Dynamics of Coastal Models - Manejo Costero

Theory of mixing 231initial height of warm particlefinal height of cold particlecoldwarmheightwarmcoldfinal height of warm particleinitial height of cold particletemperatureFigure 7.2 Schematic of the vertical mixing of heat due to particles interchanging position in thewater column. The vertical temperature profile is shown by thicker full line at some initial time.Two hypothetical particles are shown: one warm and the other cold, with the warm particleinitially near the top of the water column and the cold particle initially near the bottom of thewater column. The particles then interchange positions and the temperature profile changes tothat represented by the thinner line.7.2.2 Fluctuations and diffusionFluids consist of a very large number of particles which have the ability to moverelative to one another in an almost infinity of ways. We say a fluid has a very largenumber of degrees of freedom. So, to describe properly the motion of a fluid we wouldneed to divide it into a huge number of small volumes, or masses, and assign a velocityvector in three dimensions at every time to each of these small masses. In a sense, this isexactly what we would do if we set up a grid for a numerical model of a coastal basinthat had infinitesimally small cells and time steps. In reality, we can only use finite cellsand finite time steps. Within each of these cells and time steps, our model is essentiallyperforming an average over the space in each cell, and over the time that elapses duringa time step. Let us start with the example of the component of velocity in the xdirection (and suppose that we have a one-dimensional world). Our assumption isthat there exists a function u(x, t) which exactly describes the variation of u with x and t.Our model performs an average of u over both x (over a cell) and t (over a time step).This average means an integral over a small volume of space and time and is indicatedby the multiple integral