MODEL BASED VOLCANIC PLUME PROPAGATION WITH PARAMETRIC UNCERTAINTY

Transcription

1 MODEL BASED VOLCANIC PLUME PROPAGATION WITH PARAMETRIC UNCERTAINTY By HONGNAN LIN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2013

2 c 2013 Hongnan Lin 2

3 I dedicate this to everyone that helped me during this process. 3

4 ACKNOWLEDGMENTS First of all, I would like to express my gratitude to Dr. Mrinal Kumar at Mechanical and Aerospace Engineering of University of Florida, who gave me a lot of help of my Master of Science Thesis Project. Thanks go to him not only for his supervision and valuable contributions concerning my work, but also for the improvement of solving problems. Moreover, I also want to thank for my parent who gave me much encouragement and confidence during the process. Besides, special thanks go to Zinan Zhao and Yifei Sun because in the lab, we contribute to good atmosphere to do research together and they gave me many helpful advice. Finally, I would like to thank all the people who help me. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION MODEL OF PLUME MOTION IN WIND Equations of dynamics of a volcanic plume Discussion about Volcanic Plume with Deterministic Parameters VOLCANIC PLUME EVOLUTION UNDER PARAMETER UNCERTAINTY Stochastic Systems Introduction of gpc expansion and orthogonal property The gpc expansion Stochastic Galerkin Method General Procedure gpc Galerkin Method to Stochastic Volcanic Plumes Discussion about Volcanic Plume with Uncertain Parameters CONCLUSION REFERENCES BIOGRAPHICAL SKETCH

6 Table LIST OF TABLES page 3-1 Correspondence between the Type of Generalized Polynomial Chaos and Their Underlying Random Variables

7 Figure LIST OF FIGURES page 2-1 diagram illustrating a volcanic plume behavior in two types of coordinates variation of mass rate variation of momentum rate variation of angle variation of enthalpy rate Relationship between rise height and axial velocity Relationship between rise height and Temperature Relationship between rise height and bulk density Eruption Column Height variation of mass rate variation of momentum rate variation of θ variation of specific enthalpy mass rate with d = 2 8 mm mass rate with d = 2 3 mm mass rate with d = 2 2 mm mass rate with d = 2 1 mm mass rate with d = 2 5 mm mass rate with d = 2 10 mm A Comparison of Eruption Column Heights in two cases

8 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science MODEL BASED VOLCANIC PLUME PROPAGATION WITH PARAMETRIC UNCERTAINTY Chair: Mrinal Kumar Major: Mechanical Engineering By Hongnan Lin May 2013 This thesis considered model based plume propagation for volcanic eruptions. The interaction between the volcanic plume system and wind is considered, accounting for enhanced entrainment of air and horizontal momentum, distortion of the plume and a decrease of maximum plume rise height at the same mass eruption rate. To obtain the system response in the presence of uncertainties in system parameters, a generalized polynomial chaos approach is used. The resulting residual is minimized using a stochastic Galerkin method, leading to a set of integration differential equations that must be solved numerically. The final column height is computed and compared for the deterministic and uncertain cases. 8

9 CHAPTER 1 INTRODUCTION Volcanic eruptions can affect our lives in a variety of ways from causing catastrophic damage to being a hindrance in our daily lives. Particle fallout from eruption columns and dispersing plumes is a major hazard to humans living close to volcanos. The hazards of volcanic plumes such as fallout of particles, mud and acid rain, the presence of noxious gases, lightning strikes and shock waves are also associated with eruptions. Many of these dangers could happen simultaneously during a single eruption. Particle fallout threatens human safety because it could generate respiratory and eye problems in humans. The most serious danger for animals is contamination of the food supply resulting from the deposits of ash bearing toxic halogen precipitates on the vegetation they feed on. For example, the poisoning of approximately 7500 farm animals followed the 1970 eruption of Hekla when fluorine contents of up to 4000 ppm were recorded on ash-contaminated vegetation. Besides, the heavy fallout from volcanic plumes can cause enormous damage to property. Taking the eruption of Eyjafjallajokull in 2010 in Iceland as an example, the volcanic plume rose to 7000 meters altitude to form a volcanic ash cloud, and proceeded to expand to a large area due to the wind. This led to a complete shutting down of air transport across Europe. Over 5 days, European airlines canceled a total of more than 70,000 flights and suffered a loss of 200 million euros per day. Considering such serious damage, there is a need to come up with various methods to both avoid destruction and accurately predict eruptions of volcanoes so as to safely evacuate people living close to volcanoes. So far significant progress has been made to predict volcanic eruption. In particular, 3 methods have had some success. First, detecting seismic waves is key to predict volcanic eruptions. Some volcanoes normally have continual low-level seismic activity, but an increase might signal an impending eruption. Second, because the similarities between volcanic and iceberg tremors 9

10 include long durations and amplitudes, as well as common shifts in frequencies, a better method for prediction eruptions through observing iceberg tremors has been developed. Finally, remote sensing can also predict volcanic eruption. A satellite s sensors detect the electromagnetic energy which is absorbed, reflected, radiated or scattered from volcano s surface or erupted material in an eruption cloud. However, there is a common drawback in that the above methods only predict when and where a volcano will erupt instead of accurately predicting how much area will be seriously affected in order to prevent or reduce as much damage as possible. As for the prediction of the distribution of volcanic plumes, although the research in this area hasn t been perfected, some progress has been made. Modeling the dynamics of a volcanic eruption plume is a good way to solve this problem. Before developing a physical model of the dynamics for volcanic plumes, it is essential to clarify plume rise theory. In a still environment, there are three basic zones of a plume based on the dominant forces that control the motion of plume. These zones are gas thrust region, convective region and umbrella region. For gas thrust region, the momentum of flow dominates the plume motion. In this region, a mixture of hot gas and pyroclasts is ejected from the volcanic vent at a high speed and atmospheric cool air could be entrained, heated by the erupted thermal energy and then is expanded. If sufficient air is able to be entrained and heated, bulk density of the plume, which is density of mixture which consist of pyroclasts and gas could become smaller than that of surrounding atmosphere. This phenomenon causes the plume motion could enter into next stage convective region. In convection region, buoyant forces which results from the entrainment and heating of atmosphere air mainly controls the behavior of the volcanic plume. The first instance of particle fallout occurs in the convection region, starting with largest sized particles. However, the majority of pyroclasts are transferred to a greater height due to the forces of buoyancy. In the umbrella region,because of variation of the density of atmosphere, the volcanic plume will rise to a height where 10

11 its bulk density is equal to that of surrounding atmosphere. Then the volcanic plume continues to rise because of its inertia as well as expand laterally. Finally, it attains a maximum height until the upward speed decreases to zero. This process is also important in that most of sedimentation takes place in this phase. The above described procedure is confirmed by recent eruptions and laboratory experiments. The motion of volcanic plumes is a very complex process that involves numerous factors. One such factor is wind entrainment that leads to plume bending. Plume bending is a complex issue which is of great importance for human beings. This issue has been addressed by several researchers who are interested in the movement of plume. Monron[1], whose work is solid foundation in this area, modeled the bulk motion of a buoyant plume in moist atmosphere in terms of conservation of mass, momentum and energy(in terms of specific enthalpy). Briggs[4], took a derivation of plume rise for buoyant plumes, under both still and windy conditions, in laboratory and nature by employing dimensional arguments. Slawson and Csandy[10] derived a plume rise relation for bent-over plumes in cross-wind by using a fluid mechanical entrainment model and compared their results with observations. Hewett[6] developed a theoretical model of movement of buoyant smokestack plumes in a stable atmosphere based on previous models to predict plume rise. Wright[11] considered the movement of buoyant jets in density-stratified crossflow to predict the maximum rise heights. Ernst G.G.J[5], developed a theoretical model which leads to a prediction for sedimentation from turbulent jets and plumes in a still environment; M.Bursik[8], considering wind entrainment to plume bending, modeled dynamics of a volcanic plume in a plume-centered coordinate system so as to predict maximum plume rise height and explain sedimentation of pyroclasts. All these studies involve a deterministic analysis of the plume dynamics. The main issues are that the dynamical system is highly nonlinear and depending on the degree of sophistication, relatively high dimensional. None of these studies consider the problem 11

12 of uncertainty in the parameters of the system. The dynamic system of volcanic plume motion contains numerous parameters, several of which are based on heurisic and or empirical models. As a result, there is room for significant uncertainty in them that may influence the propagation of the plume[7] As we know,in fact, there is uncertainty for some input parameters in the model of dynamics of volcanic plumes. In order to solve the randomness of the system, generalized Polynomial Chaos(gPC)[3] can be well employed. The development of gpc approach is based on polynomial chaos(pc). Wiener[9] firstly introduced PC expansion and used Hermite polynomials to model stochastic processes with Gaussian random variables. An important paper, Cameron and Martin s [2]showed that PC expansion converges to L 2 sense for any random process with finite second moment. However, PC was developed only for in standard Gaussian stochastic processes. So as to generalize the result of Cameron-Martin to non Gaussian and discrete distributions, Xiu and Karniadakis[3] developed generalized PC method on the basis of the correspondence between polynomial functions derived from the Askey-scheme[14] and the weight function which is the probability density function of the random variable. In this work, we present analysis of the eruption column height and other states like mass rate, momentum flux, the angle between the plume centerline and the horizon, under different wind-speeds. One is that complete dynamics is deterministically known(deterministic Case)in Chapter 2, the other is that one of parameters is assumed to be a random variable(stochastic Case)in Chapter3. A model of dynamics of volcanic plumes in a standard state-space is rederived based on M.Bursik[8] including the interaction of wind. Then a relationship between maximum rise heights of plumes and various mass eruption rates is shown under different wind speeds in deterministic case. Moreover, in random case, generalized Polynomial Chaos and Stochastic Galerkin method, Gauss Quadrature and standard 4th-order Runge-Kutta scheme are incorporated to solve the stochastic system. We use the averages of states in 12

13 random case to compare those in deterministic case and use average within 3 standard deviation to show the difference of approximations in both two cases. This paper is organized as follows: in Chapter 2, the model of dynamics of volcanic plumes considering wind effect with deterministic parameters will be introduced and reviewed. In Chapter 3, generalized Polynomial Chaos Galerkin technique will be employed to handle the parametric uncertainty of the system. In Chapter 4, conclusions are drawn. 13

14 CHAPTER 2 MODEL OF PLUME MOTION IN WIND In this chapter, we consider the deterministic analysis of volcanic plume propagation. As previously mentioned, the dynamics of plume propagation is highly complex and must capture numerous physical processes. For example, wind can affect the motion of volcanic plumes and pyroclasts falling from a volcanic plume. Small volcanic plumes can be easily bent over by moderate winds. In the following section, a model for volcanic plumes that entrain atmospheric air has been developed based on the paper by Bursik[8] to track behaviors of volcanic plumes in a plume-centered coordinate system. The first two Equations 2-1 and 2-2 below express the calculation of plume trajectory z = x = s 0 s 0 sin θ ds (2.1) cos θ ds (2.2) where z vertical direction and x represents horizon downwind direction. The variable s measures distances along the plume(i.e. tangential to it)and the angle θ represents the inclination between the plume centerline and the horizon axis. These variables are illustrated in Figure 2-1: 2.1 Equations of dynamics of a volcanic plume The basic equations of plume propagation are based on the principles of conservation of mass, momentum and energy. According to mass conservation, we have the following equation: dy (1) ds Y 1 Y 2 = 2πρ a U ϵ + ρ 23 i=5 dy (i) ds (2.3) where Y (1) = πb 2 ρu (2.4) 14

15 Figure 2-1. diagram illustrating a volcanic plume behavior in two types of coordinates. 1 ρ = 1 n σ Y 1 Y 2 = + n ρ a (2.5) Y (1) 2 Y (2) (2.6) U ϵ = α U V cos Y (3) + β V sin Y (3) (2.7) Y(1) represents mass rate of the material in the plume and Y(2) is the momentum rate of the plume(described in greater detail below), Y(i) is mass rate of pyroclasts of ith size within the plume, ρ is bulk plume density which includes entrained air and volcanic gases. Its expression is given as Equation 2-4. In Equation 2-4, σ is the density of pyroclasts and n represents the mass fraction of gas. The variable in Equation 2-5, b represents plume column radius, U is the axial plume speed, ρ a represents ambient atmosphere density. U ϵ is entrainment velocity which is a function of wind speed V and 15

16 axial plume speed U and equation is obtained from Hewett et al.,1971. In Equation 2-3, the first term on the right side is in terms of mass rate by entrainment of air while the second term is the loss of mass rate by fallout of pyroclasts. For axial momentum conservation, we have the following equation: where dy (2) ds = π Y 1Y 2 2 ρ dy (1) (ρ ρ a )g sin Y (3) + V cos Y (3) + Y (2) ds Y (1) 23 i=5 dy (i) ds (2.8) Y (2) = πb 2 ρu 2 (2.9) U = Y (2) Y (1) (2.10) Y(2) represents the axial momentum of the plume. The first term of Equation 2-8 on the right side is the change in momentum resulting from the component of gravitational acceleration, g, in the axial direction and the second term represents entrainment of momentum from the wind while the third term is the momentum of all kinds of pyroclasts. The equation for conservation of radial momentum is given by: [ ] dy (3) = 1 π (Y 1Y 2 ) 2 dy (1) (ρ ρ a )g cos Y (3) V sin Y (3) ds Y (2) ρ ds (2.11) Y(3) is θ in terms of the angle between the plume centerline and horizon axial. The change in theta resulted from both gravity which is the first term on the right-hand side and wind, the second term on the right-hand side. For specific enthalpy conservation, we have the following equation: dy (4) ds = 2π Y 1Y 2 U ϵ ρ a C a T a π Y 1 ρ ρ g sin Y (3) + C p Y (4) CY (1) 23 i=5 dy (i) ds (2.12) where Y (4) = πb 2 ρuc v T (2.13) Y(4) is specific enthalpy rate and the first term on the right-hand side represents energy of entrainment of air, the second one is the change in thermal energy caused by 16

17 conversion to gravitational potential energy, and the last term is loss of heat resulting from sedimentation of pyroclasts. C a represents heat capacity of the air, T a is the temperature of the air, and C p is in terms of the heat capacity of the pyroclasts in the plume. In Equation 2-13, C v is heat capacity of material in the plume and T is temperature of the material. For conservation of mass rate of different kinds of particles is given as: dy (i) ds = 2 PS τ (i 4) Y (i) 6 f ϵ Y (2) Y (1) + Y 1 Y 2 Y 2 Y 1 5 (Y 1Y 2 ) 2 ρ (2.14) where PS = (sin Y (3)) (cos Y (3)) 2 (2.15) f = ( 0.78 ) 6 (2.16) = ( F0 ) 1 3 LMF τ (2.17) Y(i) represents mass rate for multiple particle sizes, f is re-entrainment parameter,f 0 is specific thermal flux at the vent, τ is given as settling speed of a particle in the given size class and LMF is in terms of momentum flux at the vent. f is re-entrainment parameter and because the diameters of particles that are sufficient smaller than could be swept back at low height by the backflow into the plume after falling out from large height. 2.2 Discussion about Volcanic Plume with Deterministic Parameters Figures from 2-1 to 2-4 display variations of four states in the model with wind-speed 25m/s namely the mass rate, momentum rate, θ and enthalpy rate. The initial mass eruption rate is assumed to be kg/s with initial plume radius, b 0 = 500m and vent speed U 0 = 100m/s. Mass rate increases along the trajectory because surrounding air is continuously entrained into the plume. The momentum rate increases until the plume reaches neutral buoyancy height and then decreases. The plume rises to maximum height via its inertia. Considering wind effect, the plume 17

18 mass rate(kg/s) 3 x V=25 momentrum rate(kg.m/s) 14 x V= s(m) Figure 2-2. variation of mass rate s(m) Figure 2-3. variation of momentum rate. 1.6 V=25 7 x V=25 theta(rad) enthalpy rate(j/s) s(m) Figure 2-4. variation of angle s(m) Figure 2-5. variation of enthalpy rate. is distorted to some degree and therefore there is a continuous decrease of θ. As for enthalpy rate, it increases during the whole process V= V=25 height(m) height(m) upward axial velocity(m/s) Temperature(K) Figure 2-6. Relationship between rise height and axial velocity. Figure 2-7. Relationship between rise height and Temperature. 18

19 12000 V= height(m) Bulk Density(kg m 3) Figure 2-8. Relationship between rise height and bulk density. In Figure 2-5, the blue curve displays the relationship between rise height and axial velocity under wind-speed being 25m/s. As the height rises, axial velocity firstly decreases, then increases and finally decreases to 0. The reason for this change is that in the gas thrust region, the bulk density is larger then that of surrounding atmosphere, therefore gravity is higher than buoyancy and the axial velocity decreases firstly. As the rise height increases, the air is kept being entrained in to the volcanic plume and the column becomes buoyant so that material in the plume could accelerate. Figure 2-6 and Figure 2-7 show variations of temperature and bulk density under the same wind-speed of the plume relative to height rises. Both temperature and bulk density decrease during the whole process and finally fall below of those in environment. Because during the whole process, the volcanic plume keeps entraining ambient cool air and solid pyroclasts in the plume heat the entrained air which will be expanded, there is a decrease of temperature and bulk density. In Figure 2-8, there are 7 curves under 7 different wind-speeds. When the wind-speed is zero, the eruption column height is highest compared with that of any other wind-speed at the same mass eruption rate. Moreover, when maximum wind speed in jet case is the same as that in constant wind speed case, the eruption column height is generally higher than that of constant wind speed. Besides, in jet case,the larger the maximum speed is, the lower the eruption column height will be. Similarly, 19

20 Eruption Column Height(m) 5 x V=0 V=25 V=50 V=100 V max =25 V max =50 V max = Mass Eruption Rate(kg/s) Figure 2-9. Eruption Column Height. when the wind-speed holds a constant value, there is also an inversely proportional relationship between eruption column height and mass rate. As the wind-speeds increases, the eruption column height decreases.besides, we make a longitudinal comparison, under the same wind-speed, the eruption column height increases as mass rate increases. Actually, all volcanic plumes are finally distorted by wind to some degree. As wind-speed becomes higher, there will be more air entrained to a volcanic plume and the entrainment coefficient also will be larger. Therefore, the bulk density of a volcanic plume will reduce faster and results in a decrease of eruption column height at last. It s very obvious for small weak volcanic plumes suffering high wind-speeds, a plenty of air with horizontal momentum is entrained into the plume and the plume is easily bent over. As a result, the plume will fail to form umbrella clouds. While for large, strong volcanic 20

21 plume and low wind-speed, the plume could not only rise to maximum height but also couldn t be bent over by wind as well as that in still environment. For most plumes with a total rise height greater than 20km[12], there s hardly no wind-speed affecting plume rise and all volcanic plumes could develop umbrella clouds. However, plumes with eruption column height less than 10km will be seriously affected by wind. In summary, from Y (1) to Y (23), all the variations of 23 states are employed to well describe the evolution of volcanic plume model. In addition, several input parameters which are not very precisely modelled can make some uncertainties. If the plume system is sensitive to uncertainties of the input parameters, the sensitivity can cause variations in final results. As a result, we need a framework that can handle such parametric variations to help us draw some conclusions about nature of plume propagation. 21

22 CHAPTER 3 VOLCANIC PLUME EVOLUTION UNDER PARAMETER UNCERTAINTY As we know, in most engineering applications, we usually convert a physical problem into a mathematical model in which all the input parameters are deterministic. In previous chapter, we approximate the variations of states in a case in which all the parameters are determined. However, in reality, these input parameters like re-entrainment coefficient, water fraction coefficient, wind speed may show randomness which could affect the final solution. This randomness fails to be considered in the deterministic case. For the uncertainty in the mathematical model, it s necessary to use some probabilistic methods to solve this problem. Monte Carlo Simulation, a statistical method, needs a large number of samples. The quality of the result greatly depends on the samples we select and generalized statistics are difficult to obtain. However, generalized Polynomial Chaos(gPC) is commonly applied in frame-work for problems involving parametric uncertainty. In practical application, gpc method which was developed by Xiu and Karniadakis[3] has been widely used in fluid mechanics, optimal control areas to handle parametric uncertainty problems. That s what I m going to do as well in this chapter. 3.1 Stochastic Systems In this section, three parts will be introduced. At first, stochastic systems will be introduced. Moreover, how generalized polynomial chaos expansion and orthogonal property will be presented. Finally, specific steps to handle the parametric uncertainty of the volcanic plume system will be displayed. Stochastic mathematical models are on the basis of a probability space(, A, P) where is the event space, A 2 its σ-algebra and P its probability measure[3]. Here 22

23 a partial differential equations(pdes) is chosen as general and basic model. u t (x, t, Z ) = L(u), D (0, T ] R d, B(u) = 0, D [0, T ] R d, (3.1) u = u 0, D {t = 0} R d. where L is the differential operator which can be nonlinear, B is the boundary condition operator, u 0 is the initial condition and Z = (Z 1, Z 2,...Z d ) R d, d 1, be a set of independent random variables describing the random inputs. 3.2 Introduction of gpc expansion and orthogonal property The gpc expansion Generalized Polynomial Chaos is based on Polynomial Chaos(PC) which is introduced by Wiener in 1938[9]. Hermite Polynomials was used by him to represent a stochastic process with a random variable of standard Gaussian distribution and its PC expansion is given as: ^X = N 0 c k H k (Z ) (3.2) where Z is standard Gaussian random variable, H k (Z ) is Hermite Polynomial and c k is time-evolution coefficient of each Hermite polynomial. Moreover PC expansion is on the basis of a theory of Cameron and Martin which is given as: E ( X ^X 2 ) = X ^X 2 df Z (z) 0 (3.3) where F Z (z) = P(Z z) is probability density function. Whereas, PC expansion can only satisfy the theory of Cameron and Martin when Hermite Polynomials corresponds to Standard Gaussian random variables. In order to make the theory of Cameron and Martin be applied in a wider area, gpc was developed and the expression of gpc 23

24 expansion is shown as : Q(x, t, Z ) = j=0 q j (x, t)ϕ j (Z ) (3.3) where ϕ j (Z ) is a set of orthogonal polynomials and Z = {Z j (w )} N j=0 represents a vector of random variables and q j (x, t) is time-evolution coefficients of orthogonal polynomial basis and is what we need to determine. Considering computational relevance, stochastic processes are represented as gpc expansion with a finite order. Due to the order truncation, the last solution becomes an approximation. Before employing gpc, a prerequisite needs to be satisfied which is there is a special correspondence between distribution of random variables and orthogonal polynomials. In table 3-1[13], some of the well-known correspondences between the probability distribution of Z and its gpc basis polynomials are listed. Orthogonal polynomial functions need to satisfy the following condition E [ϕ i (Z )ϕ j (Z )] = ζ j δ ij, i, j M, (3.4) where ζ j = E [ϕ 2 j (Z )], j M, (3.5) δ ij = 0 for i j; 1 otherwise. (3.6) ζ j are the normalization factors, M = {0,1,2,...,N} is a finite index set and δ ij is Kronecker delta function. As we know, generalized polynomials chaos work well in both discrete and continuous cases. If random variable Z is continuous, its probability density function exists as df Z (z) = p(z)dz and orthogonal polynomial functions could be written as E [ϕ i (Z )ϕ j (Z )] = ϕ i (z)ϕ ( j)(z)p(z)dz = ζ j δ ij, i, j M, (3.7) 24

25 Similarly, if Z is discrete, the orthogonality can be expressed as: E [ϕ i (Z )ϕ j (Z )] = ϕ i (z i )ϕ ( j)(z i )p(z i ) = ζ j δ ij, i, j M, (3.8) i Table 3-1. Correspondence between the Type of Generalized Polynomial Chaos and Their Underlying Random Variables. Distribution of Z gpc basis polynomials Support Continuous Gaussian Hermite (, ) Gamma Laguerre [0, ) Beta Jacobbi [a, b] Uniform Legendre [a, b] Discrete Poisson Charlier {0,1,2,...} Binomial Krawtchouk {0,1,...,N} Negative binoial Meixner {0,1,2,...} Hypergeometric Hahn {0,1,...,N} Generalized Polynomial Chaos could be applied to stochastic processes represented by random variables of commonly used distributions instead of that of standard distribution. In order to effectively use gpc, the correspondence relationship between probability density function of certain random variables and the weight functions of orthogonal polynomials of Askey-scheme Stochastic Galerkin Method As mentioned in the previous section, gpc expansion serves as a complete basis of solving stochastic differential equations. In this part, gpc expansion and Galerkin method will be combined together for solving stochastic systems. We introduce Stochastic Galerkin Method via a general stochastic partial differential equation(pde).the stochastic Galerkin procedure is employed to transform the stochastic equations to a set of deterministic equations.these equations can be discretized from continuous ones by standard numerical techniques. After using this method, we can have a set of coupled diffusion equations which can be written in a vector or matrix form. 25

26 3.2.3 General Procedure Let Z = (Z 1, Z 2,...Z d ) R d, d 1, be a set of independent random variables characterizing the random inputs. The stochastic system is given as u t (x, t, Z ) = L(u), D (0, T ] R d, B(u) = 0, D [0, T ] R d, (3.9) u = u 0, D {t = 0} R d. where L is the differential operator which can be nonlinear, B is the boundary condition operator, u 0 is the initial condition. The first step for solving stochastic systems is to apply gpc expansion to each component of u individually. For the second step, according to the gpc basis functions Equation 3-3, for any arbitrary x and t,we could get u N (x, t, Z ) = N j=0 ^u j (x, t)ϕ j (Z ) (3.10) so that for all K which is less and equal than N, we will obtain[xiu] E[ t u N (x, t, Z )ϕ k (Z)] = E[L(u N )ϕ k ], D (0, T ], E[B(u N ϕ k )] = 0, D [0, T ], (3.11) where ^u 0,k = E[u 0ϕ k ] ζ k ^u k = ^u 0,k, D {t = 0}. are the initial gpc projection coefficients. The stochastic system become one of coupled deterministic equations. The dimension of the system is dim β d N = N + d (3.12) d 3.3 gpc Galerkin Method to Stochastic Volcanic Plumes In the model of the volcanic plume, f, re-entrainment coefficient, is regarded as a random parameter which satisfies Gaussian distribution f N (µ, σ 2 ) with

27 mean and variance. On the basis of previous two sections, we employ gpc Galerkin method to solve stochastic system of volcanic plumes. At first, 23 states are expanded via gpc expansion. According to correspondence between the distribution of f and orthogonal polynomials basis, we apply Hermite Polynomials {H m (Z )} to work as orthogonal polynomial basis. There are two different ways of standarizing the Hermite Polynomials: probabilist s Hermite polynomials and physicists Hermite Polynomials. Here we apply probabilist s Hermite polynomials and the expressions are given as: H 0 (Z ) = 1; H 1 (Z ) = Z ; (3.13) H n+1 (Z ) = ZH n (Z ) nh n 1 (Z ) (3.14) where Z N (0, 1) be a standard Gaussian random variable with zero mean and unit variance. Its PDF is p(z) = 1 2π e z 2 /2 (3.15) which is similar to weight function w (Z ) w (Z ) = e Z 2 /2 (3.16) Therefore, based on the Hermite polynomials and weight function, we can obtain orthogonal properties E[H i (Z )H j (Z )] = H i (Z )H j (Z )w (Z )dz = 0 for i = j 2πn!σ mn otherwise, (3.17) As we know, in Hermite polynomials, random variable is of standard Gaussian distribution but f doesn t satisfy normal distribution. As a result, we represents f in terms of Z and the expression is obtained as f = µ + σz (3.18) 27

28 where µ = 0.43, σ = 0.01 Besides,via gpc expansion with order N = 2, the 23 states in the system could be represented as Y (i) = N j=0 c i j (s)h i j (Z ) (3.19). Then take Y(i) in to the ordinary differential equations of dynamics of volcanic plume, we can obtain dy (i) ds = _ Y (i) = N j=0 _c i j (s)h i j (Z ) = L i (s, Z, f (Z )). (3.20) where L is the differential operator of the system. By using Galerkin projection according to Equation 3-7 and 3-17, the derivatives of deterministic coefficients of Hermite polynomials can be derived N ( _c i j (s)h i j (Z ))H i m (Z )w (Z )df = f j=0 f L i (s, Z, f (Z ))H i m (Z )w (Z )df (3.21) where m = 0, 1,...N, if m = j, we can obtain _c L i j (s) = f i (s, Z, f (Z ))Hj i (Z )w (Z )df f (H j i (Z ))2 w (Z )df (3.22) As s increases, the coefficients also change and could apply a standard ODE solver to solve the resulting system. In order to realise this, standard fourth-order Runge-Kutta scheme is employed in this work. 3.4 Discussion about Volcanic Plume with Uncertain Parameters Approximated mean, µ i (s), and variance, σi 2 (s), can be determined via using coefficients we obtained from solving (3.22). µ i (s) = < Y i (s) > = < N j=0 c i j (s)h i j (Z ) > = N j=0 c i j (s) < H i j (Z ) > (3.24) σ 2 i (s) = < (y i (s) µ i (s)) 2 > = N (c i j (s))2 < (ϕ i j (Z ) < ϕi j (Z ) > )2 > (3.25) j=1 28

29 mass rate(kg/s) x 109 y1 y average (1) y up (1) y low (1) s(m) Figure 3-1. variation of mass rate. momentrum rate(kg.m/s) 14 x y2 y average (2) y up (2) y low (2) s(m) Figure 3-2. variation of momentum rate. theta(rad) y3 y average (3) y up (3) y low (3) enthalpy rate(j/s) 7 x y4 y average (4) y up (4) y low (4) s(m) Figure 3-3. variation of θ s(m) Figure 3-4. variation of specific enthalpy. From Figure 3-1 to Figure 3-4, those 4 figures display evolutions of 4 states in the volcanic plume system with wind-speed being 25m/s. The initial mass eruption rate is still kg/s with initial plume raidus, b 0 = 500m and original velocity U 0 = 100m/s. In each figure, there are 4 curves and the blue curve represents evolution of a state in Deterministic Case, black one displays variation of average of a state in Stochastic Case. y up (i) = y average (i) + 3σ, y low (i) = y average (i) 3σ; respectively red and green lines, are in terms of outcomes within 3 standard deviation of a state to show how far the numbers lie from the mean. In Figure 3-1, it displays evolution of mass rate which increases across the whole process because of continuous entrainment of ambient air. We observe that gpc Galerkin solution, y a verage(1), don t deviate from that, y1, in Deterministic Case. 29

30 Moreover, y low (1) and y up (1) are almost the same and fit very well with other two curves from which standard deviation, σ is almost zero. Therefore, the gpc Galerkin approach presents good approximation results. In Figure 3-2, it presents variation of momentum rate. At first, momentum rate decreases since in the gas thrust region, the plume rises via its inertia. Besides, in the convective region, momentum rate increases until the plume reaches to neutral buoyancy height(a height which its density equals that of the atmosphere) because in this region, buoyancy which is larger than gravity leads to acceleration of the plume. What s more, the momentum rate decreases again since stratified density results in buoyancy less than gravity and the plume rises to rest by its inertia. For gpc Galerkin solutions, y average (2) converges well to y2 in Deterministic Case. Taking y low (2) and y up (2) into consideration, there is also no deviation relative to y2 which displays that standard deviation is zero. In Figure 3-3, the angle between plume-centerline and horizon keeps decreasing all the time because with such wind effect of high speed, the plume is easily bent over. The four curves are almost coincided which shows that gpc Galerkin outcomes satisfy convergence to y3. From Figure 3-5 to Figure 3-10, those 5 figures display the evolution of mass rates of different sizes of pyroclasts in the plume considering wind speed of 25m/s. When particle size are small, the pyroclasts in the plume can be held instead of escaping out of the margin of the plume. While for pyroclasts of large size, because gravity is higher than buoyancy, they are ejected at low heights. In Figure 3-5 and Figure 3-6, the mass rates of pyroclasts with diameters of 2 8 mm, 2 3 mm, and 2 2 mm almost keep the same across the whole process. However, for pyroclasts with diameter2 1 mm, the mass rate continuously decreases as plume height rises. As diameter of pyroclasts continues to increase, the degree of decrease of mass 30

31 (kg/s) y5 y average (5) y up (5) diameter = 2 8 mm y low (5) s(m) Figure 3-5. mass rate with d = 2 8 mm. (kg/s) x 10 5 y11 y average (11) diameter = 2 2 mm y up (11) 2 y (11) low s(m) Figure 3-7. mass rate with d = 2 2 mm. (kg/s) 10 x 106 diameter = 25 mm y18 y average (18) y up (18) y low (18) s(m) Figure 3-9. mass rate with d = 2 5 mm. (kg/s) 9 x 105 diameter = 2 3 mm y10 y average (10) y up (10) 1 y low (10) s(m) Figure 3-6. mass rate with d = 2 3 mm. (kg/s) x 10 6 y12 y average (12) diameter = 2 1 mm 0.5 y up (12) y (12) low s(m) Figure 3-8. mass rate with d = 2 1 mm. (kg/s) 12 x 105 diameter = 210 mm y23 y average (23) y up (23) y low (23) s(m) Figure mass rate with d = 2 10 mm. 31

32 Eruption Column Height(m) 5 x V=0 V=25 V=50 V=100 V max =25 V max =50 V max = Mass Eruption Rate(kg/s) Figure A Comparison of Eruption Column Heights in two cases. rate of pyroclasts also becomes more intense. As for pyroclasts in Figure 3-9 and Figure 3-10 with large diameters, mass rates both decrease to almost zero. In each figure, we observe that four curves are almost coincided and gpc Galerkin solutions converge well to those from Deterministic Case. As for y low (i) and y up (i), there s no deviation relative to y average (i). Figure 3-11 presents a comparison of eruption column height under different wind speeds and mass eruption rates in both two cases. The approximation results are the same. At constant mass eruption rate, as the wind speed increases, eruption column height becomes smaller. While, for the same wind speed, there is a positively proportional relationship between maximum rise height and mass eruption rate. The larger mass eruption rate is, the higher eruption column height is. For gpc Galerkin approach, it provides accurate approximations. 32

33 In summary, if the volcanic plume system is sensitive to uncertainty of re-entrainment parameter, f, there should be same evolution trends but some degree of deviation between states of the system in both two cases. While, in reality, the variation of states in both two cases are almost completely the same. As a result, uncertainty of f fails to vary the behavior of volcanic plume a lot. 33

34 CHAPTER 4 CONCLUSION The model of dynamics of a volcanic plume is highly nonlinear, coupled and of high-dimension. It involves numerous parameters, many of which are determined in a heuristic or empirical fashion. In addition, there are time-varying parameters such bulk density. Therefore, the numerical integral of this system is not a trivial undertaking. For instance, there is scope for singularities in the system of equations(as the axial velocity, U, approaches 0). These difficulties must be appropriately handled. We found that the interaction between a volcanic plume and wind can not only cause enhanced entrainment of air and horizontal momentum but also lead to plume bending. Moreover, there is an inversely proportional relationship between wind speed and maximum plume rise height. The higher the wind speed is, the smaller the maximum plume rise height will be. In addition, as mass eruption rate increases, the eruption column height also becomes larger. In the jet case, maximum plume rise height approximately remains constant across a wide range despite the mass eruption rate increases. In addition, as maximum wind speed increases in the jet case, the range over which maximum plume rise height keeps approximately the same becomes broader. As for volcanic plumes under uncertain parameter, we used a gpc Galerkin approach to simulate system dynamics as an alternative to the brute force Monte Carlo simulation. The results of Monte Carlo simulation greatly depends on quality of the selected samples and generalized statistics are difficult to obtain. Whereas, gpc is commonly used in frame-work for problems involving parametric uncertainty. Overall, gpc simply provides a more elegant solution to the problem under consideration. During the process of realization of this method, it s difficult to solve integral differential equations. The Gauss Quadrature method was used to solve non-standard integral and apply it in conjunction with the Euler Method to solve differential equations of orthogonal polynomial coefficients. However, the Euler method was not sufficiently accurate to 34

35 provide derived convergence of the integration procedure, due to the build up of round off errors from the previous steps. Therefore, a 4th-order Runge-Kutta method was employed to solve the integral differential equations and the approximation results were good. From Figure 3-1 to Figure 3-11, we can conclude that gpc Galerkin method provides a good way to deal with the system involving uncertain parameters. Moreover, randomness of re-entrainment coefficient, f, isn t so sensitive to the system and doesn t vary outcomes a lot in that there should be some error between the approximations under those two cases. In fact, re-entrainment process is significant(especially in convection region), however, the re-entrainment coefficient doesn t affect too much. Although, gpc does not give a stark result, showing high sensitivity of the dynamic system to f, it is till an important result because it shows that f is not a very sensitive parameters which was confirmed by Monte Carlo simulation. It s likely that this result could be used to derive better models for volcanic plume propagation. In future, uncertainty of other parameters or combination of parameters like wind speed, V, will be carried on to check the influence to the development of the plume propagation. 35

36 REFERENCES [1] M. B.R. and T. G.I., Turbulent gravitational convection from maintained and instantaneous sources, Proceedings of The Royal Society A, vol. 234, pp. 1 23, [2] R. Cameron and W. Martin, The orthogonal development of non-linear functionals in series of fourier-hermite functionals, Ann.Math, vol. 48, pp , [3] D.Xiu and G. Karniadakis, The wiener askey polynomial chaos for stochastic differential equations, SIAM, vol. 24, pp , [4] B. G.A., A plume rise model compared with observations, Journal of the Air Pollution Control Association, vol. 15, pp , [5] E. G.G.J, C. S.N., and S. R.S.J., Sedimentation from turbulent jets and plumes, Geo-phys, vol. 101, pp , [6] T. Hewett and J. FAY, Laboratory experiments of smokestack plumes in a stable atmosphere, Atmospheric Environment, vol. 5, pp , [7] U. Konda, T. Singh, P. Singla, and P. Scott, Uncertainty propagation in puff-based dispersion models using polynomial chaos, Environmental Modelling Software, vol. 25, pp , [8] M.Bursik, Effect of wind on the rise height of volcanic plumes, Geophysical Research Letters, vol. 28, pp , [9] W. N., The homogeneous chaos, American Journal of Mathematics, vol. 60, pp , [10] S. P.R. and C. G.T., On the mean path of buoyant, bent-over chimney plumes, Journal of Fluid Mechanics, vol. 28, pp , [11] W. S., Buoyant jets in density-stratified crossflow, Journal of Hydraulic Engineering, vol. 110, pp , [12] R. Sparks, M. Bursik, and S. Carey, Volcanic Plumes. England: John Wiley and Sons, [13] D. Xiu, Numerical Methods for Stochastic Computations. New Jersey: Princeton University Press,

37 BIOGRAPHICAL SKETCH Hongnan Lin was born in Dalian, China. He attended No. 12 high school in Dalian. He went to the Department of Automotive Engineering in 2007, and got his bachelor s degree from Wuhan University of Technology in He then went to graduate school at University of Florida in He completed his Master of Science in Mechanical Engineering in