Last week I had some days off for hiking in the area around Lake Traunsee (Map).
And, by chance, last weekend was also one of the hottest weekends ever measured in Austria. Therefore, the water temperature of Traunsee, which has a serious reputation for being one of the coldest lakes in Austria, was almost cosy 23 degrees centigrade (measured in Gmunden on July 26).
If we had all desired data, could we model (and eventually calculate) the time evolution of the Traunsee temperature? I am speaking of the forward problem, meaning we want to know if it makes sense to have our bathing trunks prepared after having attended a seminar given by the physicist Anton Zeilinger and the philosopher Franz Schuh "How real is reality?" on August 15.
Obviously, this is not an easy task. We would have to know the initial (today’s) temperature distribution of the water in the lake and of the mountains around it, the temperature of the incoming water (river Traun and possible rainfalls), and at least the thermal boundary conditions at the lake surface, influenced by air temperature and the intensity of solar radiation. After having collected the data, we would have to solve the equations of meteorology with a time horizon of two weeks. Some talented persons can do this with a reasonable accuracy, but I cannot.
As mathematical modelling and computational finance is related to the art of simplification, maybe we can solve a toy problem. So, for the sake of simplicity, the model Traunsee is assumed to be one-dimensional with a given (inhomogeneous) initial temperature u(x,0). We further assume that there is no heat transport by convection within the model Traunsee, only by conduction, and that there is no energy loss or gain across the surface of the lake but only by heat flow into the or out of the (zero-dimensional) model beach. The physical parameters of the Traunsee water (density, heat conductivity and specific heat) are assumed to be known and constant.
This is a forward heat conduction problem, which can be easily solved by finite elements or finite difference techniques. As long as you use an implicit time-stepping scheme, you don’t even have to think about stability.
In our model problem, we set the initial temperature of the left half of Traunsee to 17 degrees, and of the right half to 23 degrees.
Then we fix the boundary temperature to 20 degrees and let diffusion work for a while. (As we have not specified the dimensions of 1D Traunsee and of the physical parameters of the water in it, it does not make sense to say "for two weeks".) As we espected it, the final temperature is as follows.
What happens mathematically if we know the final temperature and want to calculate the initial one? In a forthcoming blog entry we will study this problem. The backwards heat equation is an inverse problem for a diffusion operator. Expect nasty things to happen.