in a channel, centrifugal forces (1965) in which the channel cross-section could be cause the mass of debris to rise up the outside of well constrained, as shown in Fig. 2. At cross-section bends. The runup or superelevation is the height to 2, superelevation of 18 m was calculated for Unit 3 which material rises as it banks through the curve flowing in a channel 170 m wide and with a radius of (Chow, 1959; Evans et al., 2001). It is possible to 454 m, giving a velocity of 21.7 m/s. The same calculate the minimum velocity from the following velocity was calculated for a superelevation of 43 m in relationship: cross-section 5 where the channel width and radius pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi were 105 and 117 m, respectively. There are several v ¼ ½ŠðÞgdr =b ð6Þ drawbacks to using this relationship: (1) friction is ignored and thus the velocities may be too low; (2) where g=gravitational acceleration, d=superelevation, alternatively the velocities could be too high due to r=centerline radius of curvature and b=channel width. the internal rigidity of the flowing material; and (3) it This relationship was applied at two of the cross- has not been rigorously tested on debris avalanches.

Fig. 2. Cross-sections 2 and 5 showing the channel width and superelevations used in the velocity calculations (see text). Original profiles redrawn from Fig. 10 of Crandell and Fahnestock (1965). 94 M.F. Sheridan et al. / Journal of Volcanology and Geothermal Research 139 (2005) 89–102

Evans et al. (2001) used this relationship as part of gravity, and a is the surface slope. In application to the their field investigation of the 1983 Mt. Cayley Sherman Glacier avalanche, McSaveney (1978) used avalanche and debris flow in the British Columbia, a section of the avalanche deposit where many and compared the velocities they calculated with those curvilinear ridges and troughs are identifiable. These determined using their dynamic analysis numerical ridges and troughs clearly define the margins of model DAN, which were found to be similar. Pierson several flow lobes and the directions in the lobe were (1985) suggested that Eq. (6) underestimates the flowing. Using a value of 12 m/s for V1 [the velocity velocities of debris flows by about 15%. of the interior of the flow, calculated using Eq. (4)], a coefficient of 0.11 was obtained—a value half that 3.2. Coefficient of friction using the H/L method. This method was applied to the Little Tahoma Peak The coefficient of friction (l) is a measure of the avalanches in two places: (1) high up on Emmons resistance to flow generated by a sliding avalanche at Glacier, approximately 1 km from the base of the the contact with the underlying topography. In its peak, and (2) in the area of cross-section 2. Velocity in simplest form, it is the tangent of the angle (a) these areas is approximately 57 and 27 m/s, respec- connecting the top of the source area to the most distal tively (see previous section for calculations). Using part of the flow (Heim, 1932): these velocity estimates, coefficients of 0.037 and 0.217 were obtained respectively for movement over Hmax=Lmax ¼ l ¼ tana ð7Þ ice and snow and over the unconsolidated glacial– fluvial deposits in the White River Valley between the where Hmax is the fall height and Lmax is the horizontal distance travelled. According to Crandell glacier terminus and the terminal moraine. The coefficient value calculated for the Little Tahoma and Fahnestock (1965), Hmax is 1.8 km, and Lmax is 6.8 km, giving a coefficient of 0.246 (or a=14.88). Peak avalanches over snow and ice (0.037) is very Comparing this value to those of other published H/L low, much lower, in fact, than the value for the values, it falls within the range of subaerial non- Sherman Glacier avalanche. volcanic avalanches (Hayashi and Self, 1992). For example, the 1964 Sherman Glacier avalanche (com- 3.3. Avalanche source and deposit parameters prised of highly fractured metamorphic bedrock according to McSaveney, 1978) has a coefficient of Crandell and Fahnestock (1965) estimated the total volume of the seven avalanche deposits to 0.22, while the 1883 Elm landslide (comprised of 6 3 limestone and dolomite) has a coefficient of 0.31 be 10.710 m (Table 1). This value was (Heim, 1932). Typical values for volcanic avalanches determined by resurveying a series of cross-sections are 0.106 for the 1980 Mount St. Helens avalanche originally surveyed by Fahnestock (1963) for a and 0.166 for the 1964 Shiveluch avalanche (Hayashi study of the White River. From the cross-sections, and Self, 1992). Uncertainty exists in the value for they were also able to determine that the maximum thickness was approximately 30 m and that the Hmax for Little Tahoma because the original config- uration of the buttress that collapsed is unknown. majority of the deposit (the centre-of-mass) was The coefficient of friction can also be determined located between their cross-sections 10 and 12 (see on the basis of the debris velocity in the interior and at Fig. 10 in Crandell and Fahnestock, 1965). The the margins of the flow using the following relation- avalanches are likely to have entrained much snow ship (McSaveney, 1978): and other debris during flow and thus, this estimate of rock volume is likely to be a maximum. No 1 estimate of the volume of material missing from the l ¼ tana À V 2 À V 2 =ðÞsgcosa ð8Þ 2 2 1 source area was made, as the exact configuration of the failed buttress was not known. However, where V1 is the velocity of debris as determined in Crandell and Fahnestock (1965) estimated the Eq. (4), V2 is the velocity of debris margin (V=0), s is source to be approximately 540 m high by 550 m the distanced travelled, g is the acceleration due to wide. M.F. Sheridan et al. / Journal of Volcanology and Geothermal Research 139 (2005) 89–102 95

4. Flow models (2) Volumetric parameters (source volume, flow thickness, deposit thickness) are not included amongst As stated earlier, the purpose of this research is to either the input or output data. compare the results of simulating the 1963 Little (3) Conservation of mass and momentum are not Tahoma Peak avalanches in a new geophysical mass used in the model. flow model with published data and the results of (4) An accurate measurement of planimetric extent another flow model FLOW3D. of an avalanche can not be obtained from the output. (5) As with many flow codes, an arbitrary flow 4.1. FLOW3D termination mechanism is assumed. In this case, the model stops when the flow velocity reaches 0.1 m/s The FLOW3D code (Kover, 1995) is based on on a slope smaller than the critical value. Otherwise, Coulomb resistance to a sliding block, somewhat the computations could continue for unreasonable similar to the model of McEwen and Malin (1989). amount of time. The model calculates the changes in velocity as the Nevertheless, this model closely mimics the path, block slides across a 3D digital elevation model velocity, and extent of actual avalanches. Fig. 3 shows (DEM) constructed using a Triangulated Irregular two results from applying the FLOW3D model to the Network or TIN. The block trajectory is traced in 1963 Little Tahoma Peak avalanches. small increments of time until it stops. The velocity and position of the block at each time step is recorded 4.2. Titan2D model and can be plotted to show the trajectory and runout of a large number of blocks. The driving force of the Titan2D (Pitman et al., 2003; Patra et al., submitted modelled bflowsQ is given by a gravitational accel- for publication) is a code for incompressible Coulomb eration vector unique to each of the triangles in the flow based on the work of Savage and Hutter (1989), TIN. Resistance to flow in the model is calculated Iverson (1997), Iverson and Denlinger (2001) and using the formula of Mellor (1978) for snow Denlinger and Iverson (2001). In essence, it is a depth avalanches averaged dshallow-waterT granular-flow model with some similarities to that of Mageney-Castlenau et al. 2 s ¼ a0 þ a1v þ a2v ð9Þ (2002). The conservation equations for mass and momentum are solved with a Coulomb-type friction where s is the resistance to flow, v is the velocity, term at the basal interface (Pitman et al., 2003). and a0, a1, and a2 are parameters that represent the Because rock avalanches have insignificant heating, resistance due to basal friction, viscosity (or internal conservation of energy can be neglected to the first friction), and turbulence, respectively. In addition to order. The governing equations are solved using a the three parameters that proxy for basal friction, parallel, adaptive mesh, Godunov scheme (Patra et al., viscosity and turbulence, the x and y coordinates of submitted for publication). The Message Passing block starting locations are needed. Initial velocities Interface (MPI) allows for computing on multiple may be input for blocks resulting from explosions or processors, which increases computational power, column collapse. The path of the block can be decreases computing time, and allows the use of large displayed graphically as a line on the DEM that is data sets. Adaptive gridding allows for the concen- coloured coded to show velocity. From this graph- tration of computing power on regions of special ical display, it is possible to determine a rough interest. Mesh refinement captures the leading edge of estimation of the extent of the affected area if a the avalanche, as well as locations where the top- sufficiently large number of blocks are included in ography changes rapidly. Mesh unrefinement is the simulation. applied where solution values are relatively constant There are several limitations to this model: or small. (1) Multiple sliding blocks do not interact with The model assumes that the debris avalanche or each other; each block moves as if it were the only debris flow starts as an ellipsoidal pile of material block traversing the slopes. with user-specified dimensions of height and width 96 M.F. Sheridan et al. / Journal of Volcanology and Geothermal Research 139 (2005) 89–102

Fig. 3. Examples of output generated using FLOW3D. (A) Basal friction of 0.246 and a1 of 0.01. (B) Basal friction of 0.15 and a1 of 0.01. Shaded area in panel A corresponds to the energy cone (energy line swept through 3608). Results in panel B compare favorably in terms of extent, velocity history, and flow behavior. The simulated avalanche is bent back by the terminal moraine. M.F. Sheridan et al. / Journal of Volcanology and Geothermal Research 139 (2005) 89–102 97

(designating two radii in the x and y planes), as The model stops when either of the two limits is well as the starting location coordinates. The two reached. This stopping mechanism differs from that other input parameters are the internal friction angle of the Mageney-Castlenau et al. (2002) model that and the basal or bed friction angle. Several uses a kinetic stopping scheme. mechanisms are incorporated for stopping the The Titan2D model has several useful features model. The basic mechanism is that the model including the effects of erosion, variable basal stops when the flows cannot overcome the resist- friction angle keyed to different bed surface ance forces acting on them. However, this generally materials, and a visualisation platform for displaying requires an inordinate amount of computational time the flows. The effects of internal pore pressure and with no significant movement within the pile of particle interactions are not included in the current material. An alternative method of stopping the model. model require the user to input a maximum number Results from Titan2D are displayed as an anima- of time steps and a maximum run time in seconds. tion using the user specified number of time steps.

Fig. 4. Results from TITAN2D showing the thickness of the avalanche and the resulting deposit. (A) Time step 1. (B) Time step 2001. (C) Time step 4001. (D) Time step 10001. Heavy black outline is the mapped extent. Light black lines are 100-m contours. See text for discussion of locations A, B and C. 98 M.F. Sheridan et al. / Journal of Volcanology and Geothermal Research 139 (2005) 89–102

Fig. 5. Results from TITAN2D showing the velocity distribution within the avalanche during emplacement. (A) Time step 1. (B) Time step 2001. (C) Time step 4001. (D) Time step 10001. Note changing scale in legend. Heavy black outline is the mapped extent of area over which avalanches passed. Light black lines are 100-m contours. See text for discussion of locations A, B and C.

The results can be viewed in the 2D (as shown in information in that summary concerning both the Figs. 4 and 5) or in 3D over the realistic topography dynamics (e.g., velocity, runup heights) and features represented by the grid DEM. The animations show (deposit thickness, flow behavior), a series of criteria the progression of flows through a series of time were developed as a basis for evaluation of the model steps. The distribution and thickness of the flowing and are as follows: material can be clearly seen. Likewise, the history of velocity and momentum with time and space could (1) The runup heights should fall within the range be displayed. measured in the field. (2) Deposit thickness should fall within the range in 4.2.1. Titan2D evaluation criteria the published cross-sections. Section 2 contains a summary of published data on (3) Planimetric distribution should be comparable to the 1963 Little Tahoma Peak avalanches. Based on the that seen in the field. M.F. Sheridan et al. / Journal of Volcanology and Geothermal Research 139 (2005) 89–102 99

(4) Titan2D should be able to replicate the diversion runout distance of the model flows is comparable to of the avalanche through the gap between the that of the actual avalanches; the longest flow terminal moraine and the lower slopes of Goat travelling 6960 m. Maximum velocities range from Island Mountain. 77 to 81 m/s, and the travel time varies from 133 to (5) Velocities determined from TitanN2D should 173 s. The simulated flows also ran up the lower match those calculated by FLOW3D and those slopes of Goat Island Mountain, as did the actual calculated from field observations. avalanches. While it is not possible to determine accurate velocities and runup heights in the two locations [the northwest slopes (Location 1 in Fig. 3B) 5. Discussion of results and the north facing slope opposite the terminal moraine (Location 2)], it appears that the velocities Figs. 3–5 show results obtained from FLOW3D and are comparable to those determined by Crandell and Titan2D simulations of the 1963 Little Tahoma Peak Fahnestock (1965) (see Table 1) as the purple shading avalanches. Table 1 also compares some results from of the flow vector corresponds to a velocity of the two models. All simulations were run on a DEM/ approximately 20–40 m/s. In the best simulation TIN constructed from the post-avalanche topography. calibrated on the runout distance of 6 km, the value Because most of the deposit lies in a depression near of basal friction is 0.15 (8.538), which is much lower the terminus of its extent and the thickness is generally than might be expected for this size of avalanche. less than the resolution of the Dem, the effects of changed topography are only minor. 5.2. Titan2D model results

5.1. FLOW3D results Figs. 4 and 5 show the results from one simulation of the Little Tahoma Peak Avalanches using Titan2D. Fig. 3 shows two screenshots of runs done using Table 1 lists the parameters used for the simulation two different values for the basal friction. The show in Figs. 4 and 5. A volume of 1.0106 m3 was viscosity parameter (a1=0.01) was similar in both used in the simulation because this is approximately runs and no value for turbulence was used because the average value for the seven avalanches. The other these avalanches are not considered to be turbulent parameters shown in Table 2 were the result of flows. At first FLOW3D was run using basal calibrating the model to fit the runout length of the friction coefficient of 0.246 derived using the H/L original avalanches as mapped by Crandell and method as described earlier in Eq. (7); this is Fahnestock (1965). A sensitivity analysis of the model equivalent to an angle of approximately 148.Asis was carried out to see how the values of the boundary clearly shown in Fig. 3A, the flows stop short of the conditions and input parameters affected the model. terminal moraine that marks the furthest extent of the These are discussion in Section 5.3. avalanches as mapped by Crandell and Fahnestock Fig. 4A–D shows the distribution of the simulated (1965), the longest flow travelling approximately 5 avalanche pile as it descends Emmons Glacier and km. Consequently, a variety of values for both bed friction and viscosity were tried in order to get a good representation of the avalanches. Fig. 3B shows the Table 2 Parameters used in the simulation of the Little Tahoma Peak results using values of 0.15 for bed friction and 0.01 Avalanche with TITAN2D shown in Figs. 3 and 4 for viscosity. These values produce flows that are Parameter Value comparable to the actual events in terms of behavior and aerial distribution. Boundary conditions Pile dimensions (x,y,z) (m) 959075 One striking feature of the actual avalanches is the Volume (106 m3) 1.0 way Unit 3 was funnelled through the gap between the crest of the terminal moraine and the lower slopes of Input parameters Bed friction (M)12 Goat Island Mountain. This deflection is clearly seen M in the simulation of Fig. 3B. For this simulation the Internal friction ( )33 100 M.F. Sheridan et al. / Journal of Volcanology and Geothermal Research 139 (2005) 89–102 into the area of deposition in the White River Valley. lines are overlaid on the velocity distributions (fine After descending the steep north face of Little Tahoma black lines shown in Fig. 5), runup height is appro- Peak, the simulated avalanche proceeds to flow across ximately 60 m. This is comparable to the 52 m of the glacier and across the topographic slope, before runup measured by Crandell and Fahnestock (1965). heading down the glacier. As the flow moves down The basal friction angles used in the Titan2D the glacier, it ramps up the lateral moraine on the simulations (128), like that used in FLOW3D (8.58), north side of Emmons Glacier (location A, time step are lower than H/L values expected for actual 2001, Fig. 4B). Having reached the floor of the avalanches of this volume, typically 25–338. How- unglaciated White River Valley in the first 3000 time ever, Emmons Glacier, in the upper reaches of the steps, the simulated flow slows down considerably avalanche course, represents a surface that provides and travels the remaining 2 km to the terminal very little resistance to flowing debris, perhaps moraine over the course of the next 7000 time steps, accounting for the lower values. However, by using spreading out across the valley floor and forming a the basal friction coefficient of 0.037 that was deposit up to 3 m thick, with the thickest section of calculated for the avalanches as they travelled over the deposit forming at the base of the north-facing the glacier, the simulated flows would travel too far. slope of Goat Island Mountain (location B, time step The use of a geographically distributed matrix of basal 4001, Fig. 4C). A small amount of the debris friction values would help to resolve this problem and continues to move, however, and flows through the provide more realistic velocity and runup values for gap between the crest of the terminal moraine and the the simulated flows. lower north facing slopes of Goat Island Mountain (location C, time step 10001, Fig. 4D), as did 5.3. Sensitivity analysis avalanche 3. However, the simulated avalanche does not run up the slopes to the same height as the actual It is important to test the sensitivity of a model avalanche. The small part of the simulated avalanche to changes in input parameters and boundary that flows through the gap along the White River conditions. For Titan2D, we examine the role of channel forms a deposit up to 2 m thick. friction angles, pile dimensions, pile volume and As described above, Titan2D simulations will stop starting location. An informal method of determin- in one of two ways, either when the resisting forces ing the sensitivity to changes in a parameter uses cannot be overcome, or when the number of time the following relationship: steps or maximum run time is reached. The sequence Sensitivityu P1 À P2 = P1 4100% of time steps in Fig. 5 shows the velocity distribution i i i within the simulated avalanche. As is clearly seen in 1 2 1 4 Y Oi À Oi = Oi 100% ð10Þ Fig. 5D (time step 10001), most of the avalanche has 1 2 very low or zero velocity. The color shading indicates where Pi and Pi are two slightly different values 1 2 that only the edges of the deposit are moving. This for the same input parameter and Oi and Oi are two clearly shows that resisting forces are causing the slightly different values for the same model (output) avalanche to stop flowing, a result of a gentle parameter. topographic slope. Running the simulation with more Increasing the bed friction value by 25% (from 128 time steps confirms this. to 158) produced a 54% decrease in runout length The maximum velocity reached by the simulated (from 6.8 to 3.1 km) in this simulation. Increasing bed avalanche is 72 m/s at approximately time step 1501 friction further by 66% (from 128 to 208), produced an (not shown in Figs. 4 and 5), and is comparable to the 80% decrease in runout length (from 6.8 to 1.3 km) maximum velocities (82 m/s) calculated by FLOW3D. Reducing bed friction by 16% (from 128 to 108) Velocities recorded during runup at location A (time produced a greater than 33% increase in runout step 2001, Fig. 5B) decrease by about 20%, to length. Reducing internal friction by 24% (from 338 approximately 55 m/s. The height of runup at location to 258) resulted in a 14% decrease in runout length A, as calculated by rearranging Eq. (5) and using a (from 6.8 to 5.8 km). Moving the initial position of velocity of 60 m/s is 183 m. However, when contour the pile 500 m upslope affected the runout by 10%. M.F. Sheridan et al. / Journal of Volcanology and Geothermal Research 139 (2005) 89–102 101

6. Conclusions References

The purpose of this study was to evaluate the Chow, V.T., 1959. Open-Channel Hydraulics. McGraw-Hill, New Titan2D model developed at University at Buffalo and York, 680 pp. compare it with results of another flow model, Crandell, D.R., Fahnestock, R.K., 1965. Rockfalls and avalanches from little Tahoma Peak on Mount Rainier, Washington. U.S. FLOW3D. The results from FLOW3D are comparable Geol. Surv. Bull. 1221-A, A1–A30. to those of the actual avalanches in terms of the runout Denlinger, R.P., Iverson, R.M., 2001. Flow of variably fluidized length, velocity history, and flow behaviour (runup on granular material across three-dimensional terrain: 2. Numer- Goat Island Mountain and funneling through the gap ical predictions and experimental tests. J. Geophys. Res. 106, in the terminal moraine). The aerial distribution of the 553–566. Evans, S.G., Hungr, O., Clague, J.J., 2001. 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