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This explain a little the notations used in the Hydrology Real Books

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0-RealBookStyleAndNotation

  1. 1. The Real Books: On Style and Notation R. Rigon- Il tavolo di lavoro di Remo wolf Riccardo RigonTuesday, February 26, 13
  2. 2. “Standards are nice if each one of us has his own” Sandro MaraniTuesday, February 26, 13
  3. 3. Notation Objectives Each set of these slides contains a summary, or description, of the communication objectives that want to be achieved. •These slides will explain what a Real Book is •The layout of these slides is explained •They will explain how to write and comment the formulae •The various parts of the single slides are also explained 3 R. RigonTuesday, February 26, 13
  4. 4. Notation Notes on Style For these slides I have chosen to use the Lucida Bright font, at 24 point size, with justified text. The titles have been centred and they have been written in a 36 point Lucida Bright font. The notes are in 18 point Lucida Bright. The references are in 14 point Lucida Bright. The choice of font is linked to the formulae, which are pdf images created with LaTeX (specifically LaTeXit! for Mac), using the Computer Modern font, which is very similar to Lucida Bright. The formulae usually use a 36 point font size. There follows an example. dM H f =P dt f 4 R. RigonTuesday, February 26, 13
  5. 5. Notation Notes on Style Experience teaches that, in order to reproduce the communicative effect of writing by hand on a blackboard, the formulae need to commented. For these slides I have chosen the following method: the formula is “boxed” in red (2 pt) and a red arrow points to an explanation in italics. The three slides show how to dM H comment an equation, term by f =P term. Slowness is necessary to dt f reproduces some optimal flux of information. Conservation of mass of snow 5 R. RigonTuesday, February 26, 13
  6. 6. Notation Notes on Style: an example 1/3The evolution of the water equivalent of snow is found by solving the mass balanceequation: the liquid and solid precipitation less the flow of water due to meltingand sublimation is equal to the variation in the water equivalent during the timestep. The three slides show how to dM H comment an equation, term by f =P term. Slowness is necessary to dt f reproduces some optimal flux of information. Change of mass of the snow in the control volume per unit time 6 R. RigonTuesday, February 26, 13
  7. 7. Notation Notes on Style: an example 2/3The evolution of the water equivalent of snow is found by solving the mass balanceequation: the liquid and solid precipitation less the flow of water due to meltingand sublimation is equal to the variation in the water equivalent during the timestep. dM H f =P dt f Total precipitation 7 R. RigonTuesday, February 26, 13
  8. 8. Notation Notes on Style: an example 3/3The evolution of the water equivalent of snow is found by solving the mass balanceequation: the liquid and solid precipitation less the flow of water due to meltingand sublimation is equal to the variation in the water equivalent during the timestep. The three slides show how to dM H comment an equation, term by f =P term. Slowness is necessary to dt f reproduces some optimal flux of information. Heating of snow divided by the enthalpy of fusion of ice 8 R. RigonTuesday, February 26, 13
  9. 9. Notation Notes on Style: The slides have some The centre of the standard information: a general index slide is white: this is for improved visibility and to avoid wastage of toner The slides have some when printing. The The slide number: standard information: cover slide, on the gives the audience a authors other hand, is all blue reference point with an image. Rigon, 2013 The slides have some For these slides a standard information: Creative Commons the authors of the License has been contribution used (http.cc) 9 R. RigonTuesday, February 26, 13
  10. 10. Notation Other Notes: The formulae have been written using LaTeXit, and they are alive, in the sense that dragging them back to LaTeXit, the code that generated them reappears. Generally, wherever possible, parts of the calculation code or graphic generation code are also given. 10 R. RigonTuesday, February 26, 13
  11. 11. Notation Symbols Where possible, there will be one or more tables listing the symbols used, like the one below: 11 R. RigonTuesday, February 26, 13
  12. 12. Symbols The aim, wherever possible, is to use standard symbols that are different for different quantities. 12 R. RigonTuesday, February 26, 13
  13. 13. Symbols The Name is as in the CF Conventions (http.CF), or is given in that style 13 R. RigonTuesday, February 26, 13
  14. 14. Symbols The unit of measure should always be shown 14 R. RigonTuesday, February 26, 13
  15. 15. Risorse web •http.wp - http://en.wikipedia.org/wiki/Real_Book - Last accessed May, 7, 2009 •http.cc - http://creative.commons.org - Last accessed May, 7, 2009 •http.CF -http://cf-pcmdi.llnl.gov/ 15 R. RigonTuesday, February 26, 13
  16. 16. Bibliography Each set of these slides contains a bibliography. •R. Rosso, Corso di Infrastrutture Idrauliche, Sistemi di drenaggio urbano, The Real book, CUSL, 2002 •S. Swallow - The Real Book CD, •.......... 16 R. RigonTuesday, February 26, 13
  17. 17. Basic Notation for Scalar, Vector and Tensor Fields, and Matrices Bruno Munari - Libri illeggibiliTuesday, February 26, 13
  18. 18. Objectives •In these slides the notational rules used in the Real Books are defined. •In particular, explanation is given on how to write the formulae so that the indices and various graphic aspects can be interpreted univocally. •However these are guidelines that can be violated in practical cases in favor of simplicity of notation. 18 R. RigonTuesday, February 26, 13
  19. 19. Basic Basics Let Ulw be a space-time field. Then Ulw (⌥ , t) = Ulw (x, y, z, t) x is a scalar field. The field can be independent of some space variable or time, which is then omitted. Whether the vector is 2-D or 3-D depends on the context. On the other hand ⌥ x ⌥ Ulw (⌥ , t) = Ulw (x, y, z, t) is a vector field. Other notations for vectors are possible, but not used. ⌥ x ⌥ Ulw (⌥ , t) = Ulw (x, y, z, t) = {Ulw (⌥ , t)x , Ulw (⌥ , t)y , Ulw (⌥ , t)z } x x x 19 R. RigonTuesday, February 26, 13
  20. 20. Basic Basics The components of the vector field can be written as: ⌥ x ⌥ Ulw (⌥ , t) = Ulw (x, y, z, t) = {Ulw (⌥ , t)x , Ulw (⌥ , t)y , Ulw (⌥ , t)z } x x x or, by omitting the dependence on the space-time variables, as: ⌥ x ⌥ Ulw (⌥ , t) = Ulw (x, y, z, t) = {Ulw x , Ulw y , Ulw z } Please take note of the space between the “lw” and coordinate index. Sometimes just the space variable, or the time variable, dependence can be omitted to simplify the notation as: ⌥ x ⌥ Ulw (⌥ , t) = Ulw (x, y, z, t) = {Ulw (t)x , Ulw (t)y , Ulw (t)z } 20 R. RigonTuesday, February 26, 13
  21. 21. Derivatives The normal derivative of the field with respect to the variable x can be expressed in the canonical form: ⇥ d d d d d Ulw (x, t) = Ulw (x, y, z, t) = Ulw (x, t)x , Ulw (x, t)y , Ulw (x, t)z dx dx dx dx dx The partial derivative of the field with respect to the variable x can also be expressed in the canonical form: ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ Ulw (x, t) = Ulw (x, y, z, t) = Ulw (x, t)x , Ulw (x, t)y , Ulw (x, t)z ⇥x ⇥x ⇥x ⇥x ⇥x The partial derivative of the field with respect to the variable x can also be expressed as: ⇥x Ulw (x, t) = ⇥x Ulw (x, y, z, t) = {⇥x Ulw (x, t)x , ⇥x Ulw (x, t)y , ⇥x Ulw (x, t)z } Other forms are possible but not used. 21 R. RigonTuesday, February 26, 13
  22. 22. Gradient and Divergence The gradient of a scalar field is expressed, in the canonical form, as: ⌃ ⇤Ulw (⌃ , t) = {⇥x Ulw (⌃ , t), ⇥y Ulw (⌃ , t), ⇥z Ulw (⌃ , t)} x x x x The divergence of a vector field is expressed, in the canonical form, as: ⌃ x ⇥ · Ulw (⌃ , t) = ⇥x Ulw (⌃ , t)x + ⇥y Ulw (⌃ , t)y + ⇥z Ulw (⌃ , t)z x x x where on the left there is the geometric (coordinate independent) form, and on the right are the gradients in Cartesian coordinates. Vector symbol above the divergence is omitted to remind that the result of the application of the operator to a vector is a scalar. 22 R. RigonTuesday, February 26, 13
  23. 23. Gradient and Divergence The divergence can also be expressed in a more compact form using the Einstein summation convention: ⌃ x ⇥ · Ulw (⌃ , t) = ⇥ i Ulw (⌃ , t)i = ⇥i Ulw (⌃ , t)i x x i {x, y, x} meaning that when an index variable appears twice in a single term, once in an upper (superscript) and once in a lower (subscript) position, there is a summation over all of its possible values. 23 R. RigonTuesday, February 26, 13
  24. 24. Discrete Representation It is interesting to see how scalar and vector fields are represented when they are discretised into a grid Ulw ij,t;k subscript symbol 24 R. RigonTuesday, February 26, 13
  25. 25. Discrete Representation It is interesting to see how scalar and vector fields are represented when they are discretised into a grid Ulw ij,t;k e m p t y space 25 R. RigonTuesday, February 26, 13
  26. 26. Discrete Representation It is interesting to see how scalar and vector fields are represented when they are discretised into a grid Ulw ij,t;k spatial index, first index refers to the cell (center) the second to the cell face, which is then j(i). If only one index is present it is a cell index. 26 R. RigonTuesday, February 26, 13
  27. 27. Discrete Representation It is interesting to see how scalar and vector fields are represented when they are discretised into a grid Ulw ij,t;k temporal i n d e x , preceded by a comma 27 R. RigonTuesday, February 26, 13
  28. 28. Discrete Representation It is interesting to see how scalar and vector fields are represented when they are discretised into a grid Ulw ij,t;k iterative index, preceded by a semicolon 28 R. RigonTuesday, February 26, 13
  29. 29. Discrete Representation Possible alternatives with the same meaning are: Subscripts and superscripts can be omitted, for simplicity, when the meaning of the variable is clear from the context. All of the above are calculated at/across face j of cell i at time step t and it is iteration k. When there is no ambiguity, also the comma can be omitted 29 R. RigonTuesday, February 26, 13
  30. 30. Discrete Representation Possible alternatives with the same meaning are: All the above quantities are calculated for cell i at time step t and it is iteration k 30 R. RigonTuesday, February 26, 13
  31. 31. Discrete Representation When a single index is presented, it can be, for instance with varying i. Therefore, a “vector”, meaning an array of data, can be built: 31 R. RigonTuesday, February 26, 13
  32. 32. Discrete Representation where the symbol is used to identify a column vector 32 R. RigonTuesday, February 26, 13
  33. 33. Discrete Representation where the symbol is used to identify a row type of vector 33 R. RigonTuesday, February 26, 13
  34. 34. Discrete Representation The two symbols or “harpoon” are used for distinguishing this type of vector from the spatial euclidean vectors that have certain particular transformation rules upon rotations in space. 34 R. RigonTuesday, February 26, 13
  35. 35. Discrete Representation If the cell in which the system is discretized is a square in a structured cartesian grid, then the same as above applies, but the cell is identified by the row and column numbers enclosed in ( ): As in the previous cases the comma can be omitted 35 R. RigonTuesday, February 26, 13
  36. 36. Discrete Representation If the cell is a square in a structured cartesian grid, then the same as above applies, but the cell face is identified by the row and column numbers enclosed in ( ) with +1/2 (or -1/2) 36 R. RigonTuesday, February 26, 13
  37. 37. Discrete Representation Cell points and face points in a structured grid: 37 R. RigonTuesday, February 26, 13
  38. 38. Discrete Representation If position or time or iteration are identifiable from the context, or they are unimportant or a non-applicable feature, then they can be omitted means the field Ulw at the face between position i,j and i,j+1 in a cartesian grid at a known time. Ulw i means the field Ulw at cell i in an unstructured grid at a known or unspecified time. ,t Ulw means the field Ulw at a generic cell at time t 38 R. RigonTuesday, February 26, 13
  39. 39. Discrete Representation of Vector Components These are represented with a straightforward extension of what was used with scalars: ⇤ Ulw ij,t;k = {Ulw.x ij,t;k , Ulw.y ij,t;k , Ulw.z ij,t;k } 39 R. RigonTuesday, February 26, 13
  40. 40. Tensors A tensors field is represented by bold letters (either lower or upper case) Ulw (⌃ , t) = Ulw (x, y, z, t) x In this case Ulw is a 3 x 3 tensor field with components: ⇥ Ulw (⇧ , t)xx x Ulw (⇧ , t)xy x Ulw (⇧ , t)xz x ⇤ Ulw (⇧ , t)yx x Ulw (⇧ , t)yy x Ulw (⇧ , t)yz ⌅ x Ulw (⇧ , t)zx x Ulw (⇧ , t)zy x Ulw (⇧ , t)zz x The components are not written with bold characters. 40 R. RigonTuesday, February 26, 13
  41. 41. Tensors However, a tensor by components representation is preferable. So U becomes: Or, when the notation is not ambiguous (not to be confounded with the (ij) element of a grid) simply: The context says if the subscripts refer to a grid point or to the component of a tensors. This is deemed necessary to avoid extra 41 R. RigonTuesday, February 26, 13
  42. 42. Tensors All the rules given for scalars and vectors apply consistently to tensors Tensors are matrices, and matrix notation applies to tensors However, bear in mind that scalars, vectors, and tensors are geometric objects which have properties that are independent of the choice of reference system (i.e. independent of the origin, the base, and the orientation of the space-time vector space) and the coordinate system (i.e. cartesian, cylindrical or curvilinear or other). 42 R. RigonTuesday, February 26, 13
  43. 43. Tensors are matrices, and matrix notation applies to tensors Thus, while tensor indices always refers to space-time, matrix indices do not. Remember also that divergence, gradient and curl are themselves geometric objects and obey the same rules as tensors. By changing coordinate system, they change their components but not their geometric properties. These geometric properties, in fact, should be preserved in a proper discretisation, since they are intimately related to the Conservation Laws of Physics. 43 R. RigonTuesday, February 26, 13
  44. 44. When doing thermodynamics Internal energy can be written, for instance, as : U = U (S, V, Mw ) thus, its differential is: dU ( ) = T ( )dS p( )dV + µw ( )dMw where T ( ) , p( ) and µw ( ) are followed by ( ) to indicate that they are functions and not independent variables. Usually they are also functions of space and time (fields), but this dependence remains implicit. This notation is convenient since the real dependence of each function on the variables S, V, Mw depends on the system under analysis, and is unspecified a-priori. 44 R. RigonTuesday, February 26, 13
  45. 45. Thank you for your attention. G.Ulrici, 2000 ? 45 R. RigonTuesday, February 26, 13

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