Excitation mechanism of atmospheric pressure waves from the 1980

Geophys. J. R. astr. SOC. (1985) 81,445-461

Excitation mechanism of atmospheric pressure waves from the 1980 Mount St Helens eruption

Takeshi Mikumo Disaster Prevention Research Institute, Kyoto University, Uji, Kyoto 61 I , Japan

Bruce A. Bolt Seismographic Station. University of California, Berkeley,

California 94720, USA

Accepted 1984 November 5 . Received 1984 October 25; in original form 1984 July 25

Summary. Atmospheric pressure waves from the Mount S t Helens eruption 1980 May 18 have been clearly recorded by a sensitive microbarograph at Berkeley, California. The record shows three types of waves with different group velocities. The pressure waves can be interpreted in terms of direct waves A1 , antipodean travelling waves A2 and circumnavigating waves A3, all of which are composed of several acoustic-gravity modes propagated in the lower atmosphere. Synthetic barograms appropriate to the Berkeley station have been calculated on the basis of the dynamic response of the lower atmospheric structure, together with various assumptions of source properties. Comparisons between synthetic and observed barograms provide estimates for ranges of the time history of upward particle velocity at the source, source dimensions and the velocity of the source spreading over the blast zone, as well as for the average dissipation effects over the circum- ferential path. The results suggest that two major compression pulses on the A1 record correlate with the arrival of pressure waves from the first (lateral) blast and second (vertical) blast, although the inferred interblast time interval is not consistent with that estimated from seismic observations.

1 Introduction

The violent volcanic eruption of Mount St Helens on 1980 May 18 generated strong atmospheric pressure disturbances that propagated on a global scale around the Earth’s surface, and also excited long-period seismic waves that travelled to teleseismic distances within the Earth. The atmospheric waves from the eruption were recorded by barographs at local weather stations in the Washington-Oregon-Idaho region (Reed 1980), and also by sensitive microbarographs and in some cases by seismographs at world-wide stations (e.g. Ritsema 1980; Bolt & Tanimoto 1981; Donn & Balachandran 1981; Liu et al. 1982; Eissler, Kanamori & Harkrider 1983; Kanamori, Given & Lay 1984). The maximum

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446

amplitude of the recorded waves ranged from 3.7 mbar at Toledo, Washington (Reed 1980), 0.35 mbar at Berkeley, California (Bolt & Tanimoto 1981) down to less than 0.1 mbar at distant stations (Liu et al. 1982), decreasing nearly inversely proportional to the square root of the travel distance (Reed 1980; Donn & Balachandran 1981; Eissler ef al. 1983). The observed periods ranged from several tens of seconds up to about 18 min, with predominant periods of 5-8 min in typical microbarograph recordings (Donn & Balachandran 1981 ; Liu et al. 1982).

The characteristic features of the atmospheric waves recorded in the present case may be compared with those from natural sources such as of the 1883 Krakatoa eruption (Pekeris 1939), the fall of the 1908 Siberian meteorite (Whipple 1930), and of the 1964 great Alaskan earthquake ( e g Bolt 1964; Mikumo 1968), and also from artificial explosive sources in the atmosphere (e.g. Donn & Shaw 1967). Theoretical analyses leave little doubt that these waves, propagating to long distances, are composed of dispersive acoustic-gravity wave modes. The propagation of atmospheric pressure waves from the present eruption, particularly their velocity, has been interpreted in terms of acoustic-gravity waves in the lower atmosphere. However, the mechanism of eruption that produced the atmospheric waves has not yet been constructed from the analysis of their waveforms, although the recorded amplitudes provided estimates for the explosive energy of the eruption (Ritsema 1980; Reed 1980; Bolt & Tanimoto 1981; Donn & Balachandran 1981).

By contrast, seismic waves from a moderate-size (mb = 4.9) earthquake just prior to the main 1980 eruption and from the massive associated landslides, which were clearly recorded at seismograph stations, have been used to infer the type and magnitude of the reactive force exerted on the ground and also the time function and radiated energy at the seismic source (Kanamori & Given 1982, 1983; Niazi & Johnson 1984). Without specific evidence to the contrary, however, it is prudent to assume that the source characteristics inferred from the seismic waves were somewhat different from those determined from the atmospheric waves, because the volcanic energy may have been released in different ways.

It is the purpose of the present paper to estimate the general source properties of the volcanic eruption from the atmospheric pressure waves. We analyse in detail the pressure waves recorded at Berkeley, California, which is located at 927 km south of the volcano, because they have some particularly detailed features (Bolt & Tanimoto 1981). Our approach is t o calculate synthetic barograms based on the dynamic response of the atmosphere, incorporating assumed source-time histories, source dimensions and shock wave speeds in the source area, and then to compare the calculated waveforms with the observed records. Because the Berkeley microbarograph distinctly recorded the direct waves Al, the antipodean travelling waves A2 and the circumpropagating waves A3, the present analysis also provides estimates on the average attenuation of the pressure waves propagated in the lower atmosphere.

T. Mikumo and B. A. Bolt

2 The characteristics of the Berkeley barogram A preliminary description and analysis of the records from the microbarograph in the Byerly seismographic vault at the University of California, Berkeley, were given by Bolt & Tanimoto (1981). An electric flow sensor, in an air-line between the atmosphere and a thermally insulated reservoir, detects variations in the resistance of two thermisters placed symmetrically about a heated thermister (Bolt 1964). The original records are reproduced in Fig. 1. They show the occurrence of three separate pressure trains or phases, denoted by Al, A2 and A3. A brief description of these pressure waves is now given with several numerical corrections to typographical errors published in the 1981 preliminary report. (Note that the paper chart speed is 5 mm min-' - not 6 mm min-' as given in 1981.)


Atmospheric waves from Mt St Helens 447 16 00 -5/18/1980 I7 00

A l

02:OO - 5/20/1980 03:OO

A 2 A 3

Figure 1. Berkeley microbarogram showing atmospheric pressure waves recorded after the 1980 Mount St. Helens eruption. The top record shows the direct wave A l , and the bottom record shows the antipodean wave A2 and the circumpropagating wave A3. The sensitivity is 0.11 mmpb-' on the record. The recorded amplitude of the first compression peak of the A1 wave is 0.35 mb.

The A1 train is complex, with a first onset pulse of period about 5 min, group velocity for a direct path of 308ms-I and amplitude 0.35mb (corrected value). The more monochromatic A2 train arrived at Berkeley about 33 hr later with period 6 min, for an antipodean path, a group velocity of 314ms-', and amplitude 0.lOmb. The A3 train at about 03 hr on Fig. 1 has a maximum amplitude of 0.13mb and period 6.5min (corrected values). This train is probably the dispersed A1 train after it has propagated completely around the great circle; the group velocity of the maximum pulse is about 320 m s-'.

3 The adopted atmospheric model

In order to calculate the synthetic barograms to be compared with the observations, a detailed model of a realistic atmospheric structure is required. The vertical temperature structure of the atmosphere and hence the speed of sound vary rapidly as a function of height. The lower atmospheric structure from the troposphere up to a part of the F-region has often been represented by the ARDC standard model (e.g. Press & Harkrider 1962), which is characterized by two temperature minima at 18 and 85 km and rapidly increasing temperatures above 110 km. The structure of the upper atmosphere in the F-region such as represented in the CIRA model (e.g. Yeh & Liu 1974) does not significantly affect acoustic- gravity waves with periods shorter than 15 min, as observed in the present case, and will not be discussed here.

Theoretical studies of pressure waves propagated in the lower atmosphere with realistic thermal structures have been developed by Press & Harkrider (19621, Pfeffer & Zariclmy (1 963), Harkrider (1 964), Midgley & Liemohn (1 966), Harkrider & Press (1967), Francis (1 973) and others. These studies have shown that various dispersive acoustic-gravity modes can propagate in the lower atmosphere with appropriate velocities. The phase and group velocities and the dynamic response of the atmosphere to a surface source and receiver have been computed using a matrix formulation by Harkrider (1964) and Harkrider & Press (1967) for a standard non-dissipative, flat earth model. The approximation had 39

15*


448 horizontally-stratified isothermal layers terminated at an altitude of 220 km with an isothermal half-space or a free surface. In the present paper, we use the atmospheric response obtained by Harkrider (1964) to calculate the synthetic barograms, as in the case of pressure waves from the 1964 Alaskan earthquake (Mikumo 1968).

The above calculations do not incorporate the effects of dissipation. The presence of dissipation, due to viscosity and thermal conduction in the atmosphere, which are the results of molecular processes, and also ion drag which arises from Lorentz forces, attenuates the acoustic-gravity waves propagated in the atmosphere. The effects of viscous and thermal dissipation and of ion drag have been discussed in some detail (Pitteway & Hines 1963; Midgley & Liemohn 1966; Volland 1969; Klostermeyer 1972; Francis 1973; Lighthill 1978). In particular, Francis (1973) has given attenuation distances as a function of period for various acoustic-gravity modes, on the basis of a model atmosphere with realistic thermal and dissipative structures. (The attenuation distance is defined as the horizontal distance each of the modes can propagate before its amplitude decays by a factor of l/e.) Theoretically, these results should be taken into account to calculate the synthetic barograms. However, the results of Francis (1 973) indicate that the attenuation distances for the acoustic modes with periods ranging from 1.5 to 3.5 min are of the order of a few to 10 times the circumference of the Earth, and that the distance for the lower atmospheric, fundamental gravity mode with periods longer than 4.5 min ranges between 10 and 100 times the circumference. Since the attenuation of these lower atmospheric modes is thus quite small, we do not directly incorporate the theoretical results into the present analysis. Instead, we estimate from our observations the average dissipation factor Q for the combined effects of various modes, which is analogous to the Q estimates for seismic waves propagated within the Earth.


4 The source characteristics

The source of the pressure waves associated with the present eruption is a strong disturbance applied to some localized area of the bottom surface of the atmosphere, due to sudden injection of energy, momentum and mass into the atmosphere. The properties of the Mount St Helens eruption were quite complex, as have been described in the literature (e.g. Crosson et al. 1980; Christiansen 1980; Christiansen & Peterson 1981; Kieffer 1981a, b ; Voight et al. 1981; Eicherberger & Hayes 1982; Kanamori et al. 1984). The best evidence indicates that the sequence of events started at 15 : 32 : 11.4 GMT with an earthquake of a magnitude mb = 4.9 at a depth of about 3 km beneath the volcano; this earthquake triggered, within 7-20 s, a massive landslide on the north slope of Mount St Helens. More than half a minute after the initiation of the landslide, a violent lateral blast took place directed northward and upward, probably due to the unloading of overburden rock mass by the landslide and the exposure of a magma or hydrothermal reservoir. A few to several minutes later, a second, vertical Plinian eruption occurred a few kilometres north of the caldera, before the first one was fully developed; this eruption was accompanied by another earthquake.

The extent of the blast zone in the first lateral eruption has been assessed from the devastated area (the area in which all trees were destroyed), which covers a nearly semi- circular northward sector of the volcano. This zone is 30 km across from west to east and extends outward more than 20 km from the summit (Christiansen & Peterson 1981). A more detailed estimate of the blast zone has been given by Kieffer (1981a, b) using a quasi-steady flow model. In this estimate, the limit of the blast zone is about 20-25 krn from the volcano, and there are an inner direct blast zone (where virtually no trees remained) and an outer channelized zone. Strong pressure disturbances were recorded at local barograph


Atmospheric waves from Mt St Helens 449 stations, which indicate that the strongest propagation occurred in the north-westward and northward directions and the minimum propagation in the southward azimuths (Reed 1980). Kieffer's (1981b) estimates show that excess pressures due to the landslide may be too small to be detected even at these nearby stations. This indicates that the atmospheric pressure waves recorded at the local and world-wide stations (or, at least, their first portions) may be attributed to the first, lateral blast. The second, vertical eruption formed a central crater along the northern flank, yielding a great amphitheatre of 1.5 x 3 km (Christiansen 1980; Christiansen & Peterson 1981). This dimension may almost define the horizontal extent of the second eruption.

The source-time history of the first eruption has been calculated from photographs on the behaviour of expanding clouds that emerged after the explosion (Voight 198 l), regarding the cloud edge as the boundary between eruption debris and clear air. The velocity and displacement time-histories derived for the eruption, which are based on the analysis by Voight (1981), show that the eruption development took longer than 20 s and the peak velocity in an early stage reached about 130-140 m s-'. The estimated expansion velocity should be regarded as the maximum value for expanding clouds in a limited area, but not exactly that of the source of pressure waves, because the acoustic pulse may have been generated by shock disturbances applied over a wider area.

The source-time function of the eruption (or, more exactly, of its reactive force) has been estimated also from seismic body and surface waves radiated from moderate-size earthquakes that occurred during the eruption. The observed P- and S-waves with periods of 20-30 s indicate a nearly vertical single force consisting of two groups of several events each with the source duration of about 25 s (Kanamori et al. 1984), while a longer characteristic time of about 150 s has been given for a nearly horizontal single force which may correspond to the landslide (Kanamori & Given 1982). A more complex source time function composed of a long asymmetrical waveform and two superimposed shorter pulses has been suggested for the landslide and two major blasts, respectively (Niazi & Johnson 1984).

We use all the above information on the source characteristics of the eruption as plausible bounds for the excitation function and source dimension of the atmospheric pressure waves.

5 Calculation of synthetic barograms The problem of the pressure perturbation in an isothermal atmosphere excited by a volcanic eruption was first theoretically treated by Pekeris (1939, 1948) as a point source located on the ground. The solution has been obtained on the condition that the upward particle velocity of the air at the ground is exactly the same as the outflow velocity of volcanic gases during the eruption. We modify his 1948 solution to a more general case with a realistic atmospheric structure composed of horizontally isothermal layers, including the effects of the source-time history and source dimension described in the foregoing section.

By reference to the solution obtained by Harkrider (1964) and Ben-Menahem & Singh (1981) for explosive sources and to that by Harkrider & Press (1967) and Mikumo (1968) for a volcanic or earthquake source, the frequency domain expression of the pressure pertui- bation on the ground (except at the source area) may be written by

2 iso CIJ Po

Sw,=-- . I o ( K r ) K exp(iwt)


450

where 6w, is the change in the vertical particle velocity integrand at the source, N:), N(qZ) and FA are functions of layer matrices for the atmosphere, and So implies the source coeffi- cient in the frequency-wavenumber domain (Harkrider 1964). In the present case, So may be given by


P A t ) = P o c o i o ( t )

where p , ( t ) is the excess pressure generated at the source close to the ground, and a means the effective height of the bottom surface of the atmosphere perturbed by volcanic blasts. The far-field solution for each mode (kj root of the period equation FA = 0) may be derived from the residue contribution of equation ( I ) as,

When both source and receiver are located at the ground, N $ ) = - A21, N y ) = 1 and FA = Az2, where A21 and Azz are the layer matrices. If we use an asymptotic expansion for

Earth’s surface (r/ae sin 0)112 is included, equation ( 3 ) can then be rewritten as, large arguments of H , (2) (,ti‘), and also if an approximate curvature correction for the

When the source is distributed over a finite dimension, (4) should be integrated in space and will then include the source finiteness factor. The pressure variation that would be recorded on a microbarograph at a station may therefore be expressed in the following form (Mikumo 1968),

1 r m

F ( w ) exp [-i4(w)] given in this form includes the effects of the source pressure-time function, source finiteness, dynamic atmospheric response and the microbarograph response.

For the upward source velocity, we assume the following four different time histories,

i o ( t ) = a/r (0 < t < 7)

= (a/7) exp ( - t / ~ )

= (a/T)( 1 - t / ~ ) exp [( 1 - t /7)]

= ( n a / 2 7 ) sin (nt/.r) (0 < t < 27).


Atmospheric wuves from Mt St Helens 45 1

Their Focrier transforms are

i o (w) = (u/T) sin ( w 7 / 2 ) / ( 0 ~ / 2 ) , &(w) = exp ( - iw7/2)

= (U/T)(I i- w272)-1’2, =exp [- iw tan-’(aT)]

= u w T e ( 1 -t U ’ T ’ ) - ~ , = exp I[-iotan-’[(l - w272)/2w7]j (7c)

= n2u sin (wT) / (w ’T~ - n’), ( 7 4 = exp (- ~ O T )

corresponding to equations (6a) to (6d). The source finiteness factors D ( w ) exp [-i@~(o)] for circular and rectangular sources

have been given in equations (1 1) and (12) in Mikumo (1968). In the present case, however, it has been suggested (Kieffer 1981a, b) that pressure disturbances directly from the lateral blast (initiated from 1 km north of the summit of the volcano) rapidly expanded to the north as a Mach disc shock. This model requires that the bottom of the overlying atmosphere be successively disturbed with a finite speed by the vertical component of shock waves. If the source propagation velocity is included, the source finiteness factor may be rewritten as,

and

TL = L [ l / u - cos /3/Cj(o)], Tw = W sin P/Cj(w)

where u is the average speed of the unilaterally propagating pressure disturbances, and 2 L and 2 W are the length and width of the source area.

The atmospheric transfer function A (a) exp [-i@a ( a ) ] is redefined here by,

A ( w ) = k:./’AAj(w) M ( o ) , AAj(u) = A21(aA22/ak)-1, M ( u ) = exp [- a ( u ) r ] (9) GA(w) = wro/Cj (w) - n/4 - mnI2.

Aai (w) , the dynamic response of the realistic atmosphere to a surface source and receiver, and q(w) , the phase velocity for each mode, have been computed by Harkrider (1964) and Harkrider & Press (1967), and we use their results. Because the above expression is for a flat earth model, it is valid only for the A1 waves after an approximate curvature correction is applied. For the A2 waves which travelled by a longer great circle path through the antipode and for the A3 waves which travelled along all the circumference, a direct application of (4) and (9) is not justified. We tentatively assume, however, that these expressions may also be applied to these waves, as a first approximation, if the distance r is taken as their appropriate travel distance through the antipode and a positive polar phase shift of mn/2 is added (m = 0 fo A l , m = 1 for A2 and m = 2 for A3). Beside these, we incorporate the dissipation effects M ( o ) into the above expression. The attenuation of simple harmonic waves may be represented by a damped sinusoidal form, exp [ -a(w)r] exp [ i o ( t - r/c)]. A form for .(a) for sound waves in clear air has been given by Morse & Ingard (1968), which is proportional to the square of the wave frequency. In the present case, however, we assume the following form analogous to the case of seismic waves; a ( o ) = 0 /2C j (w) Q. Q is assumed as the average dissipation along the travel distance, and is also assumed here to be independent of frequency and the same for all modes. We note, however, that this expression may not be entirely suitable for acoustic-gravity waves, in view of the results obtained by Francis (1973). Thus, the model should be regarded only as indicating the order of overall attenuative properties.


452 T. Mikumo and B. A. Bolt

A - I W A V E G R O

The frequency response B(w) exp [-I’@B(w)] of the Berkeley microbarograph may be taken as the same as given in Mikumo (1968), but its maximum sensitivity around the time of the 1980 Mount St Helens eruption had been changed to 0.11 mm pb-’ (Bolt & Tanirnoto 1981). With these expressions, synthetic barograms can now be calculated by assigning various values to the parameters in equations (5) to (9).

A - 2 W A V E G R O

s o Ih

6 Comparison between synthetic and observed barograms

(1) Green’s functions. As a first step, we compute theoretical Green’s functions for the impulse response of the non-dissipative atmosphere and the recording system for the Berkeley station. These correspond to synthetic barograms for an impulsive point source model ( L = W = 0 in 8; T = 0 in 6a). Fig. 2(a, b) shows the Green’s functions thus calculated for four acoustic-gravity wave modes, the fundamental gravity mode GRO, the fundamental, first and second acoustic modes SO, S1 and S2 for A1 waves ( r = 9 2 7 km, @ = 180P4) and A2 waves (r = 39 097 km, @ = 0’?4), respectively. The amplitude scale here is taken arbitrarily, but numerals in the abscissa indicate theoretical travel times of these modes for the atmospheric model adopted here. We see that the GRO mode is a predominant component in both of the A1 and A2 waves but contributions from the acoustic modes SO and S1 may also be significant.

The composite Green’s functions from the four modes are shown in the middle trace in Fig. 3(a, b), respectively, in comparison with the observed microbarograms. Fig. 3(c) gives the record for A3 waves (r = 40 95 1 km, @ = 180P4) and the corresponding Green’s function. Also included in Fig. 3 (the lowest traces) are the synthetics for a point source having a different source-time function with positive and negative impulses (T -+ 0 in 6d). The absolute times for the synthetics have been slightly adjusted to match the first peak to that of the corresponding records. The comparison in Fig. 3 shows that the Green’s function and similar synthetics with short source-time durations cannot give a satisfactory explanation to general features of the observed A1 record. A most remarkable discrepancy is that the first large trough and second peak with half-period of about 5 min recorded on the microbarogram do not appear on the calculated traces. Also, the shorter-period waves (2-3 min)

s 2

I 7 IOrn in .7

S I -

s 2

1 ,-- 10rnin--,

Figure 2. Green’s functions for the impulse response of a non-dissipative lower atmosphere and the Berkeley microbargograph. GRO indicates the fundamental gravity mode, and SO, S1 and S2 indicates the fundamental, first and second acoustic modes. The vertical scale is taken arbitrarily. (a) Direct wave A l , (b) antipodean wave A2.


Atmospheric waves from Mt St Helens 453

cm 3-

A - l W A V E n

0-I-

-

o-l+Pv C A S E 2

I A - 2 W A V E

(b)

I - O b s . 0 -

I f I O m i n . 7

( c )

I A - 3 W A V E

L I i- IOrnin.- j

involved in the synthetics are not seen on the records. On the other hand, the agreement between the observed and calculated traces for the A2 and A3 waves, particularly up to the time of the second peak, is quite satisfactory. These features of the A2 and A3 waves appear to be governed mainly by the atmospheric response, although some information of the source characteristics should still be involved there.

(2) Effects of the source time duration and source dimension. The next step is to estimate the effects of source properties on the direct wave A l , because we would expect that variations in the time constant of the upward source, and in the source dimension, will have considerable effects on the synthetic waveforms. Fig. 4(a) gives synthetic barograms for a point source with three different time constants (7 in 6d is varied from 0 to 1 min). It may be seen, as might be expected, that as the time constant becomes longer the amplitude of the short-period components diminishes. Thus, the waveform of the first large compression on the records may be more satisfactorily explained by longer time constants. In Fig. 4(b) are also shown the calculated barograms for three different source dimensions with a faed time constant (7 = 0 in 6d). The uppermost, middle and lowest traces correspond to a point source, to the area of the direct blast zone and to the channelized zone (Kieffer 198 la, b), respectively. In this calculation, we assume that



C A S E 2

A A - I W A V E ( T I

0-

0 -

C A S E 3 1 J--l 0 -

I\ A - I W A V E ( S ) I C A S E 2

C A S E 7

- IOrnin.-; I

Figure 4. (a) Synthetic barograms for a point source with three different source time constants, (1) T = 0 , (2) T = 0.3 min, (3) T = 1.0 min in equation (6d). (b) Synthetic barograms for three different source dimensions with a time constant T = 0, (1) 2L = 2 W = 0, (2) 2 L = 10 km, 2 W = 9 km, (3) 2 L = 20 km, 2 W = 1 5 krn.

the bottom of the atmosphere over the source area was subjected to simultaneous disturbance with no time delays (LJ = 00 in 8). From these results, we conclude that the barogram waveform is not very sensitive to the source dimension if shock pressure disturbances were transmitted at very high speeds as compared with the dimension. It should be noticed, however, that none of the calculated traces in Fig. 4 accounts for the second large peak on the observed record.

( 3 ) Possibility of doubZe sources. Next, we take into account all possible source parameters in order to interpret the observed waveform of A l . A number of trial calculations of synthetic barograms with various assumptions for the source parameters suggest that Q = 30 m, T = 0.2 min (in 6b) and IJ = 0.66 km S ', 2 L = 20 km and 2 W = 15 km provides a favourable combination to match the calculated amplitude and waveform of the first compression waves with those on the record, as illustrated as source 1 in Fig. 5. There are also alternative plausible combinations of these parameters, which are listed in Table 1.

Figure 5. Observed barogram (uppermost trace) and the synthetic barograms from two sources. Source 1 for the first, lateral blast, souIce 2 for the second, vertical eruption, assuming that the second eruption took place 6.5 min after the first one. Source 1 + 2 indicates the composite trace with the assumed time lag.


Atmospheric waves from M t St Helens 455 Table 1. Assumed source parameters for the two blasts.

1st Blast 2nd Blast

Case ZL 2W a 2 Vr 2L 2W a Z Vr Q rig. No. (km)(km) (.)(inin) (krnls) (krn)(km) (m) ( m i n ) (km/s)

10000 2 , 3 1 0 0 30 0.0 a - 2 0 0 30 0.0 d e-

3 0 0 30 0.3 d m

4 0 0 30 0.5 d

5 0 0 30 1.0 d - 6 10 8 30 0.0 d m

7 20 15 30 0.0 d -

10000 3,4

10000 4

10000

10000 4

10000 4

10000 4

8 20 15 30 0.1 a - 10000

9 20 15 30 0 . 2 b 0.66 4 4 90 0 . 2 b 0.66 1500 5

10 20 15 30 0.2 b 0.66 4 4 90 0.2 b 0.66 6000 6

11 20 15 30 0.2 b 0.66 4 4 90 0.2 b 0.66 3000 6

1 2 20 15 30 0.2 b 0.66 4 4 90 0.2 b 0.66 2000 6

13 20 15 30 0.2 b 0.66 4 6 90 0.2 b 0.66 1000 6

14 20 15 30 0.2 b 0.33 4 4 90 0.2 b 0.66 1500

15 20 15 30 0.2 b 0.99 4 4 90 0.2 b 0.66 1500

16 20 15 30 0.11 b 0.99 4 4 90 0.1 b 0.33 1500 7

17 20 15 30 0.17 b 0.66 4 4 90 0.1 b 0.33 1500 7

18 20 15 30 0.34 b 0.33 4 4 90 0.1 b 0.33 1500 7

19 20 15 30 0.67 b 0.17 4 4 90 0.1 b 0.33 1500 7

20 20 15 30 0.17 b 0.66 4 4 90 0.1 b 0.66 1500 8

21 20 15 30 0 . 3 d 0.66 4 4 90 0.1 d 0.66 1500

22 20 15 30 0.6 d 0.66 4 4 90 0.1 d 0.66 1500 8

23 20 15 30 2 .6 c 0.66 4 4 90 2.6 c 0.66 1500

It is obvious, however, that the second large positive peak on the record still cannot be closely simulated by the computed trace.

As mentioned before, the velocity-time function for expanding clouds from the eruption has been estimated from photographs (Voight 1981 ; Banister 1984). Assuming that this function might be an approximate expression of the source-time history for the excitation of pressure waves, we examine next the incorporation of this function into our calculations of synthetic barograms. The other parameters are those adopted above. The numerical results (not shown here) indicate that the calculated barograms for cloud A and B models (Voight 1981) have almost exactly the same waveform as given by source 1, although the absolute amplitude is somewhat different. We conclude that minor fluctuations with short periods involved in the empirically-determined source-time function do not yield significant effects on the recorded waveform.

On the other hand, Bolt & Tanimoto (1981) have suggested the hypothesis that the second positive pressure peak could be due to the arrival of waves from a second, vertical eruption that followed the first lateral blast by about 6 min. An independent suggestion with a time interval between the first and second eruptions of about 1.5-2 min, came from observations of seismic waves (Kanamori et al. 1984). We find, however, that calculations of pressure wave synthetics with a time lag less than 2 min do not yield a second peak coincident with the observations. For this reason, we successively assume several different time intervals. It was found that a best fit can be obtained for a lag of about 6-7 min. Synthetics for source 2 (a = 90 m, T = 0.2 min, u = 0.66 km s-', and


456

2L = 2 W = 4 km) and the superposed trace (sources 1 + 2) with this lag are displayed in Fig. 5 . We see that the lowest composite trace shows some features consistent with the observed record, except for late portions with longer periods. If we take into account the second source, the agreement between the calculated and observed A3 waves (not shown here) has also been improved to some extent as compared with that in Fig. 3(c). The above calculations appear to favour the hypothesis of double extended sources corresponding to the first lateral and second vertical blasts. The hypothesis will be further discussed in a later section.

(4) Dissipation effects. Now we evaluate the average Q value related to overall dissipative properties in the lower atmosphere. Bolt & Tanimoto (1981) attempted to estimate both the mean attenuation of pressure waves and the source propagation velocity from the spectral amplitude ratio between the A1 and A2 waves recorded at Berkeley. They obtained a preferred Q of about 2000 and a subsonic velocity. A weakness of this method is that it assumes that the atmospheric waves are mainly composed of a single mode. The super- positions of several acoustic and gravity modes with different propagation velocities and those from the first and second pulses contaminate the spectral amplitudes. Consequently, these could yield spurious spectral zeros and changes of spectral slope against frequency.

As an alternative, the method we use here is to calculate synthetic barograms of the A1 and A3 waves for variously assumed Q values, and to compare the maximum amplitude ratios in the time domain between the two types of waves with those from the observed record. Fig. 6(a) gives the calculated barograms for the A3 waves with Q values ranging from 6000 to 1000. Although the waveform which is composed of a well-dispersed train does not change appreciably within the above range of Q, the amplitude decreases significantly. For the A1 waves, the variations in Q values affect only slightly the calculated amplitude and waveform. In Fig. 6(b), the maximum double amplitude ratios between the A3 and A1 waves are plotted as a function of Q. Because the corresponding ratio from the record is about 0.45, a possible range of the average Q falls between 1300 and 1700. It should be remembered, however, that the curvature correction adopted here for the divergence effect for the A3 waves is probably incomplete. It is therefore possible that the computed Q value might not measure only intrinsic average dissipation but reflect the degree of departure from the present assumptions. In subsequent calculations, Q is set equal to 1500.

(5) Effects of source propagation velocity and time constant. We now return to the problem of the source, and we make a further attempt to estimate plausible ranges in the


0.8

0.7

0.6

0.5

0.4

A - 3 W A V E 0=6000

2-

-

-

-

-

-

0-

I I i- IOrnin.-,

2-

0-

I 0.3

2-

0- 0-1000

M a x . P - P

A 3 / A I

Figure 6. (a) Synthetic barograms for the A3 waves for four different average dissipation factors. (b) Maximum double amplitude ratio ( p - p ) between the A3 and A1 waves as a function of average Q.


Atmospheric waves from Mt St Helens 457

C A S E 16

2-

0-

2- C A S E 19

0 -

(a) . 1 ,- IOrnin.-l

, 2 h / b

I I

I 1 I I I 2 3 Mach V r

Figure 7. (a) Synthetic barograms from two possible sources with different source propagation velocities and time constants (see Table 1). (b) Variations of b and 2h/b as a function of Vr, where b / 2 and h indicate the calculated half-breadth and amplitude of the first peak of the A1 wave, respectively, and Vr is the source propagation velocity normalized by the sound speed.

propagation velocity of the pressure shock and in the time constant at the source. It should be noted that these two parameters, however, might not be well resolved. Let the source dimensions of the first lateral blast and the second vertical eruption be those estimated before, and the source time history have the form of equation (6b). We assume successively various source propagation velocities and time constants for the first eruption as indicated in Table 1, and fixed values for the second blast. The calculated barograms of the A1 waves for four different cases are shown in Fig. 7(a). Notice from these traces that the amplitude of the first positive peak decreases and its pulse shape broadens, as the velocity decreases and the time constant becomes longer. These situations are also illustrated in Fig. 7(b), where b/2 and h indicate the calculated half-breadth and amplitude of the first peak, respectively, and Vr is the source propagation velocity normalized by the sound speed. We see that b decreases rather slowly but 2h/b is sensitive to the variations in Vr. Because the observed value of 2h/b falls between 2.3 and 2.7, Vr may be of the order of 1.7-2.4 which is supersonic, about twice the sound velocity near the ground. This result gives support to the shock wave speed estimated by Kieffer (1981a, b) from a steady flow model applied to the devastated area, and also appears consistent with satellite data which have been reported to show up to 450 m s-' northward directed velocities of the destructive surge (Rice 1981). The source-time constant corresponding to this case ranges from T = 0.2 to 0.15 min, although slightly longer and shorter constants would give almost the same waveform as well. The estimated time constant in the form of equation (6b) implies that the source velocity drops to less than 5 per cent within 37 = 0.6-0.45 min (36-27 s). This is almost the same order as have been estimated by Banister (1984) from the analysis of expanding cloud data (Voight 1981) and also from seismic body wave data (Kanamori et al. 1984).

As the final step, we may now provide improved synthetics to compare with the Berkeley microbarogram. By the use of the best estimated source parameters, we again calculate the synthetic barograms. These parameters are given as cases 20 and 22 in Table 1. Fig. 8(a, b, c) show the calculated traces for the A l , A2 and A3 waves, respectively, for two different time histories, in comparison with the records. Note that the characteristic features of the A1 waves can generally be well explained by the calculated traces for cases 20 and 22. It may also be noticed that the synthetic barograms of the A2 and A3 waves agree closely with the corresponding recorded waveforms.



A - l W A V E

I -

(b)

C A S E 22 0---

I ,- 1Ornin.-,

2 - cmr-==-l O b s

0 -

I A - 3 W A V E crn

C A S E 22

- 1Ornin.- “I: (C)

Figure 8. Comparison between the observed microbarogram (uppermost trace) and two different synthetic barograms for cases 20 and 22 (see Table 1). (a) A1 wave, (b) A2 wave, ( c ) A3 wave.

It should be noted that the oscillations with periods of about 1 min superposed on a long-period wavetrain, which appear between the first and second large compression peaks and even after the second peak on the observed record (see Fig. l), cannot be accounted for by these synthetic barograms. Possible sources for these oscillations might be attributed either to propagation path effects or to strong excitations yielding the corresponding periods at the volcanic area.

In addition, the late-arriving waves with periods of about 10-18 min recorded on the A1 barogram are also not explained by any of the synthetic calculations. We did not make further attempts to reconcile this discrepancy; perhaps the long-period waves may be the gravity modes GR1 and GR2 which are associated more with the upper atmospheric structures. There remains a doubt about such an interpretation, however, because a negative phase with periods of about I3 min was also recorded at a nearby weather station (Y = 54 km) at Toledo, Washington (Reed 1980).

7 Discussion

Close examinations of the acoustic-gravity waves from the 1980 Mount St Helens eruption, well recorded at global stations, show that some of them recorded remarkably similar waveforms (including a second positive pressure pulse) to those on the Berkeley


Atmospheric waves from Mt St Helens 459 microbarogram. These stations are Boulder, Colorado (r = 1530 km), Palisades, New York (r = 3950 km). Honolulu, Hawaii (r = 4070 km), Kushiro (r = 6945 km), Tokyo (r = 7780 km), and Yonago (r = 8210 km), the latter three being in Japan (taken from the data reported by Donn & Balachandran 198 1, Liu et al. 1982, and Eissler et al. 1983).

The frequency responses of the microbarographs used at the above stations are not known accurately to us, but they appear to be similar to that of the Berkeley instrument, in view of the overall recorded waveforms. The half-period of the second pulse ranges between 3 and 4min. Some of the records have shorter-period waves superposed on the peak of the second long-period pulse, but if the shorter ones are filtered out, the time delay of the peak from the first one is nearly constant, about 6.5 min. Other stations such as Washington, Hamburg, Buchholz and Wajima show somewhat different waveforms on the records, perhaps due to different instrumental responses, but the time interval between the first and second peaks is almost the same as that at the first group of stations. The travel time-distance curve of the second peak for all these stations, when added to fig. 7 in Liu et al. (1982), runs almost parallel to that of the first peak, which gives a horizontal phase velocity of about 305 m s-'. This evidence strongly suggests that the second pulse may have been generated at the source, about 6-7 min after the first pressure disturbance caused by the lateral blast. The alternative hypothesis that, instead, the second pulse was from the propagation effects in the lower atmosphere, would entail that the time delay after the first one would gradually change with distance due to dispersion. In order to examine this point, synthetic barograms have also been computed for two stations at r = 37" and 65", assuming that the barograph response is the same as that at Berkeley. The results confirm the above inferences.

A question arises, therefore, as to why the second pressure disturbance (or corresponding seismic pulse) has not been recorded at the expected time on any seismograms, although that from the first lateral blast was clearly registered on the long-period seismograph at Longmire ( r = 68 km, q5 = 26') (Kanamori 1983, private communication). As already noted, an analysis of seismic body and surface waves shows the occurrence of a moderate-size earthquake following the first eruption within 2min (Kanamori et al. 1984). One possible explanation would be that the second pressure pulse did not come directly from the second, vertical eruption but from the delayed impulse applied to some height of the atmosphere, or due to refractions and reflections of shock waves near the source area; and that the seismograph at Longmire, located north of Mount St Helens, was sensitive to pressure perturbations from the northward blast but not to those from the vertical eruption. This speculation needs further detailed study, and there could be different interpretations.

The kinetic energy emitted from the volcanic eruption has been evaluated from various observations. Bolt & Tanimoto (1981) estimated the energy in the A1 train to be at least 10" erg. We re-estimate the energy using the source parameters obtained in the above modelling. The kinetic energy at the source would be

where M / p is the volume of the virtually uplifted atmosphere due to pressure disturbances from the eruption, and u ( t ) is the velocity function as expressed in equations 6(a)-6(d). In the present source model, M corresponds to the mass of ejecta p x 2L x 2 W x a. Jf we take the simplest form (6a) for u ( t ) = i o ( t ) , then,


460

Using the estimated parameters in case 20, the total energy for the first and second eruptions may be estimated as about 4 x erg. This value is the same order of magnitude but somewhat low compared with other estimates from the atmospheric waves (Donn & Balanchandran 1981 ; Eissler et al. 1983) and from seismic waves (Kanamori & Given 1982).

T. Mikurno and 3. A. Bolt

8 Conclusions

The main purpose of this study was to explore the inverse problem of estimating the source mechanism of excitation of atmospheric pressure waves from large volcanic eruptions and also to estimate the overall dissipative properties of atmospheric acoustic-gravity waves. The main conclusions drawn from the comparisons between the synthetic and observed barograms are as follows:

(1) There is a strong likelihood that the two major compression pulses with a time interval of 6-7 min and periods of 5-8 min on the direct A1 barograph record are the pressure waves radiated directly from the first, (lateral) blast and indirectly from the second (vertical) eruption, respectively.

(2) The preferred velocity-time history of the first eruption has a duration time of about half a minute. The major sources of excitation of the pressure waves were strong disturbances successively applied vertically to the bottom of the atmosphere. The disturbances appear to have propagated with supersonic velocity of about twice the sound speed over the source region corresponding to the devastated area.

(3) On the assumption that the attenuation of the circumpropagating waves A3 are attributable to the dissipative properties of the lower atmosphere, the average dissipation factor Q is of the order of 1500.

Acknowledgments

We wish to thank Drs Hiroo Kanamori, H. Eissler, Mansour Niazi, J. R. Banister and J. E. Reed for providing us preprints of their papers in advance of publication. The first. author has greatly benefited from discussion on the present problem particularly with Drs Kanamori and Eissler. Much of the research was done while the first author was at the Seismographic Station, University of California, Berkeley. Kind assistance by Norman Abrahamson and Yoshinobu Hoso in drawing the figures is also acknowledged.

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Excitation mechanism of atmospheric pressure waves from the 1980