Layout 6 1 ANNALS OF GEOPHYSICS. 61, 5, SE560, 2018; doi: 10.4401/ag-7837 “AN ANALYTIC STUDY OF FRICTIONAL EFFECT ON SLIP PULSES OFEARTHQUAKES„ Jeen-Hwa Wang1,* (1) Institute of Earth Sciences, Academia Sinica, Nangang, Taipei, Taiwan 1. INTRODUCTION Heaton [1990] first found the existence of slip pulses of earthquakes. Let TR and TD be the rise time of displacement at a site on a fault and the duration time of ruptures over the entire fault, respectively. He observed TR<>1; while the CS shows a crack- like rupture for VW friction when υ is not too large. For SW friction, TR and TR/TD decrease when the slip pulse propagates in advance along the fault and when ω0 and γ increase. TR and TR/TD also depend on vR (rupture velocity) and increasing L (fault length). For the PS, Tp/TD is a good indication to show the existence of pulse-like oscillations at a site, because Tp (the predominant period of oscillations at a site) is slightly longer than TR. Results show the existence of a pulse-like oscillation at a site for the two types of friction. A pulse-like oscillation is generated when Δ>1.6 for SW friction and when υ>0.6 for VW friction. Tp/TD decreases with increasing Δ. For the two types of friction, To/TD decreases when vR and L increase. Beeler and Tullis, 1996; Cochard and Madariaga, 1996; Perrin et al., 1995; Zheng and Rice, 1998; Nielsen et al., 2000; Lapusta et al., 2000; Ben-Zion and Huang, 2002; Nielsen and Madariaga, 2003; Coker et al., 2005; Rice et al., 2005; Ampuero and Ben-Zion, 2008; Urata et al., 2008; Ando et al., 2010; Garagash, 2012]. Fric- tion used by those authors includes VW friction, SW friction, velocity- and state-dependent friction, and thermal-pressurized friction. Results suggest that rup- ture modes are controlled by several factors, including friction laws, fault strengths, stress conditions on faults, energy and heat generated by faulting, scaling laws of faults, and spatial-temporal complexity of fault slip. In addition, some researchers considered ge- ometrical heterogeneity of slip is a mechanism to stop earthquake rupture. Beroza and Mikumo [1996] sug- gested that the short TR could yielded by pre-existing stress with heterogeneous fault strengths. The slip pulses have also been studied by some au- thors [Wu and Chen, 1998; Chen and Wang, 2010; El- banna and Heaton, 2012] based on the 1-D spring-slider model (abbreviated as the 1-D BK model hereafter) proposed by Burridge and Knopoff [1967]. From analytic studies by using SW friction, Wu and Chen [1998] claimed that SW friction can result in the self-healing slip pulse and the width of a pulse de- pends on vR and friction strength. From numerical studies by using VW friction Chen and Wang [2010] found the propagation of slip pulses with TR/TD<0.1 along the model. Their simulation results are in agree- ment with Heaton’s observations. Elbanna and Heaton [2012] pointed out the differences between the BK model and crack models. According to linear elastic fracture mechanics, slip pulses are seldom generated in the continuum models because slipping region inside of a fault cannot release applied stress without con- tinuous slip while rupture is extending. On the other hand, in the BK model slip pulses can be produced due to the following reason. Each slider can completely re- lease stress exerted by the leaf springs by going back to their equilibrium position even while rupture is ex- tending. In other words, each slider does not transfer stress after their stoppage and information on the length of rupture does not feedback. Laboratory experiments also provide significant in- formation on generation of slip pulses. Coker et al. [2005] observed the existence of both pulse-like and crack-like ruptures under certain conditions. Lyko- trafitis et al. [2006] observed the pulse-like shear rup- tures with self-healing. Lu et al. [2007] found that the rupture modes depend on the level of fault pre-stress and VW friction is important for earthquake dynam- ics. Biegel et al. [2008] found that off-fault damage can affect the slip-pulse velocity. As mentioned previously, in order to generate slip pulses some authors prefer to VW friction, while other favor SW friction. In this study, I will investigate the frictional effects on the generation of slip pulses using the continuous form of the 1-D BK model with linear SW friction or linear VW friction. Hence, it is signifi- cant to examine which friction (SW or VW friction) is more capable than the other for yielding slip pulses. 2. ONE-DIMENSIONAL SPRING-SLIDER MODEL 2.1 MODEL Burridge and Knopoff [1967] proposed the 1-D BK model (see Figure 1), in which there are N sliders and springs. A slider with mass, m, is connected to its near- est two neighbors by a coil spring of stiffness, Kc. Of course, the two end sliders are only connected to the respective one nearest slider. A moving plate with a constant velocity, Vp, pulls each slider through a leaf spring of stiffness, Kl. Each slider rests in its equilib- rium state at time t=0. The position of i-th slider (i=1, …, N) is denoted by Xi, which is measured from its ini- tial equilibrium position, along the horizontal axis rep- resented by the coordinate y. Hence, Xi is in a function of y and t. Each slider is exerted by a frictional force between it and the moving plate. The frictional force is usually a function of displacement, Xi, and particle ve- locity, Vi (=dXi/dt), of the slider and shown by the func- tion Fi(Xi;Vi), which has a static frictional force of Fsi=Fi(Xi;0) at rest. The equation of motion is m(∂2Xi/∂t 2)=Kc(Xi+1-2Xi+Xi-1)-Kl(Xi-Vpt)-Fi(Xi;Vi) (1) In Equation (1), there is an implicate parameter ‘a’ which is the space between two sliders in the equilib- rium state. The ratio κ=Kc/Kl has been defined by Wang [1995] to be the stiffness ratio of the system. This ratio represents the level of conservation of energy in the system. Larger κ is equivalent to stronger coupling be- Jeen-Hwa WANG 2 FIGURE 1. An N-degree-of-freedom dynamical spring-slider sys- tem. tween two sliders than between a slider and the moving plate, thus leading to a smaller loss of energy through the leaf spring or a higher level of conservation of en- ergy in the system, yet opposite for smaller κ. Since the fault system is dynamically coupling with dissipation, κ must be a non-zero finite value. The Vp is in the order of ~10-12 m/s. The moving plate pulls a slider and thus gradually increases the elastic force, KcVpt, on it. When KcVpt is slightly higher than static frictional force, Fsi, at the i-th slider, the two forces are cancelled out each other and can be ignored during ruptures. After a slider moves, Fsi drops to Fdi (i.e., the dynamic frictional force). 2.2 FRICTION The frictional force between two contact planes is a very complicated physical process. Laboratory experi- ments have exhibited time-dependent static frictional strength of rocks [Dieterich, 1972] and velocity-depen- dent dynamic friction [Dieterich, 1979; Shimamoto, 1986]. Dieterich [1979] and Ruina [1983] proposed em- pirical, rate- and state-dependent friction laws. The detailed description of friction laws and the debates concerning the laws and their application to earthquake dynamics can be found in some articles [e.g., Marone, 1998; Wang, 2009; Bizzarri and Cocco, 2006a; Bizzari 2011]. Several simple friction laws have been taken to the- oretically and numerically study earthquake dynamics [see Wang, 2016]. The laws are: the velocity-dependent, weakening-hardening friction law [Burridge and Knopoff, 1967]; the slip-dependent friction law [Cao and Aki, 1984/85]; the nonlinear VW friction law [Carl- son and Langer, 1989a,b; Carlson, 1991; Carlson et al., 1991; and Beeler et al., 2008]; and the piece-wise, lin- ear velocity-weakening and weakening-hardening fric- tion [Wang, 1995, 1996]. Purely velocity-dependent friction could yield unphysical phenomena and mathe- matically ill-posed problems as pointed out by Madariaga and Cochard [1994]. Ohnaka [2003] stressed that the pure velocity-dependent friction law is not a one-valued function of velocity. The problem has been deeply discussed by Bizzarri [2011]. Nevertheless, for a purpose of comparison the single-valued linear veloc- ity-dependent friction law is still considered below. Friction is an important factor in controlling earth- quake dynamics. Based on the 1-D BK model in the presence of linear VW friction with a decreasing rate, rw, of friction force with velocity, Wang [1996] found three types of rupture propagation: (1) subsonic type with rw>2(Klm) 1/2; (2) sonic type with rw=2(Klm) 1/2; and (3) supersonic type with rw<2(Klm) 1/2. Supersonic-type ruptures are non-causal, because vR is greater than the sound speed. Knopoff et al. [1992] stated that the sys- tem is asymptotic to dispersive-free elasticity in the continuum limit when rw=2(Klm) 1/2. They also found that large rw is more capable of generating large events than small rw. Carlson and Langer [1989a,b] used F(v)=1/(v+vc) where vc is the characteristic velocity. The related decreasing rate is 1/vc(1+v/vc) 2 with the val- ues in the range of from 1 to 0 when vc varies from 0 to ∞. Hence, their friction law basically exhibits super- sonic behavior with rw<2(Klm) 1/2, and thus is poten- tially capable of producing very large events. Wang [1997] also stressed the effect of frictional healing on earthquake ruptures. Several authors [Nur, 1978; Carl- son and Langer, 1989a,b; Carlson, 1991; Carlson et al., 1991; Knopoff et al.,1992; Rice, 1993; Wang and Hwang, 2001] stressed the influence on earthquake rup- tures due to heterogeneous fault strengths on the fault. Carlson and her co-workers emphasized that de-local- ized events can be generated when the friction strengths over the fault plane is uniform. In this study, I will analytically study the frictional effects caused by VW friction or SW friction on the generation of slip pulses by using the 1-D BK model. In order to perform analytic manipulation, only the linear laws are taken into account. The SW friction law (see Figure 2a) is: F(X)=Fo(1- X/Xc) (2) where X and Xc are, respectively, the displacement and the characteristic distance. The VW friction law (see Fig- ure 2b) is: F(V)=Fo(1- V/Vc) (3) where V=dX/dt is the velocity and Vc is the character- istic velocity. The breaking strengths are uniform over the model. This means that only steady travelling waves are taken into account. 3 AN ANALYTIC STUDY OF SLIP PULSES FIGURE 2. (a) For linear, slip-weakening friction law: F(X)=1- X/Xc (Xc=characteristic displacement) and (b) for lin- ear, velocity-weakening friction law: F(V)=1-V/Vc (Vc =characteristic velocity). Jeen-Hwa WANG 4 4. ANALYTICAL MANIPULATION 4.1 EQUATION OF MOTION Define xi=Xi-Vpt. This gives Xi=xi+Vpt and Vi=dXi/dt=dxi/dt+Vp=vi+Vp. Hence, Equation (1) becomes m(∂2xi /∂t 2)=Kc(xi +1-2xi +xi -1)-Klxi -Fi(xi +Vpt;vi +Vp) (4) After a slider moves, Vpt and Vp can be neglected because of Vpt<Δ, Equation (7) can work only for u<Δ. When the driving force reaches the static strength of the friction whose val- ue is unit in Equation (7), stability at the slider is determined by the competition between the rate of friction |∂Fstrength/∂u|=1/Δ and the rate of stress-relax- ation between the slider and the leap spring |∂Fstress/∂u|=|∂[h2(∂2u/dξ2)-u]/∂u|=|h2∂(∂2u/dξ2)/∂u-1| at u=0. With the condition 1-1/Δ>|∂[h2(∂2u/dξ2)-u]/∂u-1|, we have |∂Fstrength/∂u|> |∂Fstress/∂u|. This means that sta- ble motions cannot exist, and is in contrast with the known source time function for dynamic ruptures where an initial acceleration phase should exist. This again makes Equation (7) work only when u<Δ. Under SW regime, Equation (7) means that max{xp}=D0/γ 2=Xc/(Δ-1). This means Δ>1. To solve Equation (7), the Laplace Transformation (LT, denoted by L), which can be seen in numerous text- books [e.g., Papoulis, 1962], is used to transform it to a different form. The LT of Equation (7) is h2∂2U/∂ξ2+(s2+1-Δ)U=1/s (8) The solution of U includes the complementary solu- tion, Uc, and particular solution, Up, that is, U=Uc+Up. According to the method given in Johnson and Kioke- meister [1968], the solution of Equation (8) is U(ξ,s)=C1e -ψξ/h+C2e ψξ/h-1/sψ2 (9) where ψ=(s2+1-Δ-1)1/2. There are two types of waves from Equation (9): The first one is the travelling wave represented by the first term along the +ξ direction and the second one along the -ξ direction in its right-handed-side, i.e., Uc(ξ,s)=C1e -ψξ/h+C2e ψξ/h. The second one is the oscilla- tion at a site given by the third term, i.e., Up(ξ,s)=-1/sψ 2. The second term of the first type with ξ<0 can be re- written as e-ψ|ξ|/h. The Inverse Laplace Transformation (ILT, denoted by L1) of Uc(ξ,s) with |ξ|/h>0 is uc(ξ,τ)=C{1-γ(|ξ|/h)J1[γ(τ 2-(|ξ|/h)2)1/2]/ (10) [τ2-(|ξ|/h)2]1/2}H(τ-|ξ|/h) where C=C1 or C2, γ=(1-Δ -1)1/2, J1[…] is the first-order Bessel function, and H(τ-|ξ|/h) is the unit step function (H(z)=0 as z<0 and H(z)=1 as z≥0) representing a trav- elling plane wave. Since τ=ω0t and ξ=y/D0 are, respec- tively, the normalized time and normalized rupture dis- tance, h=vR/D0ω0 is the normalized rupture velocity. When the rupture propagates from 0 to ξL, which is the normalized rupture length and equal to L/D0 (L=the rupture length), the normalized duration time is τD=ξL/h, and thus the duration time is TD=τD/ω0=ξL/ω0h, thus giving TD=ξL/ω0h=(L/D0)/ω0(vR/D0ω0)=L/vR. Let tr be the arrival time of the travelling wave at a site y, that is, tr=|y|/vR. Substituting uc(ξ,τ)=xc(y,t)/D0, ξ=y/D0, τ=ω0t, and tr=|y|/vR into Equation (10) gives xc(y,t)=CD0{1-(γω0tr)[γω0(t 2-tr 2)1/2]]/ (11) (t2-tr 2)1/2}H[ω0(t-tr)] Equation (11) shows a propagating wave which is usually represented by a function of the form G(t-|y|/vR), where t’=t-|y|/vR is known as the retarded time for situations where causality holds [e.g., Perrin et al., 1995; Nielsen et al., 2000]. The rise time, TR, is mea- sured from t’=0 or t=tr=|y|/v to larger t’=t* when the wave amplitude and the particle velocity reach their re- spective peak values. The quantities inside {…} multi- plied by CD0 of Equation (11) show the wave amplitude. In order to further understand the properties of Equa- tion (11), we only need to examine the function Ss(y,t) =1-(γω0tr)J1[γω0(t 2-tr 2)1/2]/(t2-tr 2)1/2. Define θ to be γω0(t 2-tr 2)1/2, and thus Ss(y,t) can be represented by 1- (γ2ω0tr)J1(θ)/θ. When t=tr, θ=0 and J1(θ)=0. This makes the value of J1(θ)/θ be indefinite. From the l’Hospital the- orem [see Johnson and Kiokemeister, 1968], we have lim θ→0[J1(θ)/θ]=limθ→0 [dJ1(θ)/dθ]=limθ→0[J0(θ)-J1(θ)/θ]. According to the recurrence relation of Bessel functions [Abramowitz and Stegun, 1972]: J1(θ)/θ=[J0(θ)-J2(θ)]/2, we have lim θ→0[J1(θ)/θ]=1/2 because of J0(θ)=1 and J2θ)=0 when θ=0. Hence, xc(y,0)=CD0/2 and C=2xc(y,0)/D0. The value of constant C depends on the initial value of xc(y,0). The waveform xc(y,0) appears before the main rup- ture and can behaves like the nucleation phase. From the observations, they are very small [Beroza and Ellsworth, 1996; Ellsworth and Beroza, 1995,1998; Mori and Kanamori, 1996; Wang, 2017]. This suggests xc(y,0)=CD0/2<<1, and thus C could be small. The first- order Bessel function J1(θ) is positive, but vibrates and decreases with increasing θ, and (t2-tr 2)-1/2 or θ-1 also de- cays fast with time when t>tr. From TR=t*-tr, t* is the time when Ss(y,t) reaches its peak value. This time is commonly calculated from the necessary condition, i.e., dSs(y,t)/dt=0. This can be ob- tained by taking dSs(y,t)/dt=[dSs(θ)/dθ]dθ/dt=0. Since dθ/dt=γω0(t 2-tr 2)-1/2 cannot be zero when t>tr, we only need dSs(θ)/dθ=0 or d[J1(θ)/θ]/dθ=J0(θ)-J1(θ)/θ=0. The condition of dSs(θ)/dθ=0 at θ*=γω0(t* 2-tr 2)-1/2 leads to the following equality: J0(θ*)-J1(θ*)/θ*=0. Actually, nu- merous values of θ* can make this equality hold. Among them, the first value is 1.975 [Abramowitz and Stegun, 1972], thus giving γω0(t* 2-tr 2)1/2=1.975 and t*=(tr 2+3.901/γω0) 1/2. This makes TR be t*-tr=tr{[1+ 3.901/(γω0tr) 2]1/2-1}. Figure 4 demonstrates the plots of TR versus tr for various values of γ, and ω0: (a) for γ=0.1, 0.25, 0.5, and 1.0 (from top to bottom) when ω0=1 Hz; and (b) for ω0=0.1, 0.2, 0.5, 1.0, and 2.0 Hz (from top to bottom) when γ=1. Obviously, TR decreases rapidly with increasing tr, thus meaning that TR de- creases when the slip pulse propagates along the fault. It is interesting to investigate the variation in Ss(y,t*) with y=vRtr. Inserting t* into Ss(y,t) leads to Ss(y,t*)=1- γω0trJ 1[γω0(t* 2-tr 2)1/2]/(t*2-tr 2)1/2. Hence, Ss(y,t*)=1- γω0trJ 1{γω0[tr 2+3.901(γω0) -2-tr 2]1/2}/[tr 2+3.901(γω0) -2 -tr 2]1/2| due to t*=(tr 2+3.901/γ2ω0 2)1/2 as mentioned above. This gives Ss(y,t*)=1-(γω0) 2(|y|/vR) J1(1.975)/1.975, where 3.9011/2=1.975, and thus Ss(y,t*)=1-0.294(γω0) 2(|y|/vR) because of J1(1.975)= 0.5782 [see Abramowitz and Stegun, 1972]. When Ss(y,t*)=0, we have 0.293(γω0)2(|y|/vR)=1. This gives |y|=yc=3.413vR/(γω0) 2 and tc=yc/vR=3.413/(γω0) 2, where tc is tr related to yc. Clearly, the value of yc (>0) is a function of vR, γ, and ω0. Hence, Ss(y,t*) changes from positive (when |y|yc or tr >tc) by passing through zero (when |y|=yc or tr =tc). Figure 3a displays an example of a normalized waveform Ss(y,t)/Ssmax, where Ssmax is the value of Ss(y,t) at t* as defined before in the figure, of the rup- ture wave having a propagation velocity of vR=2 km/sec and the predominant angular frequency of ω0=1 Hz at a position on the y-axis when the wave propagates with a travelling time tr=3.607 sec under the action of slip- weakening friction force with γ=0.9. This plot shows a 5 AN ANALYTIC STUDY OF SLIP PULSES FIGURE 3. Figure shows Ss(y,t)/Ssmax (Ssmax=Ss(y,t*)) and Ss(y,t)/Ssvmax (Ssvmax=Sv(y,t*)) at y=vRtr with vR=2 km/sec and tr=3.607 sec: (a) for Ss(yy,t)/Ssmax with γ=0.9 and ω0=1.0 Hz; and (b) for Sv(y,t)/Ssvmax with σ=0.9 and ω0=1.0 Hz. pulse-like wave with short TR. Since the value of tc of this case is 4.216 sec, the related value of t*, which is not displayed in Figure 3a, is positive as expected be- cause of tr=3.607 sec1 sec, which appears very soon after an earthquake starts to rupture. Since the slip pulses have frequencies in a small range of 0.5 to 2 Hz, ω0 is not the main factor. Since the value of γ varies in a large range from 0+ to 1, γ must be the main factor in controlling the slip pulse. Considering an example of a slip pulse with ω0=1 Hz and γ=1, we have (1+3.901/tr 2)1/2<1.3. When tr >2.378 sec, the inequality TR/TD<0.3 always holds and thus the slip pulse appears. Figure 6 displays the plots of tr ver- sus γ for ω0=0.1, 0.2, 0.5, 1.0, and 2.0 Hz (from top to bottom). The area between the curve and vertical line denotes the solution regime. Clearly, tr decreases with increasing ω0. The example shown in Figure 3a can meet the condition. Figure 5 displays the plots of TR/TD versus tr with vR=2 km/sec for various values of γ, ω0, and L: (a) for γ=0.1, 0.25, 0.5, and 1.0 (from top to bot- tom) when L=50 km and ω0=1 Hz; (b) for ω0=0.1, 0.2, 0.5, 1.0, and 2.0 Hz (from top to bottom) when L=50 km and γ=1; and (c) for L=10, 30, 50, 70, 100 km (from top to bottom) when γ=1 and ω0=1 Hz. Obviously, TR/TD decreases very rapidly with increasing tr, thus meaning that TR decreases when the slip pulse propagates along the fault. Meanwhile, TR also decreases when γ, ω0, and L increase. The third term of Equation (9), i.e., Up(ξ,s)=-1/s(s 2+1- Δ-1), is not a function of locality and thus only represents the oscillations at all sites on the fault. But, it is still sig- nificant to examine if it behaves like a pulse-like oscil- lation or not. Its ILT is up(ξ,τ)=L 1[Up(ξ,s)]=[1-cos(γτ)]/2γ. The solution exists when Δ>1. For Δ>1, we have up(ξ,τ)=[1-cos(γτ)]/2γ=sin 2 (γτ/2)/γ2 (13) The particular solution up(ξ,τ) comes from the ILT of -1/s(s2+1-Δ-1). Equation (8) leads to h2∂2U/∂ξ2=-(s2+1- Δ-1)U+1/s, in which -(s2+1-Δ-1)U behaves like the LT of an elastic force and 1/s is the LT of 1 (see Equation (8)) normalized from the static friction force, F0. When h2∂2U/∂ξ2=0, i.e., the divergence of U along the ξ-axis is zero, we have (s2+1-Δ-1)U+1/s. This indicates a bal- Jeen-Hwa WANG 6 FIGURE 4. The plots of TR versus tr for various values of γ and ω0: (a) for γ=0.1, 0.25, 0.5, 1.0, and 2.0 (from bottom to top) when ω0=1 Hz; and (c) for ω0=0.1, 0.2, 0.5, 1.0, and 2.0 Hz (from bottom to top) when γ=1. FIGURE 5. The plots of TR/TD versus tr for various values of γ, ω0, and L: (a) for γ=0.1, 0.25, 0.5, and 1.0 (from top to bottom) when L=50 km and ω0=1 Hz; (b) for ω0=0.1, 0.2, 0.5, 1.0, and 2.0 Hz (from top to bottom) when L=50 km and γ=1; and (c) for L=10, 30, 50, 70 km. ance between the elastic force and the static friction force and its ILT shows a vibration on the fault. The two forces behave like pre-existing forces on the fault plane. Since the two forces are of locality- indepen- dence, the related particular solution is also of local- ity-independence. Inserting Equation (13) into Equation (7) results in ∂2U/∂τ2=-(1-Δ-1)up-1. Since the initial value of up is zero, we have ∂ 2u/∂τ2=-1, thus implying a negative initial acceleration at a slider. There are two cases in Equation (13): one with 1-Δ-1<0 or Δ<1 and the other with 1-Δ-1>0 or Δ>1. The first case is sin[(1-Δ-1)1/2τ/2]= i sinh[(Δ-1-1)1/2τ/2], and thus sin2[(1-Δ-1)1/2τ/2]=-sinh2[(Δ-1-1)1/2τ/2]. This solution exhibits a negative hyperbolic sine-type function, which cannot be a common wave, and can be ignored. The second case represents a positive sine-type func- tion and thus can represent a vibration at all sites be- cause it is site-independent from Equation (13). Replacing up(ξ,τ) by xp(y,t) and τ by ω0τ into Equation (13) leads to xp(x,t)=D0sin 2(γω0t/2)/γ 2 (14) Define ω*=γω0/2=(1-Δ -1)1/2ω0/2 to be the predom- inant angular frequency of the oscillation, and thus Tp=2π/ω* is its predominant period. This gives T0/Tp= 2/(1-Δ-1)1/2. The plot of T0/Tp versus Δ is shown in Fig- ure 7a. T0/Tp increases with Δ. The increasing rate of T0/Tp with Δ is high when Δ<2 and becomes low when Δ>2. Figure 8 shows the typical velocity waveform (in the solid plus dotted curve) and the typical displacement waveform (in the solid curve). Since the wave propa- gates from -y to +y, it stops at the time instant when v=0. In Figure 8, Tp is the period of velocity waveform. Since TR≈Tp/2, the ratio of Tp over TD, i.e., Tp/TD, is a good indication to show whether a slip pulse can ex- ist or not. The ratio Tp/TD is 2T0(1-Δ -1)-1/2/(L/vR)=2(1- Δ-1)-1/2vRT0/L, where vRT0 denotes a characteristic dis- tance of wave propagation in the natural period. For real M≥6 earthquakes vR varies from 1.5 to 4 km/s, L from 30 to 300 km, and T0 from 10 -1 to 10 sec. Figure 9 shows the plots of Tp/TD versus Δ: (a) for vRT0=0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0 km (from bottom to top) when ω0L=80 km/s; and (b) for L=20, 40, 60, 80, 100, 120, 140, and 160 km (from top to bottom) when vRT0=2 km. As mentioned above, the quantitative criterion to confirm the existence of a slip pulse is TR/TD<0.3. Now, TR is almost equal to Tp. Hence, a pulse-like os- cillation can exist when Tp/TD<0.3. In Figure 9, the upper bound of Tp/TD is 0.3. Meanwhile, the solution 7 AN ANALYTIC STUDY OF SLIP PULSES FIGURE 6. The plots of tr versus γ for ω0=0.1, 0.2, 0.5, 1.0 and 2.0 Hz (from top to bottom) based on SW friction. FIGURE 7. (a) The plots of T0/Tp versus Δ; and (b) the plots of T0/Tp versus υ for SW friction. The vertical line shows the value of Δ or υ at which the curve becomes flat. FIGURE 8. The solid plus dotted curve represents the velocity waveform and the solid curve denotes the displace- ment waveform. Tp and TR are, respectively, the pre- dominant period of the velocity waveform and the rise time of displacement waveform. exists only when Δ>2 (as displayed by a vertical line in the figure). Obviously, the major portion of each curve is below the upper bound, thus suggesting the exis- tence of a pulse-like oscillation in the solution regime. 4.3 SOLUTIONS BASED ON VELOCITY-WEAKENING FRICTION The normalized VW friction law from Equation (3) is φ(v)=1-v/υ where υ=Vc/D0ω0 is the dimensionless char- acteristic velocity. Equation (6) becomes ∂2u/∂τ2=h2(∂2u/∂ξ2)-u-(1-v/υ) (15) When v>υ, φ(v) is negative and unreasonable. Hence, Equation (15) works only for v<υ. The LT of Equation (15) is s2U(ξ,s)=h2∂2U(ξ,s)/∂ξ2-U(ξ,s)-1/s+sU(ξ,s)/υ (16) This gives h2∂2U/∂ξ2-(s2+1)-sυ-1)U=1/s (17) Let ζ=(s2+1-sυ-1)1/2. The general solution of Equa- tion (17) is U(ξ,s)=C1e -ζξ/h+C2e -ζξ/h-1/sζ2 (18) There are two types of waves from Equation (18): the first one is the travelling waves represented by the first term (travelling along the positive direction along the ξ- axis) and second term (travelling along the negative di- rection along the ξ-axis) in its right-handed-side, i.e., Uc(ξ,s)=C1e -ζξ/h+C2e -ζξ/h, and the second one is the os- cillation at site shown by the third term, i.e., Up(ξ,s)= -1/sζ2. The second term of the first type of waves with ξ<0 can be re-written as e-ζ|ξ|/h. The ILT of Uc(ξ,s), i.e., L1[Uc(ξ,s)] with |ξ|/h>0 is uc(ξ,τ)=C{1-(σ|ξ|/h)eτ /2υJ1[σ(τ 2-(|ξ|/h)2)1/2]/ (19) (τ2-(|ξ|/h)2)1/2}H(τ-|ξ|/h) where σ=(1-1/4υ2)1/2. Because of σ>0, 1-1/4υ2 must be larger than 0 and thus υ>1/2. Equation (19) exhibits a travelling plane wave. The quantities inside {...} before H(τ-|ξ|/h) denote the wave amplitude. Unlike Equation (9) for SW friction, this equation has an extra term eτ/2υ, which increases exponentially with time. When the wave propagates from 0 to ξL, the normalized duration time is τD=ξL/h and thus the duration time is TD=τD/ω0=ξL/ω0h=(L/D0)/ω0h. Let tr be the arrival time when the rupture arrives at a site y, that is, tr=|y|/vR. Substituting uc(ξ,τ)=xc(y,t)/D0, ξ=y/D0, τ=ω0t, tr=|y|/vR, and ϖ=ω0/2υ into Equation (19) leads to xc(x,t)=CDo{1-(σωotr)e ϖtJ1[σωo(t 2-tr 2)1/2]/ (20) (t2-tr 2)1/2}H[ωo(t-tr)] When ϖ<<1 Hz, eϖt≈1 makes Equation (20) be sim- ilar to Equation (9). The results are similar to those for SW friction as mentioned above. This condition is equivalent to ω0<<1 Hz or υ>>1. Since the wave with ω0<<1 Hz is a very long wave, it cannot be a slip pulse. When ϖ is not a small value, we examine the proper- ties of xc(x,t) by using the function Sv(y,t)=1- (σω0tr)e ϖtJ1[σω0(t 2-tr 2)1/2]/(t2-tr 2)1/2. Define θ to be σω0(t 2-tr 2)1/2. Sv(y,t) can be represented by 1- (σ2ω0 2tr)e ϖtJ1(θ)/θ. When t=tr, θ=0 and J1(θ)=0. This leads to an indefinite value of J1(θ)/θ. Like the above- mentioned mathematical manipulation for SW friction, we have lim θ→0[J1(θ)/θ]=1/2. Hence, xc(y,0)=CD0/2 and C=2xc(y,0)/D0. As mentioned above, the value of con- stant C depends on the initial value of xc(y,0) and thus C must be in general small. The function J1(θ) is posi- tive, but vibrates and decreases with increasing θ, and (t2-tr 2)-1/2 or θ-1 also decays fast with time when t>tr. The value of TR of a slip pulse at a site is measured from the arrival time, tr, at a site to the time, t*, when the amplitude of slip pulse reaches its peak value. This gives TR=t*-tr. To obtain t*, we must calculate the time when Sv(y,t) reaches its peak value from the necessary condition: dSv(y,t)/dt=0. This can be obtained by taking dSv(y,t)/dt=[dSv(θ)/dθ]dθ/dt=0. Since dθ/dt=t(t 2-tr 2)-1/2 cannot be zero when t>0, we only need to consider if dSs(θ)/dθ equals zero. Mathematical manipulation leads to dSv(θ)/dθ=-(σ 2ω0 2tr)e ϖt{ϖJ1(θ)-[J0(θ)-J1(θ)/θ]}. From Jeen-Hwa WANG 8 FIGURE 9. The plots of Tp/TD versus Δ: (a) for vRT0=0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0 km (from bottom to top) when ω0L=80 km/s; and (b) for L=20, 40, 60, 80, 100, 120, 140, and 160 km (from top to bottom) when vRT0=2 km. the rule of Bessel functions that not any two Bessel functions have the same zero, the function ϖJ1(θ)-[J0(θ)- J1(θ)/θ] cannot be zero. When J1(θ)=0 at θ=θ*, J0(θ*) is not zero. This results in ϖJ1(θ*)-[J0(θ*)-J1(θ*)/θ*]= -J0(θ*)≠0. When J0(θ)=0 at θ=θ*, J1(θ*) is not zero. This re- sults in ϖJ1(θ*)-[J0(θ*)-J1(θ*)/θ*]=(ϖ+1/θ*)J1(θ*)≠0. Mean- while, eϖt is equal to 1 at t=0 and larger than 1 when t>0. Hence, dSv(θ)/dθ cannot be zero and there is not an extremum for Sv(y,t). This cannot yield TR/TD<0.3 and thus slip pulses do not exist for VW friction. Unlike Ss(y,t*), Sv(y,t) does change from positive to negative passing through zero. An example of a normalized waveform Sv(y,t)/Svmax, where Svmax is the maximum value of Sv in the figure, of the rupture wave having a propagation velocity of vR=2 km/sec and the predomi- nant angular frequency of ω0=1 Hz at a position on the y-axis when the wave propagates with a travelling time tr=3.607 sec under the action of velocity-weakening friction force with σ=0.9. This plot which shows an in- crease in the amplitude with time, thus suggesting a crack-like rupture with long TR. Hence, SV friction can yield pulse-like ruptures only when υ>>1 or Vc>>1 and in general produces crack-like ruptures. The third term of Equation (18), i.e., Up(ξ,s)= -1/s(s2+1-sυ-1), is not a function of locality and thus only represents the oscillations at all sites on the fault. But, it is still significant to examine if it behaves like a pulse-like oscillation or not. The function Up(ξ,s) can be re-written as -1/s-[1-(1-4υ2)1/2]/2(1-4υ2)1/2(s-b)- [1+(1-υ2)1/2]/2(1-υ2)1/2(s-c), where b=(1/2υ)[1+(1- 4υ2)1/2], and c=(1/2υ)[1-(1-υ2)1/2]. The solution does not exist when 1-4υ2=0 or υ=0.5. When 1-4υ2≠0 or υ≠0.5, L-1[Up(ξ,s)] is up(ξ,τ)=—{1+(e τ/2v/e)[eτ/2v-eτ/2v+eτ/2v)/2] (21) where e=(1-4υ2)1/2. For uc(ξ,τ),υ must be larger than 0.5 as mentioned above. Let q=(4υ2-1)1/2=ie. Consider a right triangle with three sides: the longest side with a length of R=(12+q2)1/2=2υ, and the other two with lengths of q and 1, respectively. The angle between the longest side and the side with L=1 unit is set to be θ. Hence, we have cos(θ)=1/2υ, sin(θ)=q/2υ, and tan(θ)=q. The tangent term gives θ=tan-1(q). Define sin(qτ/2υ)=(eiqτ/2υ-eiqτ/2υ)/2i and cos(qτ/2υ)=(eiqτ/2υ+ e-iqτ/2υ)/2. Replacing up(ξ,τ) by xp(y,t)/D0, ξ by y/D0, τ by ω0t, and ϖ=ω0/2υ into Equation (21) leads to xp(y,t)=-Do[1+e ϖtsin(qϖt-θ)/sin(θ)] (22) Obviously, the initial value of xp(y,t) is 0 when t=0. This xp(y,t) shows a sine-function-type oscillation at a site. Unlike Equation (14), this equation has an extra term eϖt, which increases with time. When t≤0, xp(y,t) is zero and thus Equation (22) becomes xp(y,t)H(t). Define ω*=qϖ=(1-1/4υ2)1/2ω0 and Tp=2π/ω* to be, respec- tively, the predominant angular frequency and the pre- dominant period of the oscillation. From Equation (22), the particle velocity is dxp(y,t)/dt=-Doϖe ϖtsin(qϖt)/cos(θ)sin(θ)] and the parti- cle acceleration is d2xp(y,t)/dt 2=-D0ϖ 2eϖt,sin(qϖt,+θ)/ cos2(θ)sin(θ)]. When qϖt+θ=π, we have d2xp(y,t)/dt 2=0. Hence, max{dxp(y,t)/dt}=D0ϖe (π-t)/q. This holds before the first zero-crossing of dxp/dt. Hence, the condition of D0ϖe (π-t)/q1.744. Although Tp is determined by T0 (0.11.744 (as displayed by a vertical line in the figure). Obviously, the major por- tion of each curve is below the upper bound, thus sug- gesting the existence of a pulse-like oscillation in the solution regime. 5. DISCUSSION The theoretical analyses for the conditions of gener- ating slip pulses are made based on the continuous form of the 1-D BK model in the presence of the linear fric- tion laws: f(u)=1-u/Δ for SW friction as well as f(v)=1- v/υ for VW friction. The parameters Δ and υ are, respectively, the characteristic distance and the charac- teristic velocity of the respective law. First, it is neces- 9 AN ANALYTIC STUDY OF SLIP PULSES sary to consider physical implications of the results. For SW friction, variable transformation of w=u+γ2 re- duces Equation (7) to the Klein-Gordon equation [Polyanin, 2002]: ∂2w/∂τ2=h2(∂2w/∂ξ2)-γ2w. This im- plies that the strength of the leaf spring has a factor of γ2 due to presence of friction, so that γ2 per stored en- ergy within the leaf spring is conserved and 1-γ2 per the energy is dissipated. We can see that 1-γ2=D0/Xc is a ratio of actual frictional work dissipated due to slid- ing from an area under SW line in Figure 2a. Since the stiffness of a leaf spring multiplied by γ-2 controls en- ergetics, dependence of behavior of the system on k/γ2 should be discussed. For VW friction, the restoring force should always have the same sign for stability of the system. Variable transformation of w=u+1 reduces Equation (15) to ∂2w/∂τ2=h2(∂2w/∂ξ2)-w+w’/υ, where w’=dw/dτ. With- out loss of generality, the restoring force can be as- sumed as negative, i.e., -w+w’/υ≤0. Let F[w] be the Fourier transformation of w. Then the following holds: so that should be required, otherwise the friction is not restoring force but repulsive force in high-frequency content. Such a compact supported spectrum in fre- quency domain is, however, not able to be a compact supported signal in time domain. There are the complementary solution, xc(y,t) and particular solution, xp(y,t), of the equation of motion. For SW friction, the function xc(y,t) shows a slip pulse when 1>γ=(1-Δ-1)1/2>0. TR and TR/TD both decrease rapidly with increasing tr, thus meaning that TR and TR/TD are both reduced when the slip pulse propagates along the fault. This means that the slip pulse is gen- erated when the ruptures are far away from the nucle- ation point. Meanwhile, TR and TR/TD also both decrease when γ, ω0, and L increase and when vR is re- duced. Higher γ associated with a low decreasing rate of friction with slip is easier to produce a slip pulse than lower γ related to a higher decreasing rate. A de- creases in TR and TR/TD with increasing ωo means that slip pulses should have higher angular frequencies and cannot be long-period rupture waves. The decreases in TR and TR/TD with increasing L mean that a slip pulse can be more easily produced in a long fault than in a short one. Hence, it would be difficult to detect a slip pulse on a short fault. Equation (21) shows that lower vR can more easily result in small TR and TR/TD than higher vR. This indicates that the slip pulse has a rela- tively slow propagation speed. However, vR is less im- portant than other parameters, because the value of vRtr/L in Equation (21) is always smaller than 1. For VW friction when υ>>1; while it shows a crack- like rupture for VW friction when υ is not too big. The present results suggest that unlike Heaton [1990], SW friction is easier to produce a slip pulse than VW fric- tion. Although Perrin et al. [1995] assumed that not all friction laws result in steady travelling pulses, the pre- sent result exhibits that the two types of friction in use can generate a slip pulse under their respective ranges of model parameters. For the particular solution, the ratio of Tp over TD of a rupture is a good indication to show the possible existence of a slip pulse, because the TR of slip at a site is shorter than Tp. For SW friction with Δ>1, the solu- tion clearly shows oscillations with a predominant pe- riod of Tp. Computational results in Figure 9 reveal a decrease in Tp/TD with increasing Δ. Longer (shorter) Δ means a slower (faster) decay of friction with slip. A slower decay of friction with increasing slip is easier to generating a pulse-like oscillation than a faster decay of friction. Based on the definition of Δ=Xc>D0, Δ>1 gives Xc>D0. This suggests that when the characteris- tic length is longer than the characteristic distance, the pulse-like oscillation can be generated. Figure 9a shows Tp/TD<0.4 (or TR/TD<0.2) when Δ>1.2. Figure 9b shows that except for L=20 km, Tp/TD<0.4 (or TR/TD<0.2) when Δ>1.5. Theoretical results suggest that Δ>1.5 is a significant condition for generating a pulse-like oscillation at a site under SW friction. For VW friction, when υ>0.5 this solution shows a sine-function oscillation at a site. Because of υ=Vc/D0ω0, the inequality υ>0.5 leads to Vc>D0ω0/2. This means that a pulse-like oscillation at a site can be generated when Vc>D0ω0/2. Since Vc is the character- istic velocity of VW friction law, higher Vc indicates a slower decay of friction with velocity. The present re- sult suggests that a slower decay of friction is more capable of generating the pulse-like oscillation than a Jeen-Hwa WANG 10 FIGURE 10. The plots of Tp/Tp versus υ: (a) for vRT0=0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0 km (from bottom to top) when L=80 km; and (b) for L=20, 40, 60, 80, 100, 120, 160, and 180 km (from top to bottom) when vRT0=2.0 km. faster one. Unlike the pulse-like oscillation produced by SW friction, the amplitude of pulse-like oscillation generated by VW friction could increase with time due to an extra term eτ/2υ. This suggests that VW friction can cause higher wave energy than SW friction. Smaller υ will cause a faster increase in wave ampli- tude than larger υ. Figure 10 reveals a decrease in Tp/TD with increasing υ. Higher (lower) Vc means a slower (faster) decay of friction with velocity. Clearly, a slower decay of friction with velocity is easier to generating a slip pulse than a faster decay of friction. Because of υ=Vc/D0ω0, υ>1 means Vc>D0ω0/2. When the characteristic velocity of the model is higher than the rate of a slider in the characteristic distance, a pulse-like oscillation can be generated. Figure 10 shows TR/TD<0.4 (or TR/TD<0.2) when υ>0.6. Hence, υ>0.6 could be a significant condition for generating a pulse-like oscillation at a site under VW friction. In addition, Figures 9a and 10a both show a de- crease in Tp/TD with decreasing vRT0, which is the char- acteristic rupture distance. Lower vR is more capable of generating a pulse-like oscillation than higher vR. Meanwhile, shorter T0 is more capable of generating a pulse-like oscillation than longer T0. Because of T0= (m/Kl) 1/2/2π=(ρA/κl) 1/2/2π, strong coupling (denoted by κl) between the moving plate and a fault system is ca- pable of producing a pulse-like oscillation than weak coupling. On the other hand, lighter or lower-density fault rocks are easier to producing a pulse-like oscilla- tion than heavier or higher-density fault rocks. Figures 9b and 10b exhibit a decrease in Tp/TD with increasing L, that is, longer L is more capable of generating a pulse-like oscillation than shorter L. However, the dif- ference in the values of Tp/TD between two sequential rupture lengths decreases with increasing L. This means that when L is longer than a certain value, the effect of rupture length on Tp/TD is reduced. Of course, the L-ef- fect also depends on vR and T0. A comparison between Figure 9 and Figure 10 shows that Tp/TD is in general smaller for VW friction than for SW friction, thus suggesting that the former is more ca- pable of producing a pulse-like oscillation at site than the latter. In addition, under VW friction a pulse-like oscillation can be produced even though L is short. Based on Equation (7), Wang [2016] obtained that the complementary solution exhibits ω-1 scaling in the whole range of ω for SW friction. But, for the partic- ular solution SW friction results in spectral amplitudes only at three values of ω. Based on Equation (15), Wang [2016] obtained that for VW friction with υ>0.5, the spectral amplitude versus ω exhibits almost ω0 scaling when ω is lower than the corner angular fre- quency, ωc, which is independent on υ and increases with ω0. When ω>ωc, the spectral amplitude mono- tonically decreases with ω following a line with a slope value of -1, which is the scaling exponent. This again confirms that crack-like rupture is generated by VW friction. Hence, it is easier to yield slip pulses from SW friction than from VW friction. 6. CONCLUSIONS Seismological observations show the existence of slip pulses with TR/TD<0.3. For the present model, there are complementary and particular solutions of the equation of motion. For SW friction, the complemen- tary solution shows a slip pulse when γ>1. TR and TR/TD both decrease rapidly with increasing tr, thus meaning that TR and TR/TD both decrease when the slip pulse propagates along the fault. TR and TR/TD also both decrease when γ, ω0, vR, and L increase. For VW friction, a slip pulse is yielded when υ>>1 or Vc>>1 and the crack-like ruptures is generated when υ is not too big. When υ>>1 or Vc>>1, the results produced by VW friction are essentially similar with those by SW friction. For the two types of friction, a slower decay of friction is more capable of generating slip pulses than a faster one. Lower vR is more capable of gener- ating a slip pulse than higher vR. Longer L is more ca- pable of generating slip pulses than shorter L. Of course, the importance of vR and L are lower than γ and ω0. For the particular solution, the ratio of predominant period (Tp) of oscillations at a site is slightly longer than TR, and thus Tp/TD, is a good indication to show the existence of pulse-like oscillations at a site because of TR1.6 (or Xc>1.6D0) and for VW friction when υ>0.5 (or Vc>0.5D0ω0). Tp/TD decreases with increasing Δ for SW friction and with increasing υ for VW friction. For the two types of friction, T0/TD and Tp/TD both decreases when vR and L increase. For the two types of friction, a slower decay of friction is more capable of triggering pulse-like os- cillations at a site than a faster one. Shorter T0 is more capable of generating pulse-like oscillations than longer T0. In other words, strong coupling between the moving plate and a fault system is capable of produc- ing a slip pulse than weak coupling. Lighter or lower- density fault rocks are easier to producing pulse-like oscillations than heavier or higher-density fault rocks. 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