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Temp profiles through partial derivatives

  1. Apr 16, 2013 #1
    1. The problem statement, all variables and given/known data

    The separation of layers is considered to occur at the thermocline, which is defined as the location of the steepest slope in the temperature gradient. Mathematically, this occurs at the inflection point – so the position of the thermocline can be found from the following criterion:

    (1)∂2T/∂y2=0

    where y is the depth (measured from the lake surface) and T is the temperature.

    A mathematical model for temperature as a function of depth y (in m) and time t (in days) is:

    (2)T(y,t)-T0/Tsurf(t)-T0=exp(-y2/4αt)


    where Tsurf(t) is the water temperature of the lake surface at time t, α is a property called the “eddy thermal diffusivity” and T0 is the lake temperature at time zero. Time zero must be chosen to be on a day when the lake temperature is more or less uniform.

    Here are the specific tasks:

    (1) Apply equation (1) to equation (2) and develop an expression for the location ytc of the thermocline as a function of time.

    (2) The speed at which the thermocline moves vtc is defined as

    vtc=∂ytc/∂t

    Use your results from (1) to obtain an expression for vtc as a function of time.


    2. Relevant equations



    3. The attempt at a solution
    First I expressed the function as T(y,t)=e^(-y2/4αt)(Tsurf(t)-T0)+T0....

    from this I know that only the exponential expression contains a "y" so everything else becomes a constant and the last T0 drops off...

    so for the first partial derivative this becomes (Tsurf(t)-T0)e^(-y2/4αt)(-2/4αt)

    then for the second partial its pretty much the same as the first one only now we also have a value of (-2/4αt) so this becomes: (1/4α2t2)(Tsurf(t)-T0)e^(-y2/4αt) ...
    Im not sure if this is the way to go so I just wanted to check to see if I was in the right path. Also for number 2 im not sure how to differentiate with respect to t when I have Tsurf(t) as a function of time. Any help would be appreciated.
     
  2. jcsd
  3. Apr 16, 2013 #2

    haruspex

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