Heated Earth Domain Model

Transcription

1 Heated Earth Domain Model 10/24/14 Understanding the geometry of the Earth and the physics of solar radiation are required to build a simulation of the heated Earth To reduce your research burden, I have collected here some background information and formulas that you might find useful Note that I am not mandating an implementation Note also, that the formula have not been thoroughly vetted You still need to make sure that they are correct and make sense in your situation If you find errors or find other useful background information that you want to add, please post to the forum Inputs Requested input grid spacing in degrees: 1 gstentative 180 Simulated time interval for recomputations, in minutes: The Earth The radius of the Earth in meters: R = The circumference of the Earth in meters: C = 2 R = The surface area of the Earth in meters 2 : A 4 R 2 14 = = The surface area of the Earth visible to the Sun at any one time in meters 2 : A Day = 2 R 2 = The Grid 14 The surface of the Earth is modeled as a grid The actual grid spacing in degrees: gs = argmax( i, ( 1 i gs tentative ) ( mod( 180, gs) = 0) ); that is, the largest integer less than or equal to gs tentative that evenly divides 180 Note that gs {, 1 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180} The proportion of the Equator used by one unit of grid spacing: p = gs 360 The number of columns in the grid: cols = ( 360 ) ( gs) The number of rows in the grid: rows = ( 180 ) gs Number of cells in the grid: N = rows cols The grid itself can be represented as an array: GRID[ 0 ( rows 1), 0 ( cols 1) ] Note that rows count Northward from the South Pole Columns count Westward from the Primary Meridian West neighbor of GRID[,] i j : GRID[, i mod( j + 1, cols) ] East neighbor of GRID[,] i j : GRID[, i mod( j 1 + cols, cols) ] 6 7

2 North neighbor of GRID[,] i j : ( i + 1 rows 1) GRID[ i + 1, j] ; GRID[ i; mod( j + ( cols 2), cols) ] (In this and subsequent formulas, the notation C V 1 ; V 2 is a conditional expression (if-then-else), with C being a Boolean expression and V 1 and V 2 being arithmetic expression of the same type If C evaluates to true, then the overall result is the value of V 1 ; otherwise, it is the value of V 2 ) South neighbor of GRID[,] i j : i 1 0 GRID[ i 1, j] ; GRID[, imodj ( + ( cols 2), cols) ] Latitude Longitude (latlon) Mapping to Grid Indices The Earth is subdivided by parallels of latitude and meridians of longitude One way of specifying the position of a point on the surface of the Earth is by giving its latitude and longitude Latitude ( ) is given in degrees North or South of the Equator We will denote North as the positive direction and South as the negative Longitude ( ) is given by degrees East or West of the Primary Meridian We will denote East as the positive direction and West as the negative The GRID row i corresponding to latitude : gs + ( rows 2) The GRID column j corresponding to longitude : if we define = ( + gs) ( gs) 1 then j can be computed as: < 0 ( 1) ;( cols ( + 1) ) enables the formula to work for longitudes any place in a cell, not just at its origin Cell Properties Each cell in the grid has certain properties used in support of the simulation To specify these, we use the following terminology The grid consists of cells in the shape of isosceles trapezoids The origin of a cell is at its lower left hand corner Its base is the horizontal side of the trapezoid that goes through its origin The cell s row index in GRID: i Its column index in GRID: j The latitude of the origin of cells in row i: = ( i ( rows 2) ) gs The longitude of the origin of cells in column j: = ( j < (( cols) 2) ) d; 360 d, where d = ( j + 1) gs Length of each of the vertical sides of the cell in meters: l v = C p Length of the base of the cell in meters: = cos( ) l Note that this and all subsequent formulas in this section are dependent on the grid coordinates i and j Length of the top of the cell in meters: l t = cos( + gs) 2 The height or altitude of the cell in meters: h = l v 1 4 ( l b l t ) 2 l b v l v

3 The perimeter of the cell in meters: pm = l t + l b + 2 l v From this can be computed the proportion of the perimeter accorded to each of the sides The area of the cell in meters 2 : a = 1 2 ( l t + l b ) h The proportion of the Earth s surface area taken by the cell: ra = a A X-coordinate of the origin of the cells in GRID column j as the number of meters West of the Primary Meridian: x = ( mod( j + ( cols 2), cols) (( cols 2) 1) ) C cols Y-coordinate of the origin of the cells in GRID row i as the number of meters North of the Equator: y = ( i ( rows 2) ) C rows Rotation The Earth rotates West to East Rotational period of the Earth is 24 hours or 1440 minutes Rotational angle in degrees as a function of time t in minutes since the simulation s start: t = mod( t, 1440) GRID column under the Sun at time t: j t = mod( cols ( t 360 ) + ( cols 2), cols) Number of grid cells per (24-hour) time zone (at the Equator): gcptz = cols 24 The GRID columns in daylight at time : j + 5 gcptz j 6 gcptz Physical Units Newton (force): 1 kilogram * meter / sec 2 Joule (energy): one newton meter Watt (power): one joule per second Kelvin (temperature): basic unit of temperature Solar Heating I The Earth is heated primarily by the Sun, which provides energy in the form of radiation This process can be summarized with the following equations The total power the Earth receives from the Sun: P = watts The amount of solor power reaching the Earth per unit of area, also called the Solar Constant: S = watts per meter² The Earth s aldebo (average fraction of energy reflected): a = 03 The Earth s emissivity (average fraction of energy reradiated): e = 0612 The Stefan-Boltzmann constant: = J K 4 m 2 s 1 The average temperature of the Earth in kelvins due to Solar radiation: ( 1 a) S T E = = e 3 17

4 Heat Attenuation The energy received by a cell is attenuated due both to the cell s latitude and the time of day For a planet without tilt, the cell s heat attenuation at noon at latitudes ± : = cos( ) For a planet without tilt, the cell s heat attenuation on the Equator at the Primary Meridian at time t is tprimarymeridian, = t < 90 cos( t ); 0 That is, if at time t, the Sun is visible at the Primary Meridian, then the attenuation there is cos( ); otherwise it is 0 More generally, we need to know what the attenuation is at the Equator at longitude at time t To compute this, we first determine the number of degrees of rotation d that corresponds to: d = ;, where d is always positive and less than or equal to 360 The number of degrees of rotation that is from the spot on the Equator that is currently at noon: d Noon = d t Hence, for a planet without tilt, the cell s heat attenuation on the Equator at longitude at time t is t, = d Noon < 90 cos( d Noon ); 0 For a planet without tilt, when a particular cell at latitude and longitude is in daylight at time t, the cell s combined attenuation is: = cos t, ; otherwise it is 0 Cell Temperature The temperature at a cell is affected by several factors: its current temperature, T, the power it receives from the Sun,, the amount it loses due to cooling,, and the affect of its neighbors, T Neighbors T Sun will always be positive, while T Cool will always be negative Although will always be a non-negative value, its affect on the cell may be to heat it or cool it depending on whether its net contributed temperature is greater or less than T Solar Heating II T Neighbors T Sun T Cool The Earth is heated primarily by the Sun Total power reaching the Earth per meter 2 in watts: S, (from above) Power radiated from the Sun gets converted to heat depending on the thermal conductivity of the Earth Because the Sun is only heating the day side of the Earth, we would expect cells in the simulation to show a marked difference in temperatures between cells on the night and day sides of the grid t

5 Cell Cooling Because the overall temperature of the Earth remains constant, and because the Sun is heating the Earth, this means that the individual cells are responsible for shedding the excess heat As indicated in the previous section, this heat has the average value of Thus, each cell on average must shed this much heat However, cells loose energy T Sun in proportion to their areas and to their current temperatures That is, larger cells emit more energy in a unit of time than smaller ones do, and hotter cells shed more heat than cooler ones do Average grid cell size: = A N Actual grid cell size: a (from above) Relative size factor: = a Diffusion We can think of each cell in the GRID as sharing borders with its neighbors Heat will flow among these cells depending on their relative temperatures and the lengths of there common borders For cells at the poles, one of the neighbors of the cell will be directly across the pole from it, but the amount of shared border is zero This means that even though these cells are triangular, we can go ahead and treat each cell as having four neighbors Proportion of a cell s border shared with its neighbors to the East and West: = p W = l v pm p E Proportion of a cell s border shared with its neighbor to the North: p N = l t pm Proportion of a cell s border shared with its neighbor to the South: p S = l b pm T Neighbors in kelvins can now be computed as follows: T Neighbors = ( p N T N ) + ( p S T S ) + ( p E T E ) + ( p W T W ), where T N, T S, T E and T W are the temperature of the cell s neighbors to the North, South, East and West, respectively Note that there is no need to divide by four because the shared border proportions have already been factored in References