Reply to Discussion of “Perceptual models of uncertainty for socio-hydrological systems: a flood risk change example”*

ABSTRACT

Ertsen discusses the representation of reality and uncertainty in our paper, raising three critical points. In response to the first, we agree that discussion of different interpretations of the concept of uncertainty is important when developing perceptual models – making different uncertainty interpretations explicit was a key motivation behind our method. Secondly, we do not, as Ertsen suggests, deny anyone who is not a “certified” scientist to have relevant knowledge. The elicitation of diverse views by discussing perceptual models is a basis for open discussion and decision making. Thirdly, Ertsen suggests that it is not useful to treat socio-hydrological systems as if they exist. We argue that we act as “pragmatic realists” in most practical applications by treating socio-hydrological systems as an external reality that can be known. But the uncertainty that arises from our knowledge limitations needs to be recognized, as it may impact on practical decision making and associated costs.     

Evolution of Planetesimal Velocities Based on Three-Body Orbital Integrations and Growth of Protoplanets

We obtain the viscous stirring and dynamical friction rates of planetesimals with a Rayleigh distribution of eccentricities and inclinations, using three-body orbital integration and the procedure described by Ohtsuki (1999, Icarus137, 152), who evaluated these rates for ring particles. We find that these rates based on orbital integrations agree quite well with the analytic results of Stewart and Ida (2000, Icarus 143, 28) in high-velocity cases. In low-velocity cases where Kepler shear dominates the relative velocity, however, the three-body calculations show significant deviation from the formulas of Stewart and Ida, who did not investigate the rates for low velocities in detail but just presented a simple interpolation formula between their high-velocity formula and the numerical results for circular orbits. We calculate evolution of root mean square eccentricities and inclinations using the above stirring rates based on orbital integrations, and find excellent agreement with N-body simulations for both one- and two-component systems, even in the low-velocity cases. We derive semi-analytic formulas for the stirring and dynamical friction rates based on our numerical results, and confirm that they reproduce the results of N-body simulations with sufficient accuracy. Using these formulas, we calculate equilibrium velocities of planetesimals with given size distributions. At a stage before the onset of runaway growth of large bodies, the velocity distribution calculated by our new formulas are found to agree quite well with those obtained by using the formulas of Stewart and Ida or Wetherill and Stewart (1993, Icarus106, 190). However, at later stages, we find that the inclinations of small collisional fragments calculated by our new formulas can be much smaller than those calculated by the previously obtained formulas, so that they are more easily accreted by larger bodies in our case. The results essentially support the previous results such as runaway growth of protoplanets, but they could enhance their growth rate by 10-30% after early runaway growth, where those fragments with low random velocities can significantly contribute to rapid growth of runaway bodies.     

The Effect of Tidal Interaction with a Gas Disk on Formation of Terrestrial Planets

We have performed N-body simulation on final accretion stage of terrestrial planets, including the effect of damping of eccentricity and inclination caused by tidal interaction with a remnant gas disk. As a result of runway and oligarchic accretion, about 20 Mars-sized protoplanets would be formed in nearly circular orbits with orbital separation of several to ten Hill radius. The orbits of the protoplanets would be eventually destabilized by long-term mutual gravity and/or secular resonance of giant gaseous planets. The protoplanets would coalesce with each other to form terrestrial planets through the orbital crossing. Previous N-body simulations, however, showed that the final eccentricities of planets are around 0.1, which are about 10 times higher than the present eccentricities of Earth and Venus. The obtained high eccentricities are the remnant of orbital crossing. We included the effect of eccentricity damping caused by gravitational interaction with disk gas as a drag force (“gravitational drag”) and carried out N-body simulation of accretion of protoplanets. We start with 15 protoplanets with 0.2M⊕ and integrate the orbits for 10⁷ years, which is consistent with the observationally inferred disk lifetime (in some runs, we start with 30 protoplanets with 0.1M⊕). In most runs, the damping time scale, which is equivalent to the strength of the drag force, is kept constant throughout each run in order to clarify the effects of the damping. We found that the planets' final mass, spatial distribution, and eccentricities depend on the damping time scale. If the damping time scale for a 0.2M⊕ mass planet at 1 AU is longer than 10⁸ years, planets grow to Earth's size, but the final eccentricities are too high as in gas-free cases. If it is shorter than 10⁶ years, the eccentricities of the protoplanets cannot be pumped up, resulting in not enough orbital crossing to make Earth-sized planets. Small planets with low eccentricities are formed with small orbital separation. On the other hand, if it is between 10⁶ and 10⁸ years, which may correspond to a mostly depleted disk (0.01-0.1% of surface density of the minimum mass model), some protoplanets can grow to about the size of Earth and Venus, and the eccentricities of such surviving planets can be diminished within the disk lifetime. Furthermore, in innermost and outermost regions in the same system, we often find planets with smaller size and larger eccentricities too, which could be analogous to Mars and Mercury. This is partly because the gravitational drag is less effective for smaller mass planets, and partly due to the “edge effect,” which means the innermost and outermost planets tend to remain without collision. We also carried out several runs with time-dependent drag force according to depletion of a gas disk. In these runs, we used exponential decay model with e-folding time of 3×10⁶ years. The orbits of protoplanets are stablized by the eccentricity damping in the early time. When disk surface density decays to ?1% of the minimum mass disk model, the damping force is no longer strong enough to inhibit the increase of the eccentricity by distant perturbations among protoplanets so that the orbital crossing starts. In this disk decay model, a gas disk with 10⁻⁴-10⁻³ times the minimum mass model still remains after the orbital crossing and accretional events, which is enough to damp the eccentricities of the Earth-sized planets to the order of 0.01. Using these results, we discuss a possible scenario for the last stage of terrestrial planet formation.     

Effect of hydrostatic pressure on the lattice parameters of Fe2SiO4 olivine up to 70 kbar

The pressure dependence of the three lattice parameters and unit cell volume of fayalite (Fe₂SiO₄ olivine) was determined by X-ray diffraction under hydrostatic pressures up to 70 kbar. In order to eliminate stress inhomogeneity within a composite material consisting of a specimen mixed with an internal-pressure standard, a liquid (1 : 1 mixture of ethanol and methanol) was used as a pressure-transmitting medium. The isothermal bulk modulus calculated on the basis of the second-order Birch-Murnaghan equation of state gives the values K₀ = 1.19 ± 0.10 Mbar and K₀′ = 7 ± 4, and if we assume K₀′ = 5: K₀ = 1.24 ± 0.02 Mbar. Three axes of fayalite were found to be compressible in the following order, b >c >a. Comparisons with the results obtained under non-hydrostatic compression are made.     

A new technique for measuring diffusion coefficients of cations and anions in ionic crystals

A new technique has been developed for measuring the diffusion coefficient in ionic crystals. Based on Einstein's formula expressing the relation between diffusion coefficient and electric mobility, the electrical impedance of a diatomic ionic crystal is derived theoretically as a function of frequency of the applied electric field. In this method, the diffusion coefficients of both cations and anions are determined simultaneously by fitting the measured impedance to the theoretical relation. This method was applied to NaCl single crystals in the temperature range 370–780°C. The impedance was determined over the frequency range 0.01 Hz to 1 kHz, at constant temperature. The diffusion coefficients thus obtained for NaCl agree reasonably well with previous data by means of a radioactive isotope technique. The activation energies for Na⁺ and Cl^? obtained are 1.97±0.03 eV and 2.08±0.06 eV, respectively, in the intrinsic region, and 0.92±0.02 eV and 1.06±0.02 eV in the extrinsic region. It was discovered that there are diffusion blocks of approximately 2 μm width, which obstruct free migration of ions in a single crystal.     

Protracted Active Surge Phase in the High Arctic (Svalbard): Dynamic Development and Morphological Impact

Glacial surges in Svalbard are protracted and characterized by individual dynamic evolution, in contrast to many other areas, which calls for a subdivision of the classic two‐phased surge cycle. A dominating part of the ice masses seem to have a surge potential and this represents a considerable challenge for palaeoclimatic studies. Glaciological and geological models therefore need to be coupled. The issue is discussed with Fridtjovbreen glacier as an example. This ice mass is one of few glaciers studied throughout a surge cycle. It was active for 12 years (1991–2002) and represents the most protracted surge documented. The maximum advance rate was 4.2 m day^?1, its maximum extent was reached after seven years, its run‐out distance was 4 km, and the relocated ice filled 5 km² of the fjord. Intense subglacial thrusting occurred during various stages, including part of the ice‐front retreat, as shown by sub‐bottom profiling data from 2002. A six‐stage model is presented and processes are discussed with emphasis on the ice‐front retreat with transition to the quiescent phase. Although the surge mechanism itself is unrelated to climate, climatic conditions obviously play a major role in the course of a surge. During the surge, the ice mass made a dramatic impression in the landscape, but 10 years after the maximum extent, there is little onshore evidence of the event.     

The micrometers of the Carlsberg Automatic Meridian Circle

A new moving slit star micrometer has been developed for the 18 cm Carlsberg Automatic Meridian Circle on La Palma. A zenith mean error of 0 _inf. ^sup 15 for one transit and a limiting magnitude of 14 _inf. ^supm 8 is obtained. It has a high efficiency and is well suited for improving the FK5 system and extending it to much fainter stars. A new collimator alignment micrometer and the necessary calibrations are also described.     

Variation in azimuth of the Carlsberg Automatic Meridian Circle

The variation in azimuth of the Carlsberg Automatic Meridian Circle has been investigated using azimuth marks, pairs of circumpolar stars and three polarissimae. Preliminary results from these three methods are intercompared and some conclusions drawn about their reliability and future use.     

Moraine development at a small High‐Arctic valley glacier: Rieperbreen,Svalbard

Ice‐cored lateral and frontal moraine complexes, formed at the margin of the small, land‐based Rieperbreen glacier, central Svalbard, have been investigated through field observations and interpretations of aerial photographs (1936, 1961 and 1990). The main focus has been on the stratigraphical and dynamic development of these moraines as well as the disintegration processes. The glacier has been wasting down since the ‘Little Ice Age’ (LIA) maximum, and between 1936 and 1990 the glacier surface was lowered by 50–60 m and the front retreated by approximately 900 m. As the glacier wasted, three moraine ridges developed at the front, mainly as melting out of sediments from debris‐rich foliation and debris‐bands formed when the glacier was polythermal, probably during the LIA maximum. The disintegration of the moraines is dominated by wastage of buried ice, sediment gravity‐flows, meltwater activity and some frost weathering. A transverse glacier profile with a northward sloping surface has developed owing to the higher insolation along the south‐facing ice margin. This asymmetric geometry also strongly affects the supraglacial drainage pattern. Lateral moraines have formed along both sides of the glacier, although the insolation aspect of the glacier has resulted in the development of a moraine 60 m high along its northern margin. Copyright © 2001 John Wiley & Sons, Ltd.     

Formation of terrestrial planets in a dissipating gas disk with Jupiter and Saturn

We have performed N-body simulations on final accretion stage of terrestrial planets, including the eccentricity and inclination damping effect due to tidal interaction with a gas disk. We investigated the dependence on a depletion time scale of the disk, and the effect of secular perturbations by Jupiter and Saturn. In the final stage, terrestrial planets are formed through coagulation of protoplanets of about the size of Mars. They would collide and grow in a decaying gas disk. Kominami and Ida [Icarus 157 (2002) 43-56] showed that it is plausible that Earth-sized, low-eccentricity planets are formed in a mostly depleted gas disk. In this paper, we investigate the formation of planets in a decaying gas disk with various depletion time scales, assuming disk surface density of gas component decays exponentially with time scale of τ_gas. Fifteen protoplanets with are initially distributed in the terrestrial planet regions. We found that Earth-sized planets with low eccentricities are formed, independent of initial gas surface density, when the condition (τ_cross+τ_growth)/2?τ_gas?τ_cross is satisfied, where τ_cross is the time scale for initial protoplanets to start orbit crossing in a gas-free case and τ_growth is the time scale for Earth-sized planets to accrete during the orbit crossing stage. In the cases satisfying the above condition, the final masses and eccentricities of the largest planets are consistent with those of Earth and Venus. However, four or five protoplanets with the initial mass remain. In the final stage of terrestrial planetary formation, it is likely that Jupiter and Saturn have already been formed. When Jupiter and Saturn are included, their secular perturbations pump up eccentricities of protoplanets and tend to reduce the number of final planets in the terrestrial planet regions. However, we found that the reduction is not significant. The perturbations also shorten τ_cross. If the eccentricities of Jupiter and Saturn are comparable to or larger than present values (∼0.05), τ_cross become too short to satisfy the above condition. As a result, eccentricities of the planets cannot be damped to the observed value of Earth and Venus. Hence, for the formation of terrestrial planets, it is preferable that the secular perturbations from Jupiter and Saturn do not have significant effect upon the evolution. Such situation may be reproduced by Jupiter and Saturn not being fully grown, or their eccentricities being smaller than the present values during the terrestrial planets' formation. However, in such cases, we need some other mechanism to eliminate the problem that numerous Mars-sized planets remain uncollided.