Determining the Thermocline in a Stratified Lake

In summary: T0 or does it keep changing depending on when I differentiate it with respect to y?In summary, the temperature profile in a stratified lake will change over the course of the year due to the way heat transfer works. The separation of layers is considered to occur at the thermocline, which is located at the inflection point in the temperature gradient.
  • #1
rico22
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0

Homework Statement



Lakes become stratified in temperature over the course of the year because of the way that heat transfer occurs. In the spring, the temperature of a lake as a function of depth is reasonably constant. During the summer, as heat enters the lake from solar radiation, the upper layers begin to warm. Since warm fluids tend to rise and the warmest water is already on the surface, there is not much mixing in the lake during this time. As a result, a temperature profile like that attached below tends to form.


Fig 1 – Temperature profile in a stratified lake – From Thibodeaux, Environmental Chemodynamics, 2nd ed.

Later in the year, the air becomes cooler than the water and the lake loses heat. Eventually, the surface waters cool enough that they sink, causing the lake to mix during the winter and setting up a new cycle for the following year.

During the time of year that the lake is stratified it is of interest to know the depth at which the separation of layers occurs – because physical processes such as transport of pollutants or nutrients are very different in the different layers. 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.


Homework Equations



[STRIKE]if you need to see the picture it is attached here: [/STRIKE]
https://www.physicsforums.com/attachment.php?attachmentid=57866&d=1365975296

The Attempt at a Solution


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

Then I got the second partial derivative with respect to y which gave me:
ytc=1/4α2t2(Tsurf(t)-T0)e-y2/4αt=0.

I am 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 I am not sure how to differentiate with respect to t when I have Tsurf(t) as a function of time. Any help would be appreciated.
 
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  • #2
rico22 said:
Then I got the second partial derivative with respect to y which gave me:
ytc=1/4α2t2(Tsurf(t)-T0)e-y2/4αt=0.
That's wrong in several ways. Please post all your working to get to that point.
 
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  • #3
from T(y,t)=e^(-y2/4αt)(Tsurf(t)-T0)+T0 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)
 
  • #4
rico22 said:
from T(y,t)=e^(-y2/4αt)(Tsurf(t)-T0)+T0

so for the first partial derivative this becomes (Tsurf(t)-T0)e^(-y2/4αt)(-2/4αt)
No, that would be the result of differentiating wrt y2. Use the chain rule.
 
  • #5
oh wow yeah...
so the first derivative would come out to (Tsurf(t)-T0)e^(-y2/4αt)(-2y/4αt) which then simplifies to (-y/2αt)(Tsurf(t)-T0)e^(-y2/4αt), correct?

And then for the second derivate I would need to product rule this all up...
 
  • #6
rico22 said:
oh wow yeah...
so the first derivative would come out to (Tsurf(t)-T0)e^(-y2/4αt)(-2y/4αt) which then simplifies to (-y/2αt)(Tsurf(t)-T0)e^(-y2/4αt), correct?

And then for the second derivate I would need to product rule this all up...

Yes.
 
  • #7
thank you...
 
  • #8
so for the second partial I get:

(-1/2αt)[Tsurf(t)-T0]e^(-y2/4αt)+(y2/4α2t2)[Tsurf(t)-T0]e^(-y2/4αt)

and the second part asks me to get the partial of this with respect to "t"... when doing this does Tsurf(t) just becomes T'surf(t)?
 
  • #9
You first have to solve for the value of y that makes this expression for the 2nd derivative equal to zero.
 
  • #10
y=±sqrt(2αt)... so know I differentiate this wrt "t", correct?
 
  • #11
rico22 said:
y=±sqrt(2αt)... so know I differentiate this wrt "t", correct?

Yes, except that the negative value for y is located outside the lake water, so you drop that root.
 
  • #12
thank you, I really appreciate you guys help.
 

FAQ: Determining the Thermocline in a Stratified Lake

What is a temperature profile in a lake?

A temperature profile in a lake is a representation of the change in water temperature at different depths in the lake. It shows how the temperature changes from the surface to the bottom of the lake.

Why is the temperature profile in a lake important?

The temperature profile in a lake is important because it affects the distribution of organisms and nutrients within the lake. It also influences the oxygen levels and water circulation, which are crucial for the survival of aquatic life.

What factors can affect the temperature profile in a lake?

The temperature profile in a lake can be affected by various factors, including the time of year, weather conditions, depth of the lake, and the amount of sunlight reaching the water's surface. Human activities, such as pollution and climate change, can also impact the temperature profile.

How is the temperature profile in a lake measured?

The temperature profile in a lake is typically measured using a device called a thermistor profiler. This instrument measures the temperature at different depths in the water and creates a temperature profile graph. Other methods, such as using a thermometer or data buoys, can also be used to measure the temperature profile in a lake.

How does the temperature profile in a lake change over time?

The temperature profile in a lake can change over time due to seasonal variations, weather patterns, and human activities. It can also be affected by natural processes, such as stratification and mixing, which can cause fluctuations in the temperature profile throughout the day or over longer periods.

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