Help need to explain the steps to solve a partial differential equation????

[bronbron85]

A horizontal temperature gradient, ∂T/∂x, generated through differential heating can drive an exchange flow whose layer-averaged motion, u , is determined by the momentum equation:

∂u/∂t ~ g a ∂T/∂x H (Equation 1)

where a is the thermal expansion coefficient of water, g is the acceleration due to gravity, and H is the layer depth.

The layer averaged momentum increases until a balance is reached between the rate of thermal energy input and the loss of thermal energy via advection, i.e. at steady state:

∂T/∂t ~ u ∂T/∂x (Equation 2)

Combining equations 1 and 2, and approximating ∂T/∂x by ΔT/L leads to the following estimate for the layer velocity once steady-state is reached.

u ~ (g a ΔT H)^1/2 (Equation 3)

Can someone please explain how to get from Equations 1 and 2 to Equation 3? Please explain all the steps. I do not know anything about solving differential equations but would like to learn. I have probably put too much detail into the science here (I hope this doesn't distract people). It is actually the mathematics I am struggling to understand.

 

[stapel]

I do not know anything about solving differential equations but would like to learn.

The best way to do this would be to enroll in a course on differential equations (after completing college algebra, trigonometry, and calculus, of course, and perhaps a little linear algebra). Since you "do not know anything about ... differential equations", it is unlikely, even if we had all the background information, that any explanation attempted here would make any sense to you. Sorry.