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At least 1,315 records · Page 73

Computational/experimental analysis of three low sonic boom configurations with design modifications

The Euler code, designated AIRPLANE, which uses an unstructured tetrahedral mesh was used to compute near-field sonic boom pressure signatures on three modern low sonic boom configurations: the Mach 2, Mach 3, and Haglund models. The TEAM code which uses a multi-zoned structured grid was used to calculate pressure signatures for the Mach 2 model. The computational pressure signatures for the Mach 2 and Mach 3 models are compared with recent experimental data. The computed pressure signatures were extracted at distances less than one body length below the configuration and extrapolated to the experimental distance. The Mach 2 model was found to have larger overpressures off-ground-track than on-ground-track in both computational and experimental results. The correlations with the experiment were acceptable where the signatures were not contaminated by instrumentation and model-support hardware. AIRPLANE was used to study selected modifications to improve the overpressures of the Mach 2 model.

Cliff, Susan E.↗

A Regional Analysis of Cloudy Mean Spherical Albedo over the Marine Stratocumulus Region and the Tropical Atlantic Ocean

Since clouds are the largest variable in Earth's radiation budget, it is critical to determine both the spatial and temporal characteristics of their radiative properties. This study examines the relationships between cloud properties and cloud fraction in order to supplement grid scale parameterizations. The satellite data used in this study is from three hourly ISCCP (International Satellite Cloud Climatology Project) and monthly ERBE (Earth Radiation Budget Experiment) data on a 2.50 x 2.50 latitude-longitude grid. Mean cloud spherical albedo, the mean optical depth distribution and cloud fraction are examined and compared off the coast of California and the mid-tropical Atlantic for July 1987 and 1988. Individual grid boxes and spatial averages over several grid boxes are correlated to Coakleys (1991) theory of reflection for uniform and broken layered cloud and to Kedem, et al.(1990) findings that rainfall volume and fractional area of rain in convective systems is linear. Kedem's hypothesis can be expressed in terms of cloud properties. That is, the total volume of liquid in a box is a linear function of cloud fraction. Results for the marine stratocumulus regime indicate that albedo is often invariant for cloud fractions of 20% to 80%. Coakley's satellite model of small and large clouds with cores (1 km) and edges (100 in) is consistent with this observation. The cores maintain high liquid water concentrations and large droplets while the edges contain low liquid water concentrations and small droplets. Large clouds are just a collection of cores. The mean optical depth (TAU) distributions support the above observation with TAU values of 3.55 to 9.38 favored across all cloud fractions. From these results, a method based upon Kedem, et al. theory is proposed to separate the cloud fraction and liquid water path (LWP) calculations in a general circulation model (GCM). In terms of spatial averaging, a linear relationship between albedo and cloud fraction is observed. For tropical locations outside the Intertropical Convergence Zone (ITCZ), results of cloud fraction and albedo spatial averaging followed that of the stratus boxes containing few overcast scenes. Both the ideas of Coakley and Kedem, et al. apply. Within the ITCZ, the grid boxes tended to have the same statistical properties as stratus boxes containing many overcast scenes. Because different dynamical forcing mechanisms are present, it is difficult to devise a method for determining subgrid scale variations. Neither of the theories proposed by Kedem, et al. or Coakley works well for the boxes with numerous overcast scenes.

Ginger, Kathryn M.↗

Assimilation of Gridded Terrestrial Water Storage Observations from GRACE into a Land Surface Model

Observations of terrestrial water storage (TWS) from the Gravity Recovery and Climate Experiment (GRACE) satellite mission have a coarse resolution in time (monthly) and space (roughly 150,000 km(sup 2) at midlatitudes) and vertically integrate all water storage components over land, including soil moisture and groundwater. Data assimilation can be used to horizontally downscale and vertically partition GRACE-TWS observations. This work proposes a variant of existing ensemble-based GRACE-TWS data assimilation schemes. The new algorithm differs in how the analysis increments are computed and applied. Existing schemes correlate the uncertainty in the modeled monthly TWS estimates with errors in the soil moisture profile state variables at a single instant in the month and then apply the increment either at the end of the month or gradually throughout the month. The proposed new scheme first computes increments for each day of the month and then applies the average of those increments at the beginning of the month. The new scheme therefore better reflects submonthly variations in TWS errors. The new and existing schemes are investigated here using gridded GRACE-TWS observations. The assimilation results are validated at the monthly time scale, using in situ measurements of groundwater depth and soil moisture across the U.S. The new assimilation scheme yields improved (although not in a statistically significant sense) skill metrics for groundwater compared to the open-loop (no assimilation) simulations and compared to the existing assimilation schemes. A smaller impact is seen for surface and root-zone soil moisture, which have a shorter memory and receive smaller increments from TWS assimilation than groundwater. These results motivate future efforts to combine GRACE-TWS observations with observations that are more sensitive to surface soil moisture, such as L-band brightness temperature observations from Soil Moisture Ocean Salinity (SMOS) or Soil Moisture Active Passive (SMAP). Finally, we demonstrate that the scaling parameters that are applied to the GRACE observations prior to assimilation should be consistent with the land surface model that is used within the assimilation system.

GRACE↗

Nonlinear grid error effects on numerical solution of partial differential equations

Finite difference solutions of nonlinear partial differential equations require discretizations and consequently grid errors are generated. These errors strongly affect stability and convergence properties of difference models. Previously such errors were analyzed by linearizing the difference equations for solutions. Properties of mappings of decadence were used to analyze nonlinear instabilities. Such an analysis is directly affected by initial/boundary conditions. An algorithm was developed, applied to nonlinear Burgers equations, and verified computationally. A preliminary test shows that Navier-Stokes equations may be treated similarly.

Dey, S. K.↗

Initial results from a mesoscale atmospheric simulation system and comparisons with the AVE-SESAME I data set

The development of a comprehensive mesoscale atmospheric simulation system (MASS) is described in detail. The modeling system is designed for both research and real-time forecast applications. The 14-level numerical model, which has a 48 km grid mesh, can be run over most of North America and the adjacent oceanic regions. The model employs sixth-order accurate numerics, generalized similarity theory boundary-layer physics, a sophisticated cumulus parameterization scheme, and state of the art analysis and initialization techniques. Examples of model output on the synoptic and subsynoptic scales are presented for the AVE-SESAME I field experiment on 10-11 April 1979. The model output is subjectively compared to the observational analysis and the LFM II output on the synoptic scale. Subsynoptic model output is compared to analyses generated from the AVE-SESAME I data set.

Kaplan, M. L.↗

The CMC/3DPNS computer program for prediction of three-dimension, subsonic, turbulent aerodynamic juncture region flow. Volume 2: Users' manual

The CMC fluid mechanics program system was developed to transmit the theoretical solution of finite element numerical solution methodology, applied to nonlinear field problems into a versatile computer code for comprehensive flow field analysis. Data procedures for the CMC 3 dimensional Parabolic Navier-Stokes (PNS) algorithm are presented. General data procedures a juncture corner flow standard test case data deck is described. A listing of the data deck and an explanation of grid generation methodology are presented. Tabulations of all commands and variables available to the user are described. These are in alphabetical order with cross reference numbers which refer to storage addresses.

Manhardt, P. D.↗

Objective analysis of tidal fields in the Atlantic and Indian Oceans

An objective analysis technique has been developed to extrapolate tidal amplitudes and phases over entire ocean basins using existing gauge data and the altimetric measurements which are now beginning to be provided by satellite oceanography. The technique was previously tested in the Lake Superior basin. The method has now been developed and applied in the Atlantic-Indian ocean basins using a 6 deg x 6 deg grid to test its essential features. The functions used in the interpolation are the eigenfunctions of the velocity potential (Proudman functions) which are computed numerically from a knowledge of the basin's bottom topography, the horizontal plan form and the necessary boundary conditions. These functions are characteristic of the particular basin. The gravitational normal modes of the basin are computed as part of the investigation, they are used to obtain the theoretical forced solutions for the tidal constituents, the latter provide the simulated data for the testing of the method and serve as a guide in choosing the most energetic modes for the objective analysis. The results of the objective analysis of the M2 and K1 tidal constituents indicate the possibility of recovering the tidal signal with a degree of accuracy well within the error bounds of present day satellite techniques.

Sanchez, B. V.↗

Analysis of subgrid models using direct and large-eddy simulations of isotropic turbulence

Direct and large eddy simulations of forced and decaying isotropic turbulence have been performed using a pseudospectral and a finite-difference code. Subgrid models that include a one-equation subgrid kinetic energy model with and without a stochastic backscatter forcing term and a new scale similarity model have been analyzed in both Fourier space and physical space. The Fourier space analysis showed that the energy transfer across the cutoff wavenumber k(sub c) is dominated by local interaction. The correlation between the exact and the modeled (by a spectral eddy viscosity) nonlinear terms and the subgrid energy transfer in physical space was found to be quite low. In physical space, a similar correlation analysis was carried out using top hat filtering. Results show that the subgrid stress and the energy flux predicted by the subgrid models correlates very well with the exact data. The scale similarity model showed very high correlation for reasonable grid resolution. However, with decrease in grid resolution, the scale similarity model became more uncorrelated, when compared to the kinetic energy subgrid model. The subgrid models were then used for large-eddy simulations for a range of Reynolds number. It was determined that the dissipation was modeled poorly and that the correlation with the exact results was quite low for all the models. In general, for coarse grid resolution, the scale similarity model consistently showed very low correlation while the kinetic energy model showed a relatively higher correlation. These results suggest that to use the scale similarity model relatively fine grid resolution may be required, whereas, the kinetic energy model could be used even in coarse grid.

Menon, S.↗

Ground-based grasslands data to support remote sensing and ecosystem modeling of terrestrial primary production

Estimating terrestrial net primary production (NPP) using remote-sensing tools and ecosystem models requires adequate ground-based measurements for calibration, parameterization, and validation. These data needs were strongly endorsed at a recent meeting of ecosystem modelers organized by the International Geosphere-Biosphere Program's (IGBP's) Data and Information System (DIS) and its Global Analysis, Interpretation, and Modelling (GAIM) Task Force. To meet these needs, a multinational, multiagency project is being coordinated by the IGBP DIS to compile existing NPP data from field sites and to regionalize NPP point estimates to various-sized grid cells. Progress at Oak Ridge National Laboratory (ORNL) on compiling NPP data for grasslands as part of the IGBP DIS data initiative is described. Site data and associated documentation from diverse field studies are being acquired for selected grasslands and are being reviewed for completeness, consistency, and adequacy of documentation, including a description of sampling methods. Data are being compiled in a database with spatial, temporal, and thematic characteristics relevant to remote sensing and global modeling. NPP data are available from the ORNL Distributed Active Archive Center (DAAC) for biogeochemical dynamics. The ORNL DAAC is part of the Earth Observing System Data and Information System, of the US National Aeronautics and Space Administration.

Olson, R. J.↗

Four-Dimensional Oceanic and Atmosperic Data Assimilation with Tropical Rainfall Measuring Mission Data

An oceanic data assimilation system which allows to utilize the forthcoming Tropical Rainfall Measuring Mission (TRMM) data has been developed and applied to the Pacific Ocean to produce the velocity field. The assimilated data will be indispensable to examine the effects of rainfall and its variability on the structure and circulation of the tropical oceans and to assess the impact of global warming due to the increase of carbon dioxide on the ocean circulation system and the marine pollution caused by oil spill and ocean damping of radionuclide. The data will also provide the verification for the oceanic and ocean-atmosphere coupled General Circulation Models (GCM's). The system consists of oceanic GCM, analysis scheme and data. In the system the flow field has been determined to be physically consistent with the observed density field and the sea surface winds derived from the Special Sensor Microwave Imagery (SSM/I) data which drive the ocean current. The time integration has been performed for five years until the flow field near the surface attained the steady state starting from the rest ocean with observed temperature and salinity fields, and the SSM/I surface wind velocity. The resultant flow field showed high producibility of the system. Especially the flow near the ocean surface agreed well with available observed data. The system, for the first time, succeeded to produce the eastward subtropical current which has been discovered in the joint investigation on Kuroshio current (CSK) in the 1960s. To verify the quality of the flow field a trajectory analysis has been carried out and compared with the Algos buoy data. BRIEF DESCRIPTION OF THE DATA ASSIMILATION SYSTEM ## Oceanic GCM and analysis scheme--The basic equations are much the same as used for the GCM's, except for the Newtonian damping terms introduced into the prediction equations for the potential temperature and salinity to maintain these fields as observed. The C grid of 2'lat. by 2'long. in horizontal and the 11 vertical levels are applied to the entire Pacific Ocean. At the east and west ocean boundaries the periodic boundary conditions are applied creating fictitious ocean there. The SMAC Method is used to increase the accuracy of mass conservation. * Data--The JODC temperature and salinity data obtained from 1906 to 1988 are used in the system between Long.100'E. and 60'W. The surface wind data are derived from the SSM/I data by Dr-R. Atlas of NASA/GSFC. The data set contains every 6 hours data from July 1987 to June 1989 on the grid of 2'lat. by 2.5'long. The averaged for the whole period and then interpolated into the 2'lat. by 2'long. grid data are used to force the system. The sea bottom topography data was based on the General Bathymetric Chart of the Ocean (GEBCO) supplied by the Canadian Hydrographic Service under contract with the International Hydrographic Organization and International Oceanographic Commission of UNESCO.

Takano, Kenji↗

Convective Heat Transfer in the Reusable Solid Rocket Motor of the Space Transportation System

This simulation involved a two-dimensional axisymmetric model of a full motor initial grain of the Reusable Solid Rocket Motor (RSRM) of the Space Transportation System (STS). It was conducted with CFD (computational fluid dynamics) commercial code FLUENT. This analysis was performed to: a) maintain continuity with most related previous analyses, b) serve as a non-vectored baseline for any three-dimensional vectored nozzles, c) provide a relatively simple application and checkout for various CFD solution schemes, grid sensitivity studies, turbulence modeling and heat transfer, and d) calculate nozzle convective heat transfer coefficients. The accuracy of the present results and the selection of the numerical schemes and turbulence models were based on matching the rocket ballistic predictions of mass flow rate, head end pressure, vacuum thrust and specific impulse, and measured chamber pressure drop. Matching these ballistic predictions was found to be good. This study was limited to convective heat transfer and the results compared favorably with existing theory. On the other hand, qualitative comparison with backed-out data of the ratio of the convective heat transfer coefficient to the specific heat at constant pressure was made in a relative manner. This backed-out data was devised to match nozzle erosion that was a result of heat transfer (convective, radiative and conductive), chemical (transpirating), and mechanical (shear and particle impingement forces) effects combined.

Ahmad, Rashid A.↗

Application of FUN3D and CFL3D to the Third Workshop on CFD Uncertainty Analysis

Two Reynolds-averaged Navier-Stokes computer codes - one unstructured and one structured - are applied to two workshop cases (for the 3rd Workshop on CFD Uncertainty Analysis, held at Instituto Superior Tecnico, Lisbon, in October 2008) for the purpose of uncertainty analysis. The Spalart-Allmaras turbulence model is employed. The first case uses the method of manufactured solution and is intended as a verification case. In other words, the CFD solution is expected to approach the exact solution as the grid is refined. The second case is a validation case (comparison against experiment), for which modeling errors inherent in the turbulence model and errors/uncertainty in the experiment may prevent close agreement. The results from the two computer codes are also compared. This exercise verifies that the codes are consistent both with the exact manufactured solution and with each other. In terms of order property, both codes behave as expected for the manufactured solution. For the backward facing step, CFD uncertainty on the finest grid is computed and is generally very low for both codes (whose results are nearly identical). Agreement with experiment is good at some locations for particular variables, but there are also many areas where the CFD and experimental uncertainties do not overlap.

Rumsey, C. L.↗

Using NASA Environmental Data to Enhance Public Health Decision Making

The Universities Space Research Association at the NASA Marshall Space Flight Center is collaborating with the University of Alabama at Birmingham (UAB) School of Public Health and the Centers for Disease Control and Prevention (CDC) to address issues of environmental health and enhance public health decision making by utilizing NASA remotely sensed data and products. The objectives of this collaboration are to develop high-quality spatial data sets of environmental variables, and deliver the data sets and associated analyses to local, state and federal end-user groups. These data can be linked spatially and temporally to public health data, such as mortality and disease morbidity, for further analysis and decision making. Three daily environmental data sets have been developed for the conterminous U.S. on different spatial resolutions for the time period 2003-2008: (1) spatial surfaces of estimated fine particulate matter (PM2.5) exposures on a 10-km grid utilizing the US Environmental Protection Agency (EPA) ground observations and NASA s MODerate-resolution Imaging Spectroradiometer (MODIS) data; (2) a 1-km grid of Land Surface Temperature (LST) using MODIS data; and (3) a 12-km grid of daily Solar Insolation (SI) and maximum and minimum air temperature using the North American Land Data Assimilation System (NLDAS) forcing data. These environmental data sets will be linked with public health data from the UAB REasons for Geographic And Racial Differences in Stroke (REGARDS) national cohort study to determine whether exposures to these environmental risk factors are related to cognitive decline and other health outcomes. These environmental datasets and public health linkage analyses will be made available to public health professionals, researchers and the general public through the CDC Wide-ranging Online Data for Epidemiologic Research (WONDER) system and through peer reviewed publications. To date, two of the data sets have been released to the public in CDC WONDER, Daily Air Temperature and Heat Index for years 1979-2010, and Daily Fine Particulate Matter (PM2.5) air quality measures for years 2003-2008. These data in CDC WONDER can be aggregated to the county-level, state-level, or regional-level as per users need and downloaded in tabular, graphical, and map formats. The summary statistical output are available to web and app developers via the WONDER Application Programming Interface (API). The linkage of these data with the CDC WONDER system provides a significant addition to CDC WONDER, allowing public health researchers and policy makers to better include environmental exposure data in the context of other health data available in CDC WONDER online system. It also substantially expands public access to NASA environmental data, making their use by a wide range of decision makers feasible.

Al-Hamdan, Mohammad↗

Flight Performance Feasibility Studies for the Max Launch Abort System

In 2007, the NASA Engineering and Safety Center (NESC) initiated the Max Launch Abort System Project to explore crew escape system concepts designed to be fully encapsulated within an aerodynamic fairing and smoothly integrated onto a launch vehicle. One objective of this design was to develop a more compact launch escape vehicle that eliminated the need for an escape tower, as was used in the Mercury and Apollo escape systems and what is planned for the Orion Multi-Purpose Crew Vehicle (MPCV). The benefits for the launch vehicle of eliminating a tower from the escape vehicle design include lower structural weights, reduced bending moments during atmospheric flight, and a decrease in induced aero-acoustic loads. This paper discusses the development of encapsulated, towerless launch escape vehicle concepts, especially as it pertains to the flight performance and systems analysis trade studies conducted to establish mission feasibility and assess system-level performance. Two different towerless escape vehicle designs are discussed in depth: one with allpropulsive control using liquid attitude control thrusters, and a second employing deployable aft swept grid fins to provide passive stability during coast. Simulation results are presented for a range of nominal and off-nominal escape conditions.

Tarabini, Paul V.↗

Coupled Fluid-Structure Interaction Analysis of Solid Rocket Motor with Flexible Inhibitors

Flexible inhibitors are generally used in solid rocket motors (SRMs) as a means to control the burning of propellant. Vortices generated by the flow of propellant around the flexible inhibitors have been identified as a driving source of instabilities that can lead to thrust oscillations in launch vehicles. Potential coupling between the SRM thrust oscillations and structural vibration modes is an important risk factor in launch vehicle design. As a means to predict and better understand these phenomena, a multidisciplinary simulation capability that couples the NASA production CFD code, Loci/CHEM, with CFDRC's structural finite element code, CoBi, has been developed. This capability is crucial to the development of NASA's new space launch system (SLS). This paper summarizes the efforts in applying the coupled software to demonstrate and investigate fluid-structure interaction (FSI) phenomena between pressure waves and flexible inhibitors inside reusable solid rocket motors (RSRMs). The features of the fluid and structural solvers are described in detail, and the coupling methodology and interfacial continuity requirements are then presented in a general Eulerian-Lagrangian framework. The simulations presented herein utilize production level CFD with hybrid RANS/LES turbulence modeling and grid resolution in excess of 80 million cells. The fluid domain in the SRM is discretized using a general mixed polyhedral unstructured mesh, while full 3D shell elements are utilized in the structural domain for the flexible inhibitors. Verifications against analytical solutions for a structural model under a steady uniform pressure condition and under dynamic modal analysis show excellent agreement in terms of displacement distribution and eigenmode frequencies. The preliminary coupled results indicate that due to acoustic coupling, the dynamics of one of the more flexible inhibitors shift from its first modal frequency to the first acoustic frequency of the solid rocket motor. This insight could have profound implications for SRM and flexible inhibitor designs for current and future launch vehicles including SLS.

Yang, H. Q.↗

Overflow Juncture Flow Computations Compared with Experimental Data

NASA’s Transformational Tools and Technologies Project (T(cu.) is supporting a substantial effort to investigate the formation and origin of separation bubbles found on wing-body juncture zones: the Juncture Flow experiment. The first phase of the Juncture Flow experiment, performed in NASA Langley’s 14- by 22-Foot Subsonic Tunnel, has been completed. This paper documents the CFD analysis done in conjunction with the experiment. Comparisons between CFD simulations and wind tunnel experimental results will be shown. Oil flow results, surface pressure cuts, velocity profiles, and Reynolds stress profiles will be compared. Preliminary results analyzing the effect of grid resolution, wind tunnel walls, and turbulence models on the above data will be presented. The results are not meant to be a validation study, but more as an evaluation of the current status of Reynolds averaged Navier Stokes (RANS) CFD simulations.

Lee, Henry C.↗

Geometry Modeling for Unstructured Mesh Adaptation

The quantification and control of discretization error is critical to obtaining reliable simulation results. Adaptive mesh techniques have the potential to automate discretization error control, but have made limited impact on production analysis workflow. Recent progress has matured a number of independent implementations of flow solvers, error estimation methods, and anisotropic mesh adaptation mechanics. However, the poor integration of initial mesh generation and adaptive mesh mechanics to typical sources of geometry has hindered adoption of adaptive mesh techniques, where these geometries are often created in Mechanical Computer- Aided Design (MCAD) systems. The difficulty of this coupling is compounded by two factors: the inherent complexity of the model (e.g., large range of scales, bodies in proximity, details not required for analysis) and unintended geometry construction artifacts (e.g., translation, uneven parameterization, degeneracy, self-intersection, sliver faces, gaps, large tolerances be- tween topological elements, local high curvature to enforce continuity). Manual preparation of geometry is commonly employed to enable fixed-grid and adaptive-grid workflows by reducing the severity and negative impacts of these construction artifacts, but manual process interaction inhibits workflow automation. Techniques to permit the use of complex geometry models and reduce the impact of geometry construction artifacts on unstructured grid workflows are models from the AIAA Sonic Boom and High Lift Prediction are shown to demonstrate the utility of the current approach.

Park, Michael A.↗

Analysis of faults and pit chains in Noctis Labyrinthus: Implications for early extension and possible magmatic plumbing

Noctis Labyrinthus has been a region of disputed origin due to its complexity and poor understanding of how various processes and mechanisms may have combined to form it. The surface is an integrated record of intensive tectonic activity expressed by a multiple extended sets of dip-slip faults oriented in different directions, and thought to have acted on this region over its history. These faults are always coalescent to pits and pit chains displaying a complicated geological history in the region. To understand this geological history, we mapped the surface features in Noctis Labyrinthus using the High-Resolution Stereo Camera (HRSC) onboard Mars Express ND2 nadir channel basemaps, and we adapted the Digital Terrain Map (DTM) from the Mission Experiment Gridded Data Record (MEGDR) of Mars Orbiter Laser Altimeter (MOLA) onboard Mars Global Surveyor (MGS) for the topography. We have investigated the spatial distribution and trend of fault systems, the pit chains' morphology, and the correlation between these two types of features. Our results show three fault systems: i) NS and NNE-SSW, ii) EW and ENE-WSW, and iii) NNW-SSE and NW. The analysis of the faults trending, cross-cutting correlation and the superimposition led to identify multiple intersections between these faults that have been alongside with the reactivations of some inherited faults. We interpreted the first system of fault to be related to coeval lateral extension, generated by regional stress tensor, which is probably related to the slight bending of Valles Marineris within two phases of bidirectional extension. The second system of faults has been generated by the radial oblate stress tensor related to the formation of the small shield volcanoes in Syria Planum. However, the third system is likely related to the external driving process, probably in the Tharsis province. We classified pits in four evolutionary stages based on their morphometric attributes. We believe that the formation of the pit chains in Noctis Labyrinthus is related to a surface collapse after a pressure drop related to the magma chamber deflation associated with Syria Planum volcanic province. We propose a deformational model based on early extension and magmatic plumbing as driving processes for the formation of Noctis Labyrinthus.

Mayssa El Yazidi↗