SUPPLEMENTARY INFORMATION

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1 SLEMENTARY INFORMATION doi:1.138/nature9828 Supplementary Information The wide-spread occurence of low frequency tremors prior to violent volcanic eruptions is a major issue in volcanological hazard assessment and the subject of the main text by Jellinek and Bercovici (see accompanying article). Here we develop a simple theoretical model that is consistent with the current understanding of conduit flows in explosive systems and demonstrate that it can robustly predict the observed frequency spectrum of tremors. Foam annulus model As is discussed in the main text, the magma column can oscillate or wag back and forth depending on the inertia of the column, the restoring gas-spring force of the annulus, and the viscous resistance to bending of the column. We therefore construct a model that describes the displacement of the column, as well as the flow and compression of the gas in the annulus. To elucidate the essential concepts and physics, we first consider a very simplified tremor model that is akin to a magma plug waving or rattling inside of a compressible annular jacket of foam, i.e, an impermeable porous matrix with a high volume fraction of disconnected gas bubbles. We assume the foam moves with the magma column, to which we afix the reference frame. Figure S1 illustrates the basic geometry of this model. In the following section we examine the more general situation in which gas flows through a permeable annulus relative to the rising magma column; while the results for that case are a bit more complicated, the basic concepts and intuition differ little from those in the simple foam model. We assume the inner column of magma of radius R m is cylindrically axisymmetric and initially at rest and centered inside the cylindrical conduit of radius R c. The resting width of the gap between the conduit wall and the inner column is R = R c R m. When a vertical segment of the magma column at height z is displaced to the right by an amount u(z), the maximum gap width on the left is R + u and the minimum gap width on the right is R u. We seek the net pressure force of the foam annulus exerted on a vertical segment of the magma column. To find the pressure we need to know how the volume of any parcel of foam is squeezed by the displacement of the magma column to the right (or left), and for this we need the location of the magma columns cylindrical surface. In a polar coordinate system centered on the axis of the conduit cylinder, the magma column s cylindrical surface is given by (r m cos θ u) 2 +(r m sin θ) 2 = R 2 m (1) r m is the radial distance from the axis of the conduit to the magma column s surface, θ is the angle relative the line of displacement (i.e., a radial vector with θ =points to the right). We can solve for r m (θ) for this cylinder, but in so doing we will assume small displacements and that u R m, and keep terms only of zeroth and first order in u/r m, leading to the simple equation for the cylindrical surface of the magma column: r m (θ) =R m + u cos θ (2) The volume of a segment of foam in the annulus from θ to θ + dθ and of vertical height dz is V = dzdθ rdr = 1 r m(θ) 2 dzdθ ( Rc 2 Rm 2 2uR m cos θ ) (3) The undisturbed volume of this segment is V = 1 2 dzdθ(r2 c Rm); 2 hence we can write the general volume as ( V = V 1 u ) cos θ (4) = R2 c Rm 2 (5) 2R m However, (4) represents the total volume of the segment, not the gas volume. To infer the gas volume V g we note that the magma volume in the porous annulus is conserved since the magma is assumed incompressible, which implies that V V g = V V g, V g is the unperturbed gas volume; this leads to ( V g = V φ u ) cos θ (6) φ = V g /V is the unperturbed gas volume fraction. We can relate the gas volume to pressure using the ideal gas law = (M ( /m g )RT ρ C 2 1+ u ) V g φ cos θ (7) M is the mass of gas in the segment, which is conserved, and thus ρ = M /(φ V ) is the undisturbed gas density. Here, R is the gas constant, T is temperature, m g is the molar mass of the gas, and C = RT/m g is the gas sound speed. The traction (force/area) of the gas pressure pushing on the magma column in the direction of displacement is ( ˆx ˆn cos θ u ) sin 2 θ (8) R m ˆx and ˆn are the unit vectors in the direction of displacement, and normal to the magma column s surface, respectively. Integrating this traction completely around the annulus (i.e., for θ 2π) gives the net pressure force on a vertical segment of the magma column of thickness dz: F p = dz 2π (θ)ˆx ˆn r m (θ)dθ = dzρ C 2 πr m φ u (9) 1

2 RESEARCH SLEMENTARY INFORMATION u H ΔR+u Rm ΔR-u Rc Figure S1: Sketch of the Foam Annulus model of the magma-wagging volcanic tremor. Indicated variables are discussed in the accompanying text. The resistance force due to viscous bending of the column is given by the difference in shear tractions between the top and bottom of a column segment of height dz, i.e., F b =[τ xz ] z+dz z πrm 2 (1) τ xz is the shear stress resisting the magma column bending to the left or right. At any given point in the column we can approximate (for small displacement) the stress as τ xz = µ m (11) t µ m is the magma s dynamic viscosity. Newton s 2nd law for the segment of the magma column relates the column segment s mass ρ m πrmdz, 2 acceleration / t 2 and net force F p + F b via ρ m πr 2 mdz 2 u t 2 = dzρ C 2 πr m φ u +[τ xz] z+dz z πr 2 m (12) which, in the limit of small dz, and using (11), eventually becomes t 2 = 3 u ω2 u + ν m 2 (13) t ω 2 ρ C 2 = φ ρ m R m = 2ρ C 2 φ ρ m (Rc 2 Rm) 2 and ν m = µ m /ρ m (14) are the natural oscillation frequency and kinematic viscosity of magma, respectively. redicted Tremor Frequency To give one estimate of the angular frequency of oscillation ω, we use C = 7m/s, ρ m /ρ = 1 and ρ m = 25kg/m 3, and as one example, R m = 1m and R c = 2m. For initial porosity at fragmentation we assume φ =.7al- though this number is O(1) so has little influence. Therefore we obtain ω = 2ρ C 2 φ ρ m (R 2 c R 2 m) 45 s 1 (15) This value of ω implies an oscillatory frequency of ω 2π = 45/(2π) 1Hz (16) which corresponds well to typical volcanic tremor frequencies. The predicted angular frequency ω depends on various properties, the most poorly constrained of which is related to conduit geometry, i.e., on R c and R m, all other factors in the relation for ω in (15) being well constrained. However, since ω (R 2 c R 2 m) 1/2, the dependence on geometry is weak. In particular, if 1 R m 1m and.5 R c /R m 1 2

3 SLEMENTARY INFORMATION RESEARCH then the frequency of oscillations merely ranges from about.1hz to 5Hz (see Main Text), well within the range of known volcanic tremors. Dispersion and Damping We next examine how the oscillations depend on wavelength and are damped by viscous resistance to bending in the magma column. We first simplify the model equation (13) by nondimensionalizing time according to t = t /ω and height according to z =(C/ω )z, the length scale C/ω corresponds to the distance a gas sound wave travels during one oscillation 1. Substituting these scaling relationships into (13) yields (after dropping the primes) t 2 = u +2η 3 u 2 t (17) η = ν mω 2C 2 (18) is the ratio of the resistance to viscous bending of the magma column to the restoring force of gas pressue. Assuming u e ikz+st, k is the dimensionless wave number and s the dimensionless growth rate, we arrive at a dispersion relation: s = ηk 2 ± η 2 k 4 1 (19) Oscillations obviously occur if k< 1/η. If λ is the dimensional vertical wavelength of the oscillation this condition is restated as νm λ>2π (2) 2ω For viscosity of explosive silicic magma, we assume an upper limit of about µ m = 1 7 a s, although this could be as low as 1 5 a s (see main text and Gonnermann and Manga [27]). With that viscosity, ρ m = 25kg/m 3, and ω 45, we find that the wavelength λ must exced approximately 1m for the oscillation to occur; see Main Text for further discussion. To estimate typical values of λ, we assume that the magma plug waves back and forth from a nearly fixed base (i.e., the base of the gas annulus) to a free end at z = H, at the fragmentation point. That free end represents a maximum in displacement in velocity, and thus a 1/4 fundamentalmode wavelength; thus we assume the fundamental mode has λ =4H. Thus, for example, columns shorter than about 25m will not have sustained 1Hz oscillations. To maintain the minimum tremor frequency of about.5hz, the column must be taller than about 4m, as discussed in the main text. 1 While this choice of length scale is not necessarily the natural length scale of the foam model equation e.g., ν m/ω is more natural it is the better length scale for the more complex permeable annulus model presented in the following section, and thus we use it here for the sake of consistency. However, if we use a more typical value of H = 1m, and thus λ = 4m, oscillations readily occur. Indeed, in this case η 2 k , which means the oscillatory mode readily dominates with a dimensionless angular frequency of about I(s) = 1, or dimensionally of angular frequency ω. The oscillation is damped at a dimensionless rate of R(s) = ηk 2 or dimensionally decays as ν m 2 ( ) 2 2π s 1 (21) λ for λ = 4m and the maximum viscosity plug (and 1 times slower for the minimum viscosity plug). Since tremors are sustained for hours or days, the oscillations would necessarily need to be excited by some forcing mechanism to overcome damping. lausible forcings identified with various visual, seismic, geodetic and geochmical observations, as well as analog and numerical conduit flow models include explosions, fragmentation, and pressure variations from turbulent flow of gas in the annulus. ermeable Annulus Model We again construct a model of a magma column oscillating or bending back and forth inside a bubbly gas annulus, but now include the flow of gas through an annulus with interconnected pore space and finite permeability caused by the shearing of bubbles near the conduit wall (see main text and Figure S2). We consider gas flow through the annulus, being driven both by gas flux injected from below, and by pressure variations arising in response to the swaying of the magma column from side to side. We assume for simplicity that variables (gas density, porosity, velocity, etc) are independent of radial distance r from the center of the conduit, and their dependence on azimuthal angle θ is (as in the simple foam annulus model) prescribed by the horizontal displacement of the magma column from the origin. Equation of motion for the magma column As with the foam annulus model, Newton s 2nd law on a vertical segment of thickness dz of the magma column is ρ m πr 2 mdz 2 u t 2 = dzc2 2π ρ(θ)ˆx ˆn r m (θ)dθ + dzµ m 3 u 2 t πr2 m (22) u is again the displacement of the magma column in the x direction, and we have assumed the gas pressure is C 2 ρ C is gas sound speed and ρ is gas density. Again, as with the closed model r m = R m + u cos θ and ˆx ˆn = 3

4 RESEARCH SLEMENTARY INFORMATION u w w H ΔR+u Rm ΔR-u W Rc W Figure S2: Sketch of ermeable Annulus model of volcanic tremor. See also Fig. S1. cos θ (u/r m ) sin 2 θ and thus to first order in u/r m the momentum equation for the column becomes t 2 = C2 ρ m πr 2 m 2π ρ(θ)(r m cos θ + u cos(2θ)) dθ + ν m 3 u 2 t (23) as before ν m = µ m /ρ m is the magma kinematic viscosity. It is straightforward to show that if gas density and hence pressure is uniform there will be no restoring force on the column to sustain an oscillation. Thus, what remains to be determined is the gas density as a function of azimuth around the annulus, denoted by ρ(θ). Mass conservation As with the Foam Annulus model, the motion of the magma column squeezes or dilates the pores, which in turn affect the gas density; however, in this case the gas can now escape. We assume, however, that the pores are drawn out primarily into vertical tubes, thus flow is predominantly in the vertical direction z. Conservation of mass of gas in a segment of the annulus from θ to θ + dθ, from z to z + dz and r = r m (θ) to r = R c is given by dθdz t r m ρφrdr = dθdz r m ρφwrdr (24) φ is porosity or gas volume fraction and w is the vertical velocity of gas. Assuming variables are uniform across r, this relation leads to ρϕ t + ρϕw ϕ = φ (1 u ) cos θ = (25) (26) accounts for compression/dilation of the annulus segment by motion of the magma column (i.e., because r m in the above integral limits is both z and t dependent). We next use conservation of mass in the matrix surrounding the gas (i.e., the walls of the pores or tubes) to infer how porosity φ is affected by the motion of the central magma column. First, we state that the magma column rises at a constant velocity and that the coordinated system is fixed to this magma column. We also assume the magma matrix in the porous annulus remains fixed to the magma column. Finally, we assume that any wedge in a vertical slice of the porous annulus, i.e., between θ and θ + dθ and between z and z + dz, is uniformly squeezed (or dilated) by the horizontal deflection of the magma column. The mass of magma in this piece of porous annulus is conserved hence ρ m (1 φ)rdr = ρ m (1 φ )rdr (27) r m R m φ is the porosity of the undisturbed annulus. Since we assume φ remains uniform across r even when being squeezed 4

5 SLEMENTARY INFORMATION RESEARCH (or equivalently that φ represents the average porosity across r), then φ =1 R2 c Rm 2 Rc 2 rm 2 (1 φ ) (28) Again using r m = R m + u cos θ, the porosity, to first order in u is φ = φ (1 φ ) u cos θ (29) Moreover, we find that to first order in u ϕ = φ u cos θ, (3) which, once the gas velocity w is specified through momentum conservation, can be substituted into (25) to infer how density ρ changes with u and θ. Equation of motion for the porous annulus The conservation of momentum of gas in a segment of the annulus can be derived similarly to the above development for the mass of gas; the effect of annulus compression on the mass (i.e., leading to the variable ϕ) is eliminated by employing the mass equation (25) and the requirement that uniform pressure exerts no net force. The momentum equation for vertical motion of gas in the pores or tubes is therefore the standard two-phase relation [see Bercovici and Michaut, 21]. ( w ρφ t + w w ) = φc 2 ρ cw ρφg (31) c is the coefficient of drag exerted by the tube or pore walls on the gas, and is typically related to permeability in a Darcy s law modified for turbulent flow [see Bercovici and Michaut, 21]. Because the coordinate system is fixed to the magma column and matrix, magma velocity does not appear; however, there are still forces exerted on the magma pore walls by pressure, drag and gravity. Since the matrix vertical velocity is assumed constant and uniform, the force balance on the matrix is = (1 φ)c 2 ρ + cw ρ m(1 φ)g (32) ρ m is magma density (same as in the magma column) the gas and matrix pressures are assumed to be equal, and the drag of gas on magma in the pores/tubes is equal and opposite to the drag of magma on gas. Of course, (31) and (32) can be added to obtain ( w ρφ t + w w ) = C 2 ρ ρg (33) ρ = (1 φ)ρ m + φρ; or one can take a weighted difference to eliminate the pressure gradient and obtain ( w ρφ(1 φ) t + w w ) = cw + φ(1 φ) ρg (34) Equation (33) shows how gas is driven by vertical pressure gradients but against the weight of the entire mixture (i.e., not just the weight of the gas, but the weight of matrix which is dragging down on the gas). Equation (34) shows how gas is equivalently driven by buoyancy (gas weight relative to matrix weight) but is retarded by drag. Both equations are equivalent, however, one expresses pressure gradients explicity but the drag in terms of matrix weight; while the other expresses drag explicitly by pressure gradients in terms of gas buoyancy. At this point (25), (29) and (33) are sufficient to solve for density and velocity as function of discplacement u. However, we simply seek to understand small oscillations of the magma column, and thus henceforth examine the oscillations and stability of a perturbations to a steady state of flow in the porous annulus. Steady-state vertical flow in the annulus The steady equilibrium state (around which oscillations occur) is given assuming u =, such that the undisturbed porosity and density are φ = φ and ρ = ρ, and the vertical velocity is w = W. Conservation of mass of gas (25) in the steady state dicates that ρ φ W = F is a constant. Because the gas is compressible, the hydrostatic (or lithostatic) pressure variations in both magma and gas cause the steady-state gas density ρ to be depth dependent. In particular, the total steady-state momentum equation (33) implies that C 2 ρ = ρ g (35) ρ = (1 φ )ρ m + φ ρ. However the signficance of the depth dependence of ρ depends on the ratio of domain height H to the density scale height: H ρ ρ = ρ ρ gh C 2 (36) For high porosity.7 <φ <.9 and given ρ m /ρ 1 (since the gas is highly compressed after fragmentation), then ρ /ρ O(1); moreover, with H 1m, C = 7m/s and g = 1m/s 2 then the ratio of H to density scale height is small (albeit not vanishingly small). We nevertheless assume that ρ is effectively constant, and that terms proportional to ρ are negligible except when multiplied by C2 [see Bercovici and Michaut, 21], in which case we use (35). The difference momentum equation (34) implies that c W = ρ φ (1 φ )g (37) ρ = ρ m ρ and c is the drag coefficient in the steady state. In principle, we can solve for W from the above relation, except the drag coefficient c is a nonlinear function of φ and, if accounting for turbulence, a function of W also. Moreover, W is also given by the input flux F = ρ φ W, and combining these relations we would find a relation for 5

6 RESEARCH SLEMENTARY INFORMATION φ. Alternatively we can simply set ρ, φ and W as given boundary conditions, and these determine the necessary F, as well the c that permits such forced flow to continue through the conduit by gas buoyancy. In the end it is more straightforward to set ρ, φ and W, for which we have reasonable constraints, than to set F and and prescribe the relation c (φ,w ) from which we then solve for φ and W for a given ρ. Linear equations and oscillations The gas volume fraction or porosity is perturbed by the column s displacement according to φ = φ (1 φ ) u cos θ (38) and ϕ = φ u cos θ (39) assume u/ 1. The gas flow and density with small perturbations to the steady state are expressed as w = W (1 + ϖ cos θ) and ρ = ρ (1 + ϱ cos θ) (4) both the amplitudes ϖ 1 and ϱ 1 as well. We further define the dimensionless displacement υ = u/ and adopt the non-dimensionalization from the previous section in which t = t /ω and z =(C/ω )z. Substituting (38)-(4) and the nondimensionalization scheme into (23), (25), and (33) we obtain (after some algebra and dropping the primes on dimensionless quantities) 2 υ t 2 = φ ϱ +2η 3 υ 2 t (41) φ M ϖ + φ D ϱ Dt D υ = (42) Dt φ M D ϖ Dt + ϱ Bϱ + Bυ = (43) M = W /C is the Mach number, D Dt = t + M, η is as defined previously in (18), and B = ρ m(1 φ )g ρ ω C (44) We also note that we have assumed that ρ ρ m and that, prior to nondimensionalization, we take ρ to be negligible except multiplied by C 2 in which case C 2 ρ = ρ g Finally, stipulating that υ, ϖ and ϱ all go as e ikz+st, we arrive at the characteristic equation [ s 2 ikb +(s + ikm) 2 ] + φ ikb + k 2 + φ (s + ikm) 2 +2ηk 2 s = (45) Some asymptotic dispersion relations In the long-wavelength limit of k we obtain from (45) the same result as the foam-annulus model, i.e., s = ±i, which implies a dimensional angular frequency again of ω that corresponds to the 1Hz tremor oscillation. In the short-wavelength limit of k we can infer two possible solutions to (45). In one case we assume s does not grow with k and this leads to s φ M 2 2η(1 φ M 2 )k 2 (46) which corresponds to a weakly growing perturbation without any oscillation (assuming M < 1). Another solution for k can be inferred by assuming that s grows with k and we write s = ikα, i.e., the perturbation is wave-like with dimensionless speed α; this leads to k 2 α 2 + φ (α + M) 2 1 φ (α + M) 2 i2ηk3 α = (47) For large η this leads to α = i2ηk or s = 2k 2 η, and thus strong damping from viscous bending of the magma column (over short wavelengths), as expected. For η and assuming that α M, we obtain 1 α ± φ k 2 ± (48) φ Dimensionally this corresponds to a perturbation wave-speed Cα C/ φ, which represents high frequency (small wavelength) disturbances propagating as sound waves in the annulus; however, these waves travel slightly faster than the gas sound speed, due to the higher effective compressibility imposed by the magma pore walls in the annulus [see Bercovici and Michaut, 21]. General dispersion relation The general dispersion relation results from solutions to the 4th order complex polynomial from (45). Typical values for dimensionless parameters are M =.1, B =1/15 and <η<3 (corresponding to 1 5 <µ m < 1 9 a s). The solutions for all four roots to (45) are shown in Figure S3 for characteristic values of η. Again, for a reasonable range of long wavelengths (i.e., for dimensionless wavenumbers in the range <k<1, corresponding to dimensional wavelengths within 2πC/ω <λ<, or approximately 7m <λ< ) the system has an oscillation at an angular frequency of ±ω, which is associated with the tremor frequency of order 1Hz. For smaller wavelengths (higher k) oscillations propagate like sound waves. The solutions also display low frequency modes, which peak at about.1hz, but vanish for both very high and very low wavelengths. 6

7 SLEMENTARY INFORMATION RESEARCH angular frequency Im(s)/ growth rate Re(s)/ φ =.7, B =1/15 M =.1, η = wave number, k(c/ ) angular frequency Im(s)/ growth rate Re(s)/ φ =.7, B =1/15 M =.1, η = wave number, k(c/ ) Figure S3: Dispersion curve for oscillation freqency and associated growth rate versus wavenumber for perturbations in the open model of volcanic tremors. Red and magenta curves represent downward and upward propagating oscillations, respectively, which are at the 1Hz frequency for low wavenumbers (long wavelengths) and then propagate as sound waves at high wavenumbers. Blue and green curves represent up and down propagating very low frequency oscillations. The vertical gray bar marks typical values of wave numbers for the fundamental mode of the magma column wagging back and forth with a wavelength of 4H H 1km. Frequency and growth rates are normalized by the natural oscillation frequency ω, and the wavenumber is normalized by ω /C, as indicated. Results are shown for two values of the viscous resistance parameter η (see (18)), as indicated, associated with a very high viscosity magma (µ m = 1 9 a s; top frame), and a moderate viscosity magma (µ m = 1 7 a s; bottom frame), which also gives nearly identical results to the minimum viscosity case (µ m = 1 5 a s) since η 1 for both lower viscosities. Both cases assume an approximate 1Hz fundamental frequency (ω = 45). For either case the oscillations have frequency ω at small wavenumber (long wavelength), and then propagate like sound waves at large wavenumber (short wavelength). Although the permeable model accounts for more realistic gas flow through the annulus, the oscillation frequency at long wavelength is basically identical to that of the simpler closed annulus model. References Bercovici, D. and C. Michaut, 21. Two-phase dynamics of volcanic eruptions: Compaction, compression, and the conditions for choking, Geophys. J. Int., 182, Gonnermann, H. M. and M. Manga, 27. The fluid mechanics inside a volcano, Annu. Rev. Fluid Mech., 39, D.Bercovici & M. Jellinek, February 5,