Explanation and follow-on work Introduction and aim of activity In this activity, students were asked to imagine themselves as human computers in a vast forecast factory. This idea was originally conceived by one of the pioneers of numerical weather forecasting, Lewis Fry Richardson. Today, most weather forecasts run along similar lines, although the computers are digital and housed in large supercomputing centres. The aim of the activity was to introduce students to the ways in which numerical weather forecasts are made using a very simple, advection equation. The real, forecasting equations are much more complex and harder to solve. Description of what was done and achieved The students were asked to sit with one of the assistants and act as one of the human computers. Information was given to the assistants and students, they then followed a simple worksheet to predict the future temperature at their location from the local flow and horizontal gradients of temperature. When all of the students had computed the local change of temperature a contour map of the predicted temperature was produced and compared to the solution given by a computer implementation of the same problem. The worksheets in this pack resulted from a set of data we computed from a simple implementation of the same problem. As a way of starting the discussion on the activity, we suggest that you could make a comparison between the human and computer solutions and talk about the limitations of the human approach, including how errors which are introduced into the computation may propagate to other parts of the model. You can also compare the time that it took for the human computer to work out a solution and compare this to the very rapid calculation on the digital computer. We also introduce three activities that can be used to help explain the concept of gradients, vectors and algebra which are all elements of the key stage 4 curriculum. The Department of Meteorology The Department of Meteorology was established in 1965 and is internationally renowned for its excellent teaching and research in atmospheric, oceanic and climate science. Walker Institute for Climate System Research The Walker Institute for Climate System Research brings together the University of Reading s unique breadth and depth of climate expertise. The Institute aims to improve understanding of our future climate and its impacts for the benefit of society.
Results from the activity The equation solved was: This equation is known as the advection equation for temperature. The first term on the left hand side of the equation is the rate of change of temperature (T) with time (t). This term depends on the advection of temperature in the east-west (x) and north-south (y) directions by the wind (which has components of u & v in the x & y directions). The advection terms are calculated by multiplying horizontal temperature gradients by the corresponding wind components. We can convert this equation into a form which is easily solved by a computer as follows (note this is relatively simple and was done for convenience, it has some numerical errors and would not be done this way in practice): In this equation, superscripts indicate the time level and subscripts indicate the horizontal postion. Because the temperature field is two-dimensional, horizontal subscripts are only included where important, assume that the temperature is at the same gridpoint unless stated. During the activity, the students used this equation to predict the temperature at each of 4 timesteps, 60 minutes apart. The terms Δt and Δx represent the time-step and distance between grid-points, the grid spacing. We chose to make the grid spacing uniform in both x and y to make the problem simpler.
Teachers sheet Activity 1: Graphing and gradients As part of the activity, students are introduced to the gradient of a line. In this activity they are asked to produce simple graphs of the temperature at a chosen gridpoint over the four timesteps and to practice techniques for calculating the gradient and to interpret what the gradient means. Answers 1. Graph should look like: 2. Discussion should focus on methods to estimate a gradient. Draw a triangle on the graph with a line tangent to the Temperature evolution. Talk to the student about how the gradient changes with time and that this is an estimate for that point only.
3. Temperature gradients are: Time step 2 = (12.4-20.0) / (2*60*60) = -0.001 C s -1 = -0.06 C min -1 = -3.8 C hr -1 Time step 3 = (10.6-15.3) / (2*60*60) = -0.0006 C s -1 = -0.04 C min -1 = -2.4 C hr -1 These gradients are typical changes in temperature for a location in the mid-latitudes. 4. Discuss with the student how to used the gradient to estimate the points between the observation locations. You could get them to calculate a new gradient between points 2 and 3 and then multiply the gradient by 0.5 of a timestep for example: Temperature at time-step 2.5 = Temperature at time-step 2 + (12.4-10.6)/(60*60) * (30*60) = 12.4 0.9 = 11.5 degrees Celsius
Teachers Sheet Activity 2: Vectors and winds In the activity we work in two dimensions, using wind components in the east-west and northsouth directions. In this activity, students are asked to plot and consider vectors based on the two components of the wind. They are also asked to estimate a local wind speed and direction using Pythagoras theorem and trigonometric formulas. Answers 1.-4. The vector plot should look like (shown against grid-point number): 5. The students should simply measure the length of the vector and use the protractor to estimate its bearing. The answers are: Wind speed / metres per second 5.0 5.0 5.0 5.0 10.0 10.0 10.0 10.0 15.0 15.0 15.0 15.0 20.0 20.0 20.0 20.0 Wind direction / bearing from north 150 135 120 90 150 135 120 90 150 135 120 90 150 135 120 90
6. Use Pythagoras theorem (without any measurement) to estimate the wind speed, answers should be the same as those above. 7. Similarly use the trigonometric formula to work out the bearing. Some discussion is needed to work out that the bearing will be 90 degrees minus the angle calculated. This is because all of the winds come from the north-west quadrant (although this is not always the case).
Teachers Sheet Activity 3: Equations and algebra In this activity, students are asked to consider a simple equation which represents the advection equation that we really solved. They are then asked to rearrange the equation and substitute values from the practical into the equation to estimate an unknown (in this case the north-south gradient of temperature). Answers 1. Rearrange the equation to yield: The east-west gradient (the term on the left hand side) can be calculated by simply dividing the temperature gradient in time by the u component of the wind (note that the answers will need to be multiplied by -1). The students should also note that the temperature gradient in the north-south direction can be cancelled since it is zero. 2. This is simply substitution of answers into the formula. The final table should read (where all answers are multiplied by 10-4 ) the small size should be a matter for discussion. When multiplied up to kilometer scale, the values become much larger : 0 0.5 1.0 0.5 0 0.5 1.0 0.5 0 0.5 1.0 0.5 0 0.5 1.0 0.5 3. The student should identify regions of large gradient. The discussion should focus on the fact that fronts are important both because they mark large changes of air temperature and also that precipitation is associated with fronts. Introduce the students to fronts on weather charts (the lines with triangles or circles on surface pressure charts) and get them used to tracking these features online to predict when rain will occur.
Worksheet 1: Graphing and gradients During the visit day you took part in an activity which involved solving a complex mathematical equation using human computers. At the end of the activity, we compared the solution produced by the human computers with one produced by a digital computer (in our case a laptop). In this activity we will compare the two methods in more detail by looking at how temperature changes at one grid-point. Definition grid-point: to solve the equations which predict the weather we divide the atmosphere up into a series of different squares. Each square is represented by a single point, we assume that conditions are the same as at the grid-point everywhere in the square. You will need Graph paper 1. Lets look at what happened at a grid-point which had warm air over it at the start of the simulation. During the forecast, temperatures predicted by our mathematical model were: Time since start/ hours 0 1 2 3 4 Temperature / degrees 20.0 15.3 12.4 10.6 9.4 centigrade Plot a graph showing the temperature on the y-axis and time on the x-axis for the solution provided by the laptop and the solution from the human computers. Describe how temperatures change over time at your grid-point. To help you plot the graph, remember that the difference in time between each calculation is 1 hour or 60 minutes. On the graph label the time axis in seconds rather than minutes as this will be useful later. 2. The equation that we solved in our activity was complicated because it involved both observed values of some quantities and the derivative of others. The term derivative simply means how quickly or slowly a quantity changes in time or space, in other words its gradient. In terms of temperature, we are particularly interested in how fast temperature changes in time. In other words, how many degrees centigrade the temperature changes in one second at a particular location. How could you estimate the gradient of the temperature in the graph you drew for question 1? Discuss your answer with your teacher. 3. Using your method, calculate the gradient of temperature for time-step 2 and 3 on your graph in change of degrees centigrade per second. The answer you get will be very small. Recalculate your gradient for the change of temperature per minute and the change of temperature per hour. In the real world, how fast do you think that temperatures change?
4. In real weather forecasting, it is very important to test our forecasts against observations to check how our models of the atmosphere work. In many cases significant problems can be identified and corrected. Unfortunately, observations are often made at different times to the time-steps of the forecast. Imagine that you made an observation of the temperature 2.5 and 3.5 hours after the start of the forecast. How could you use the gradient to estimate what the model forecasts at these times in order to compare with the observations? Discuss your method with your teacher and then make estimates of the model prediction 2.5 and 3.5 hours after the start of the forecast.
Worksheet 2: Vectors and winds During the visit day you took part in an activity which involved solving a complex mathematical equation using human computers. The wind was a key part of this equation, because it blew warm and cold air around. In the real atmosphere, winds can blow in three different dimensions (east-west, north-south and up-down), so we need to represent them using vectors. In our example we simplified the flow to two dimensions (ignoring wind blowing up and down), but we still needed two different wind components to describe the flow. In this activity, we will draw pictures of the flow in two dimensions. Definition In meteorology we represent the wind components in each direction on the Earth s surface using particular letters of the alphabet. u - means the wind in the east-west or x direction, positive values mean the wind is coming from the east. v - means the wind in the north-south or y direction, positive values mean the wind is coming from the south. You will need Squared paper A protractor 1. The following table shows the two components of the wind at each of the 16 grid points in our model. The first number in each part of the table shows the u component, the second number shows the v component. North 2.5, -4.3 3.5, -3.5 4.3, -2.5 5.0, 0.0 5.0, -8.6 7.0, -7.0 8.6, -5.0 10.0, 0.0 7.5, -13.0 10.5, -10.5 13.0, -7.5 15.0, 0.0 10.0, -17.3 14.0, -14.0 17.3, -10.0 20.0, 0.0 West 2. Using the squared paper, plot a 4x4 grid of points. 3. At each point on your grid, plot a line the same length as the u component of the wind across the paper. From the end of this line, plot a line along the paper the same length as the v component of the wind.
4. Now plot a third line from the point where you started to the end of the second line. You should end up with a triangle which looks like: u v The third line shows the direction and speed (from the length of the line) of the total wind at each grid point. Describe what the flow looks like. 5. The length of the vector you drew is equal to the mean wind speed, on the same scale that you drew the component vectors and the angle the vector makes with due north gives its bearing. For your vector diagrams estimate the wind speed and its bearing. 6. The wind speed can also be estimated mathematically using Pythagoras theorem: The square of the hypotenuse = The sum of the squares of the two other sides Estimate the wind speed using Pythagoras theorem. 7.The bearing of the wind can be estimated using the trigonometric formula: sin (angle) = opposite/hypotenuse Discuss with your teacher how to use this formula to estimate the bearing of the wind (remember we would like to estimate the angle between due north and the wind vector). Estimate the bearing of the wind.
Worksheet 3: Equations and algebra During the visit day you took part in an activity that involved solving a complex mathematical equation using human computers. Having an equation, even as complex as this, is extremely useful as it means that we can describe several aspects of the flow at once. In this activity we will use the equation to estimate information about the fluid that we didn t record during the day. Equation and definitions The equation that we used in our forecast factory can be written: where T is the temperature t is time x is distance in the east-west direction y is distance in the north-south direction u & v are the east-west and north-south components of the wind. Terms which include the symbol are called derivatives. They represent the gradient of temperature in time or in the x and y directions. When performing algebraic transformations on the equation above, derivatives can be treated as a single quantity like x and added or subtracted from each side of the equation. 1. Imagine that we failed to record, but wanted to recover information about the temperature gradient in the east-west direction. Rearrange the equation so that the temperature gradient in the east-west direction is on the left hand side of the equation and everything else is on the right. 2. On the other side of the sheet, you are given values for the temperature at time-step 1 and 2 and the wind speeds in the x and y directions. At time step 1, the temperature gradient in the north-south direction is zero. In the equation this means that you can set anything that is multiplied by it to zero. Using this information, you can estimate the temperature gradient in the east-west direction at every grid-point. The gradient you estimated per metre is small. How large is the gradient per kilometre? 3. Using squared paper, produce a diagram showing the temperature gradient at each gridpoint. Where are north-south temperature gradients largest? In the atmosphere, regions of strong gradients in temperature, where warm and cold air meet, are known as fronts. You can follow the progress of these features on the weather forecast on television or the internet each night. Fronts are marked as red and blue lines with small circles or triangles showing their direction of movement:
Do you know why fronts are important for meteorology? Do some research on the internet to find out why. You can start at: http://www.metoffice.gov.uk/weather/uk/surface_pressure.html Data for activity 3 Here is the data you need to do the computations in activity 3. In each case the table shows the value of the quantity at each grid point. Temperature gradient in time between timestep 1 and 2 / degrees celsuis per second North 0.0-0.0007-0.0017-0.0010 0.0-0.0005-0.0013-0.0008 0.0-0.0003-0.0009-0.0005 0.0-0.0002-0.0004-0.0003 West East-West winds (u component) / metres per second 2.5 3.5 4.3 5.0 5.0 7.0 8.6 10.0 7.5 10.5 13.0 15.0 10.0 14.0 17.3 20.0