As silicic water-rich magma ascends to the surface and decompresses, volatiles exsolve. If ex-solved gas remains trapped in the magma, volume fraction of gas increases and the magma inflates. As a result, magma fragmentation occurs and the flow changes from bubbly flow to gas-pyroclast flow, leading to explosive eruptions which form massive columns (Fig. 1). In contrast, if loss of gas from the magma occurs, inflation of the magma is suppressed and the magma does not fragment up to the vent, leading to effusive eruptions which form lava domes or lava flows (Fig. 1). Therefore loss of gas from the magma is a key process which causes diverse eruption styles of silicic magmas. In this paper, the effects of vertical relative motion between gas and liquid (vertical escape of gas) on diversity of eruption styles are investigated on the basis of a model for 1-dimensional steady flow in volcanic conduits. In our model, the vertical relative motion between the gas and the liquid is allowed and a transitional region ('permeable flow region') is introduced between the bubbly flow region and the gas-pyroclast flow region. In this region, both the gas and the liquid are continuous phases, allowing the efficient vertical escape of gas through the permeable structure.

　The features of the conduit flow with the relative motion between the gas and the liquid are described by non-dimensional numbers:

where μ is the liquid viscosity, q is the magma flow rate, ρ1 is the liquid density, g is the acceleration due to gravity, rc is the conduit radius, k is the magma permeability, μg is the gas viscosity, φ(crit) is the critical gas volume fraction for fragmentation, and q(max) is the maximum mass flow rate, which is expressed by q(max)≡P(f0)/√(n0RT). Here P(f0) is the pressure at the time of magma fragmentation for the flow without the relative motion, no is the initial H2O content, R is the gas constant, and T is the magma temperature. The parameter α represents the ratio of effects of wall friction and gravitational load, the parameter γ represents the effect of conduit conductivity r2(c)/μ, and the parameter ε is defined as the ratio of effects of liquid-wall friction force and liquid-gas interaction force in the permeable flow region, and represents the efficiency of gas escape from magma. When the relative motion is taken into account, the pressure at the time of magma fragmentation (Pf) is analytically expressed by

The magnitude of Pf decreases as the magma flow rate (α) decreases or the efficiency of gas escape (ε) increases, because the effect of gas escape suppresses the increase in the gas volume fraction accompanied by magma ascent. When α is so small or ε is so large that Pf is below the atmospheric pressure (Pa), the flow reaches, the vent before fragmentation. On the other hand, when α is so large or ε is so small that Pf is greater than Pa, the flow reaches the vent after fragmentation. The steady solutions of conduit flow in which the flow reaches the vent before and after fragmentation correspond to effusive and explosive eruptions, respectively. The problem of the 1-dimensional steady conduit flow model is formulated as a problem to find a non-dimensional magma flow rate α as a function of the parameters related to magma properties and geological conditions (e.g., γ and ε) under given boundary conditions. A graphical method to systematically find α is proposed. In the graphical method, the curves which represent the variations of the length of the region before fragmentation (Lb) positive sign downwards and that of the region after fragmentation (Lg) as a function of α (Lb curve and Lg curve) are used (Fig. 2). The steady solution of conduit flow for a given total length of the conduit (L(total)) is obtained by the relationship of Lb(α) + Lg(α) = L(total). The steady solution can be graphically obtained by finding the position where a vertical bar with the length of L(total) contacts with Lb and Lg curves (Fig. 2). When the bar is located in the region where Lg = 0, the solution corresponds to effusive eruptions. On the other hand, when the bar is located in the region where Lg > 0, the solution corresponds to explosive eruptions (Fig. 2).

　The numbers and the types of the steady solutions of conduit flow (i.e., the assemblage of the steady solutions) largely depend on parameters related to magma properties and geological conditions. On the basis of the graphical method, the relationship between the assemblage of the steady solutions and the magma properties or the geological conditions is investigated through the following four steps. In the first step, the characteristics of Lg curve are described using the critical values of ε (ε*) and α (α1, α2 and α3), which are given by

When ε is larger than ε* , Lg = 0 independent of α. On the other hand, when ε < ε* , the region where Lg > 0 exists in the range of α1 < α < α3;α1 < α < α2 corresponds to the regions where Lg increases with increasing α and Lg is independent of α, and α2 < α < α3 corresponds to the region where Lg decreases with increasing α (Fig. 2). In the second step, the characteristics of Lb+Lg are described using the critical values of Lb + Lg, L1, L2, L3, L(max) and Lp. Here L1, L2 and L3 are the values of Lb+Lg at α = α1, α2 and α3, respectively, L(max) is the maximum value of Lb+Lg, and Lp is the value of Lb+Lg in the limit of α → 0. These critical length are given by

where P0 is the pressure at magma chamber, P(f2) is the value of Pf at α = α2, n(f2) is the gas mass fraction at Pf = P(f2), and n(fm) is the gas mass fraction at Pf = P(f0)/(ε + 1). In the third step, the relationships among the critical lengths (L1, L2, L3, L(max), and Lp), L(total) and the assemblage of the steady solutions are systematically investigated. Finally, in the fourth step, those relationships are summarized using a simple regime map in the parameter space of ε, γ and L(total) (Fig. 3). The regime map illustrates the complex relationship between the assemblage of the steady solutions and the magma properties or the geological conditions. According to the regime map, only a single solution of effusive eruptions exists when ε is larger than ε*. On the other hand, when ε < ε*, the assemblage of the steady solutions is classified into the following five types: (1) Ef, (2) Ex, (3) Ef+Ex, (4) Ex+Ex, and (5) Ef+Ex+Ex, where Ef and Ex represent the solutions of effusive and explosive eruptions, respectively. Here (3), (4) and (5) represent that there exist the multiple steady solutions.

　When the effect of liquid viscosity change during magma ascent is taken into account, we can define the values of ε in lower (εL) and upper (εH) regions of a conduit with low (μL) and high (μH) viscosities, respectively. According to our analytical result, when εL < ε* < εH, the effusive solution continues to exist even if the pressure at magma chamber changes substantially. In contrast, when εL < εH < ε*, the transition between the effusive and explosive solutions is possible with the slight change in the pressure at magma chamber. The observed eruption styles and the estimated magma properties and geological conditions for well-documented eruptions indicate that most values of μL and μH for the eruptions in which only a effusive eruption style was observed show that εL < ε* < εH, whereas most values of μL, and μH for the eruptions in which both effusive and explosive eruption styles were observed show that εL < εH < ε* (Fig. 4). A good consistency between .the analytical result and the observations indicates that the complex transition between effusive and explosive eruption styles observed in nature can be explained on the basis of our analytical result.

Fig.1. Mechanism for diverse eruption styles and conduit flow model in this paper.

Fig.2. Graphical method to obtain steady solutions of conduit flow using Lb curve and Lg curve.

Fig.3. Regime map for assemblage of steady solutions of conduit flow.

Fig.4. Comparison between analytical result and observations. (a) Relationships among k, μL/r2c and εL/ε*; (b) relationships among k, μH/r2c and εH/ε*. UN-Unzen (1991-1995); ME-Merapi (1986-); SH-Shivaluch (2001-2004); SN-Santiaguito (1922-2002); HE-Mt. St. Helens (1980); SO-Soufriere Hills (1995-1999); CO-Colina (1988-); SK-Salcurajima (1914); AR-Arenal (1968-).

　珪長質マグマが上昇し噴火にいたる際,マグマからの脱ガスの程度の違いにより噴火様式が爆発的であったり非爆発的であったりするのはどのような機構によるのであろうか?本論文は,この様な疑問に対し,一次元定常火道流の解析的表現を求め,様々なパラメタの組み合わせがどのような噴火様式と対応するのかを系統的に調べることを目的としている.

　本論文は7章からなる.第1章はイントロダクションであり,まず,過去の一次元定常火道流の数値的・解析的研究が紹介される.マグマとガスの大きな相対運動を許すモデルを数値的に解いた結果からは,鉛直方向の脱ガスが容易になることによって様々な噴火様式が生じることが示されているが,マグマ物性,マグマ溜りの圧力,火道の形状などの様々なパラメタと噴火様式の関係が系統的に調べられているとは言いがたい.一方,Koyaguchi(2005)は解析解を用いて各パラメタと噴火様式の関係を視覚的に理解しやすい作図法を導入したが,マグマとガスの相対運動が考慮されていなかった.本論文では,Koyaguchi(2005)の方法を発展させ,マグマとガスの相対運動がある場合の解析解を系統的に調べている.

　第2章では一次元定常火道流モデルの定式化が行われる.これまでの一次元定常流火道モデルでは,火道内の流れをマグマが破砕する前の気泡流の領域とマグマ破砕後の噴霧流の領域に分けて解析することが多かった.これに対し本論文で用いるモデルでは,Yoshida and Koyaguchi(1999)によって提案された浸透流領域を,気泡流領域と噴霧流領域の2領域の間に設けた.浸透流領域では液体成分とガス成分が大きな相対速度を持つことにより,鉛直方向の効率的な脱ガスが可能となる.浸透流領域でのガス流が層流の場合と乱流の場合の両者が考慮されている.

　第3章では,第2章で定式化された一次元定常火道流を数値的に解くことにより得られる解の一般的な性質が述べられる.数値解には爆発的噴火と非爆発的噴火に対応する2種類の解が見られる.火道壁とマグマの間の粘性応力,マグマに作用する重力,液体マグマとガスの相互作用力などが,流れの各領域において異なる割合でバランスしている.バランスする主要な力の内,寄与の大きなものを残すことによって,浸透流領域における力学的釣り合いの近似的表現が得られる.

　第4章では,Koyaguchi(2005)と同様な手続きにより,気泡流領域の長さLb,噴霧流領域の長さLg,破砕圧Pfの解析的表現が導かれる.その際,火道壁の粘性応力と重力効果の比であるα,火口での圧力と破砕圧の比であるβ,αとβの比であるγ,粘性応力と液体とガスの相互作用の比であるεなどの無次元量が導入され,それら無次元量とLb,Lg,およびPfの関係が詳細に調べられる.更に,Koyaguchi(2005)の作図法が導入される.

　第5章では,Lb,Lgと,火道の全長Ltotal,無次元パラメタα,β,γ,εの関係を系統的に調べ,さらに,パラメタ空間における様々な定常解の存在条件が一望できるRegime mapと存在条件を探索するフローチャートが導入される.このフローチャートは本論文の大きな成果である.

　第6章では,第5章で示された各パラメタと定常解の関係を用い,マグマ溜りの圧力変化が定常解をどのように変化させるかが議論される,また,粘性の効果,ガスの流速が上昇し乱流に移行した場合の解の変化,解析解を得るために導入した近似の影響が議論される.解析解によるとεがある閾値以上では爆発的噴火に対応する解が存在しないが,閾値以下では爆発的噴火の多重解や非爆発的噴火に対応する解が存在し得る.9つの噴火事例について無次元パラメタεを求め,この解析解の特徴が概ね満たされていることを示した.

　第7章は全体のまとめである.

　本論文の構成は先行研究であるKoyaguchi(2005)を踏まえているが,Yoshida and Koyaguchi(1999)を参考にして気泡流領域と噴霧流領域の間に浸透流領域を設けることにより,先行研究では欠けていた液体成分とガス成分の大きな相対運動を導入している.これにより非爆発的噴火解が存在するためには鉛直方向の脱ガスが重要な役割を果たしていることを明らかにしている.また,先行研究では考慮されていない乱流の影響の考察や無次元パラメタεを用いた観測値との比較が新たに行われている.

　なお,本論文の第2章の一部は三谷典子との共著であり,また,第2章から第6章までは小屋口剛博との共著であるが,手法の考案,計算の実行,結果の解釈および考察は論文提出者が主体となって行っており,論文提出者の寄与が十分に大きいと判断できる.

　従って,博士(理学)の学位を授与できるものと認める.