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Calculus


Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus.

 
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The Effect of Various Vertical Discretization Schemes and Horizont...

By: M. Luneva; J. Pickering; D. Storkey; G. Shapiro

Description: University of Plymouth, UK. Results of a sensitivity study are presented from various configurations of the NEMO ocean model in the Black Sea. The standard choices of vertical discretization, viz. z-levels, s-coordinates and enveloped s-coordinates, all show their limitations in the areas of complex topography. Two new hydrid vertical coordinate schemes are presented: the s-on-top-of-z and its enveloped version. The hybrid grids use s-coordinates or en...

Becker, E. and Burkhard, U.: Nonlinear Horizontal Diffusion for GCMs, Mon. Weather Rev., 135, 1439–1454, 2007.; Berntsen, J., Xing, J., and Davies, A. M.: Numerical studies of flow over a sill: sensitivity of the non-hydrostatic effects to the grid size, Ocean Dynam., 59, 1043–1059, 2010.; Besiktepe, S. T., Ünlüata, Ü., and Bologa, A. S. E.: Environmental Degradation of the Black Sea: Challenges and Remedies, Proceedings of the NATO Advanced Research Workshop, Constanta-...

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Simulation of Seismic Waves at the Earth's Crust (Brittle–ductile ...

By: A. Craglietto; B. Farina; F. Poletto; J. M. Carcione

Description: Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (OGS), Borgo Grotta Gigante 42c, 34010 Sgonico, Trieste, Italy. The earth's crust presents two dissimilar rheological behaviors depending on the in situ stress-temperature conditions. The upper, cooler part is brittle, while deeper zones are ductile. Seismic waves may reveal the presence of the transition but a proper characterization is required. We first obtain a stress–strain relation, inc...

Albertsson, A.J., Bjarnason, O., Gunnarsson, T., Balizus, C., and Ingason, K.: Part III: Fluid Handling and Evaluation, In: Iceland Deep Drilling Project, Feasibility Report OS-2003-007, 33 pp., 2003.; Arcay, D.: Dynamics of interplate domain in subduction zones: influence of rheological parameters and subducting plate age, Solid Earth, 3, 467–488, 2012.; Brogi, A., Lazzarotto, A., Liotta, D., and Ranalli, G.: Extensional shear zones as imaged by reflection seismic lines...

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Parameterization for Subgrid-scale Motion of Ice-shelf Calving Fro...

By: M. Martin; M. Haseloff; R. Winkelmann; T. Albrecht; A. Levermann

Description: Earth System Analysis, Potsdam Institute for Climate Impact Research, Potsdam, Germany. A parameterization for the motion of ice-shelf fronts on a Cartesian grid in finite-difference land-ice models is presented. The scheme prevents artificial thinning of the ice shelf at its edge, which occurs due to the finite resolution of the model. The intuitive numerical implementation diminishes numerical dispersion at the ice front and enables the...

Kenneally, J. P. and Hughes, T. J.: Calving giant icebergs: old principles, new applications, Antarct. Sci., 18, 409–419, 2006.; Bentley, C. R.: The Ross Ice shelf Geophysical and Glaciological Survey (RIGGS): Introduction and summary of measurments performed; Glaciological studies on the Ross Ice Shelf, Antarctica, 1973–1978, Antarct. Res. Ser., 42, 1–53, 1984.; Bueler, E. and Brown, J.: The shallow shelf approximation as a sliding law in a thermomechanically coupled ic...

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Modeling the Thermal Dynamics of the Active Layer at Two Contrasti...

By: J. Weismüller; X. Pan; K. Roth; J. Boike; Q. Yu; U. Wollschläger

Description: Institute of Environmental Physics, Heidelberg University, Im Neuenheimer Feld 229, 69120 Heidelberg, Germany. Employing a one-dimensional, coupled thermal and hydraulic numerical model, we quantitatively analyze high-resolution, multi-year data from the active layers at two contrasting permafrost sites. The model implements heat conduction with the de Vries parameterization, heat convection with water and vapor flow, freeze-thaw transition parameterized...

Bittelli, M., Ventura, F., Campbell, G., Snyder, R., Gallegati, F., and Pisa, P.: Coupling of heat, water vapor, and liquid water fluxes to compute evaporation in bare soils, J. Hydrol., 362, 191–205, 2008.; Buck, A.: New equations for computing vapor pressure and enhancement factor, J. Appl. Meteorol., 20, 1527–1532, 1981.; Boike, J., Roth, K., and Ippisch, O.: Seasonal snow cover on frozen ground: Energy balance calculations of a permafrost site near Ny-Ålesund, Spitsb...

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Restoring Mass Conservation to Shallow Ice Flow Models Over Comple...

By: F. S. Anslow; C. G. Schoof; A. H. Jarosch

Description: Centre for Climate and Cryosphere, Institute of Meteorology and Geophysics, University of Innsbruck, Innsbruck, Austria. Numerical simulation of glacier dynamics in mountainous regions using zero-order, shallow ice models is desirable for computational efficiency so as to allow broad coverage. However, these models present several difficulties when applied to complex terrain. One such problem arises where steep terrain can spuriously lead to large ic...

Huybrechts, P. and Payne, T.: The EISMINT benchmarks for testing ice-sheet models, Ann. Glaciol., 23, 1–12, 1996.; A{\dh}algeirsdóttir, G.: Flow dynamics of Vatnajökull ice cap, Iceland, Mitteilung 181, Versuchsantalt für Wasserbau, Hydrologie und Glaziologie der ETH Zürich-Zentrum, Ph.D. thesis, 178 pp., 2003.; Anslow, F. S., Clark, P. U., Kurz, M. D., and Hostetler, S. W.: Geochronology and paleoclimatic implications of the last deglaciation of the Mauna Kea Ice Cap, H...

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Parameterization for Subgrid-scale Motion of Ice-shelf Calving-fro...

By: M. Haseloff; R. Winkelmann; M. Martin; A. Levermann; T. Albrecht

Description: Earth System Analysis, Potsdam Institute for Climate Impact Research, Potsdam, Germany. A parameterization for the motion of ice-shelf fronts on a Cartesian grid in finite-difference land-ice models is presented. The scheme prevents artificial thinning of the ice shelf at its edge, which occurs due to the finite resolution of the model. The intuitive numerical implementation diminishes numerical dispersion at the ice front and enables the...

%Adcroft, A., Hill, C., and Marshall, J.: Representation of Topography by % Shaved Cells in a Height Coordinate Ocean Model, Monthly Weather Review, 125, % 2293–+, 1997.; Bentley, C. R.: The Ross Ice shelf Geophysical and Glaciological Survey (RIGGS): Introduction and summary of measurments performed; Glaciological studies on the Ross Ice Shelf, Antarctica, 1973–1978, Antarct. Res. Ser., 42, 1–53, 1984.; Bueler, E. and Brown, J.: The shallow shelf approximation as a slid...

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A Model Study of the Energy and Mass Balance of Chhota Shigri Glac...

By: F. Pithan

Description: University of East Anglia, Norwich NR4 7TJ, UK. The impact of climate change on Himalaya mountain glaciers is increasingly subject of public and scientific debate. However, observational data are sparse and important knowledge gaps remain in the understanding of what drives changes in these glaciers' mass balances. The present study investigates the glacier regime on Chhota Shigri, a benchmark glacier for the observation of climate change...

Bhutiyani, M., Kale, V., and Pawar, N.: Long-term trends in maximum, minimum and mean annual air temperatures across the Northwestern Himalaya during the twentieth century, Climatic Change, 85, 159–177, 2007.; Ageta, Y. and Higuchi, K.: Estimation of mass balance components of a summer-accumulation type glacier in the Nepal Himalaya, Geogr. Ann. A, 66, 249–255, 1984.; Aizen, V., Aizen, E., and Nikitin, S.: Glacier regime on the northern slope of the Himalaya (Xixibangma ...

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Numerical Mass Conservation Issues in Shallow Ice Models of Mounta...

By: A. H. Jarosch; F. S. Anslow; C. G. Schoof

Description: Centre for Climate and Cryosphere, Institute of Meteorology and Geophysics, University of Innsbruck, Innsbruck, Austria. Numerical simulation of glacier dynamics in mountainous regions using low-order, shallow ice models is desirable for computational efficiency and their capability of including ice dynamics in estimates of mountain glacier wastage worldwide. However, these models present several difficulties when applied to complex topography. One such prob...

Anslow, F. S., Clark, P. U., Kurz, M. D., and Hostetler, S. W.: Geochronology and paleoclimatic implications of the last deglaciation of the Mauna Kea Ice Cap, Hawaii, Earth Planet. Sc. Lett., 297, 234–248, doi:10.1016/j.epsl.2010.06.025, 2010.; Bueler, E., Lingle, C. S., Kallen-Brown, J. A., Covey, D. N., and Bowman, L. N.: Exact solutions and verification of numerical models for isothermal ice sheets, J. Glaciol....

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Glacier Volume Estimation as an Ill-posed Boundary Value Problem :...

By: G. Kaser; D. B. Bahr; W. T. Pfeffer

Description: Institute of Meteorology and Geophysics, University of Innsbruck, Innrain 52, 6020 Innsbruck, Austria. Estimating a glacier's volume by inferring properties at depth from properties observed at the surface creates an instability that grows exponentially with the size of the glacier. Random errors from this instability can overwhelm the volume calculation unless problematic short spatial wavelengths are specifically excluded. Volume-area scaling inh...

Bahr, D. B.: Global distributions of glacier properties: a stochastic scaling paradigm, Water Resour. Res., 33, 1669–1679, doi:10.1029/97WR00824, 1997a.; Bahr, D. B.: Width and length scaling of glaciers, J. Glaciol., 43, 557–562, 1997b.; Bahr, D. B. and Radić, V.: Significant contribution to total mass from very small glaciers, The Cryosphere, 6, 763–770, doi:10.5194/tc-6-763...

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Of the Gradient Accuracy in Full-stokes Ice Flow Model: Basal Slip...

By: N. Martin; J. Monnier

Description: INSA, 135 Avenue de Rangueil, 31077, Toulouse Cedex 4, France. This work focuses on the numerical assessment of the accuracy of an adjoint-based gradient in the perspective of variational data assimilation and parameter identification in glaciology. We quantify the ability to identify the basal slipperiness for such methods with a non-linear friction law. The complete adjoint problem is solved and a comparison with the so called self-adjoint meth...

Griewank, A., Bischof, C., Corliss, G., Carle, A., and Williamson, K.: Derivative convergence for iterative equation solvers, Optim. Method. Softw., 2, 321–355, 1993.; Balise, M. J. and Raymond, C. F.: Transfer of basal sliding variations to the surface of a linearly viscous glacier, J. Glaciol., 31, 308–318, 1985.; Christianson, B.: Reverse accumulation and attractive fixed points, Optim. Method. Softw., 3, 311–326, 1994.; DassFlow Software: Data assimilation for free-s...

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Verification of the Multi-layer Snowpack Model with Different Wate...

By: L. Schmid; O. Eisen; A. Heilig; M. Lehning; N. Wever; C. Fierz

Description: WSL Institute for Snow and Avalanche Research SLF, Flüelastrasse 11, 7260 Davos Dorf, Switzerland. The widely-used detailed SNOWPACK model has undergone constant development over the years. A notable recent extension is the introduction of a Richards Equation (RE) solver as an alternative for the bucket-type approach for describing water transport in the snow and soil layers. In addition, continuous updates of snow settling and new snow density parametri...

Avanzi, F., Caruso, M., Jommi, C., Michele, C. D., and Ghezzi, A.: Continuous-time monitoring of liquid water content in snowpacks using capacitance probes: a preliminary feasibility study, Adv. Water Resour., 68, 32–41, doi:10.1016/j.advwatres.2014.02.012, 2014.; Baggi, S. and Schweizer, J.: Characteristics of wet-snow avalanche activity: 20 years of observations from a high alpine valley (Dischma, Switzerlan...

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Suddenly, the Energy Generated by the Active Power Deficiency or L...

By: Archil Kokhtashvili
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Atmospheric Waves as Scaling, Turbulent Phenomena : Volume 14, Iss...

By: S. Lovejoy; J. Pinel

Description: Physics, McGill University 3600 University St., Montreal, Que., Canada. It is paradoxical that, while atmospheric dynamics are highly nonlinear and turbulent, atmospheric waves are commonly modelled by linear or weakly nonlinear theories. We postulate that the laws governing atmospheric waves are in fact high-Reynolds-number (Re), emergent laws so that – in common with the emergent high-Re turbulent laws – they are also constrained by scal...

Bleistein, N. and Handelsman, R. A.: Asymptotic Expansions of Integrals, Dover, Mineola, 425 pp., 1986.; Dias, J., Tulich, S. N., and Kiladis, G. N.: An object based approach to assessing tropical convection organization. J. Atmos. Sci., 69, 2488–2504, 2012.; Frank, W. M. and Roundy, P. E.: The Role of Tropical Waves in Tropical Cyclogenesis. Mon. Weather Rev. 134, 2397–2417, 2006.; Frisch, U.: Turbulence: the legacy of A. N. Kolmogorov, Cambridge University Press, Cambr...

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Quasi-geostrophic Turbulence and Generalized Scale Invariance, a T...

By: A. Tuck; I. Tchiguirinskaia; S. Lovejoy; D. Schertzer

Description: Université Paris-Est, Ecole des Ponts ParisTech, LEESU, Marne-la-Vallée, France. Lindborg et al. (2010) claim that the spectrum power law E(k)k−3 on scales ≥600 km obtained with the help of commercial jetliner trajectory deviations (GASP and Mozaic databases) could not be brought into question (Lovejoy et al., 2009a), because this spectrum corresponds to a well known theory of quasi-geostrophic turbulence developed by Charney ...

Adelfang, S. I.: On the relation between wind shears over various intervals, J. Atmos. Sci., 10, 156–159, 1971.; Bolgiano, R.: Turbulent spectra in a stably stratified atmosphere, J. Geophys. Res., 64, 2226–2229, 1959.; Charney, J. G.: On the scale of atmospheric motions, Geophys. Publ. Oslo, 17, 1–17, 1948.; Charney, J. G.: Geostrophic Turbulence, J. Atmos. Sci., 28, 1087–1095, 1971.; Charney, J. G., Fjortoft, R., and {von N}eumann, J.: Numerical integration of the baro...

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Profile of a Low-mach-number Shock in Two-fluid Plasma Theory : Vo...

By: M. Balikhin; M. Gedalin; Y. Kushinsky

Description: Department of Physics, Ben-Gurion University, Beer-Sheva, Israel. Magnetic profiles of low-Mach-number collisionless shocks in space plasmas are studied within the two-fluid plasma theory. Particular attention is given to the upstream magnetic oscillations generated at the ramp. By including weak resistive dissipation in the equations of motion for electrons and protons, the dependence of the upstream wave train features on the ratio of the dispersion l...

de Hoffmann, F. and Teller, E.: Magneto-hydrodynamic shocks, Phys. Rev., 80, 692–703, 1950.; Farris, M., Russell, C., and Thomsen, M.: Magnetic structure of the low beta, quasi-perpendicular shock, J. Geophys. Res., 98, 15285–15294, 1993.; Burgess, D. and Scholer, M.: Microphysics of Quasi-parallel Shocks in Collisionless Plasmas, Space Sci. Rev., 178, 513–533, 2013.; Arnold, V. I.: Ordinary differential equations, MIT Press, Cambridge, MA, USA, 1978.; Bale, S. D., Balik...

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Entropy Production and Multiple Equilibria: the Case of the Ice-al...

By: D. Paillard; B. Dubrulle; C. Herbert

Description: Laboratoire des Sciences du Climat et de l'Environnement, IPSL, CEA-CNRS-UVSQ, UMR 8212, Gif-sur-Yvette, France. Nonlinear feedbacks in the Earth System provide mechanisms that can prove very useful in understanding complex dynamics with relatively simple concepts. For example, the temperature and the ice cover of the planet are linked in a positive feedback which gives birth to multiple equilibria for some values of the solar constant: fully ice-covered...

Arnold, V.: Ordinary Differential Equations, Springer, New York, 1984.; Budyko, M.: The effect of solar radiation variations on the climate of the Earth, Tellus, 21, 611–619, 1969.; Cahalan, R. and North, G.: A Stability Theorem for Energy-Balance Climate Models, J. Atmos. Sci., 36, 1178–1188, 1979.; Chylek, P. and Coakley, J.: Analytical analysis of a Budyko-type climate model, J. Atmos. Sci., 32, 675–679, 1975.; Crafoord, C. and K{ä}ll{é}n, E.: A Note on the Condition ...

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A Semi-implicit, Second Order Accurate Numerical Model for Multiph...

By: T. Esposti Ongaro; A. Neri; S. Carcano; L. Bonaventura

Description: MOX – Modelling and Scientific Computing, Dipartimento di Matematica F. Brioschi, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy. An improved version of the PDAC (Pyroclastic Dispersal Analysis Code, Esposti Ongaro et al., 2007) numerical model for the simulation of multiphase volcanic flows is presented and validated for the simulation of multiphase volcanic jets in supersonic regimes. The present version of PDAC includes second-order tim...

Arakawa, A. and Lamb, V. R.: A potential enstrophy and energy conserving scheme for the shallow water equations., Mon. Weather Rev., 109, 18–36, 1981.; Carcano, S., Bonaventura, L., Neri, A., and Esposti Ongaro, T.: A second order accurate numerical model for multiphase underexpanded volcanic jets, MOX Report 50/2012, MOX – Politecnico di Milano, Milano, 2012.; Colella, P.: Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys., 87, 171–200, 1...

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Modeling the Snow Surface Temperature with a One-layer Energy Bala...

By: C. H. Luce; D. G. Tarboton; J. You

Description: School of Natural Resources, University of Nebraska – Lincoln, Lincoln, Nebraska 68583, USA. Snow surface temperature is a key control on and result of dynamically coupled energy exchanges at the snow surface. The snow surface temperature is the result of the balance between external forcing (incoming radiation) and energy exchanges above the surface that depend on surface temperature (outgoing longwave radiation and turbulent fluxes) and the transport o...

Anderson, E. A.: A Point Energy and Mass Balance Model of a Snow Cover. NOAA Technical Report NWS 19, US Department of Commerce, 150 pp., 1976.; Berg, P. W, and McGregor, J. L.: Elementary Partial Differential Equations, Holden-Day, Oakland, 1966.; Blöschl, G. and Kirnbauer, R.: Point snowmelt models with different degrees of complexity – internal processes, J. Hydrol., 129, 127–147, 1991.; Bras, R. L.: Hydrology, an Introduction to Hydrologic Science, Addison-Wesley: Re...

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Numerical Analysis of Richards' Problem for Water Penetration in U...

By: M. Omidvar; A. R. Ghotbi; D. D. Ganji; A. Barari

Description: Departments of Civil and Mechanical Engineering, Babol University of Technology, Babol, Iran. Unsaturated flow of soils in unsaturated soils is an important problem in geotechnical and geo-environmental engineering. Richards' equation is often used to model this phenomenon in porous media. Obtaining appropriate solution to this equation therefore provides better means to studying the infiltration into unsaturated soils. Available methods ...

Barari, A., Ghotbi, A. R., Farrokhzad, F., and Ganji, D. D.: Variational iteration method and Homotopy-perturbation method for solving different types of wave equations, J. Appl. Sci., 8, 120–126, 2008b.; Barari, A., Omidvar, M., Ganji, D. D., and Tahmasebi Poor, A.: An Approximate solution for boundary value problems in structural engineering and fluid mechanics, J. Math. Problems Eng., Article ID 394103, 1–13, 2008c.; Basha, H. A.: Multidimensional linearized nonsteady...

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Energy-conserving and Hamiltonian Low-order Models in Geophysical ...

By: A. Gluhovsky

Description: Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, Indiana, USA. Arbitrary truncations in the Galerkin method commonly used to derive low-order models (LOMs) may violate fundamental conservation properties of the original equations, causing unphysical behaviors in LOMs such as unbounded solutions. To avoid these, energy-conserving LOMs are developed in the form of coupled Volterra gyrostats, based on analogies between fluid ...

D'Andrea, F. and Vautard, R.: Extratropical low-frequency variability as a low-dimensional problem. Part I: a simplified model, Quart. J. Roy. Meteorol. Soc., 127, 1357-1374, 2001.; D'Anjou, A., Sarasola, C., and Torrealdea, F. J.: On the characterization od different synchronization stages by energy consideration, J. of Phys., 23, 238-251, 2005.; DelSole, T.: Optimally persistent patterns in time-varing fields, J. Atmos. Sci., 58, 1341-1356.; De Swart, H. E.: Low-order ...

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