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Consider a body of water, where the change in temperature for a given time period can be estimated by:

$$ \frac{dTw}{dt} = \frac{Q_{net} \times A}{\rho \times C_{pw} \times V}$$

where $dt$ is the time interval, $Q_{net}$ is the net heat flux through the surface of the water body, $A$ is the surface area, $\rho$ is the water density, $C_{pw}$ is the specific heat capacity, and $V$ is the volume of water being influenced by $Q_{net}$. The change in temperature can then be determined by:

$$ dTw = \frac{Q_{net} \times A \times dt}{\rho \times C_{pw} \times V}$$

However, what if I knew the change in water temperature, $dTw$ but wanted to calculate the change in air temperature that caused this change. Air temperature is included in the equation for $Q_{net}$. Therefore, I was thinking that it might be valid to re-arrange this equation to find $T_{a}$?

$Q_{net}$ can be defined as:

$$Q_{net} = Q_{s} + Q_{lin} - Q_{lout} - Q_{e} - Q_{h}$$

where $Q_{s}$ is incoming short-wave radiation, $Q_{lin}$ is incoming long-wave radiation $Q_{lout}$ is outgoing long-wave radiation, $Q_{e}$ is latent heat flux, and $Q_{h}$ is sensible heat flux.

Given that we know $Q_{s}$ and $Q_{lout}$ (let's assume we do) I can re-arrange the equation above to:

$$ \frac{dT_{w} \times \rho \times C_{pw} \times V}{A \times dt} = Q_{s} + Q_{lin} - Q_{lout} - Q_{e} - Q_{h}$$

which can then be written as:

$$ \frac{dT_{w} \times \rho \times C_{pw} \times V}{A \times dt} - Q_{s} + Q_{lout} = Q_{lin} - Q_{e} - Q_{h}$$

So, everything on the LHS is known, and the unknown air temperature (Ta) is on the RHS within these three terms.

Given that:

$$ Q_{lin} = \varepsilon \times \sigma \times T_{a}^{4}$$ $$ Q_{lin} = [clf + (1-clf)(9.36 \times 10^{-6} \times Ta^{2})]\times \sigma \times Ta^{4}$$

and

$$ Q_{h} = \rho \times Ca \times CH \times U(Tw - Ta)$$ $$ Q_{e} = \rho \times Lv \times CE \times U(qs - qa)$$

is it possible to substitute these into the initial equation to find what Ta should be? Is the method that I'm following correct? If so, how would I incorporate the last thee terms into the main equation? I would eventually need to have Ta on its own on one side of the equation.

If this method is not correct, can someone suggest a method for estimating air temperature changes if one knows the change in water temperature, wind speed, solar radiation and relatve humidity but missing air temperature?

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  • $\begingroup$ There is very little "thermal mass" in air, and a lot of heat associated with radiant heat transfer from the sun. The changes in water temperature are more likely caused by solar radiation while the changes in air temperature are more likely caused by heat transfer from the water to the air. It seems unlikely that warm air would heat ocean water to a degree that comes even close to what solar radiation heat transfer is doing. $\endgroup$ – David White Aug 6 '15 at 3:48
  • $\begingroup$ If, after reading my comment, you STILL want to calculate Ta with your 4 equations, you don't need to separate the variable to obtain a value. There are trial-and-error techniques that can get you an answer quickly without going through "algebra hell" to get the answer. $\endgroup$ – David White Aug 6 '15 at 3:52
  • $\begingroup$ Could you please elaborate on the trial and error techniques that you mention? $\endgroup$ – Emma Tebbs Aug 7 '15 at 7:41
  • $\begingroup$ Do you mean something similar to an optimization routine that would find the best value for it? $\endgroup$ – Emma Tebbs Aug 7 '15 at 10:51
  • $\begingroup$ Sort of. The equations can be set up as a least squares problem, and an Excel add-in called Solver can manipulate the unknowns to arrive at your answer, usually in a very few seconds. $\endgroup$ – David White Aug 7 '15 at 14:14
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Start by assessing the relative importance of each $Q$ term. Assume some order of magnitude values for the unknowns (based on experience and/or intuition) and see how big $Q_{lin}$ is compared to e.g. $Q_s$. Do that for all $Q$ terms and hope some will be negligible so they can be discarded. My guess is that $Q_{lin}$ and $Q_e$ are insignificant.

You are then left with an ordinary differential equation of the form: $$\alpha\frac{dT_w}{dt} = Q + \beta(T_w-T_a)$$ where $\alpha$ and $\beta$ are constants involving parameters defined in your equations.

This equation is easily solved: $$Q+\beta(T_w(t)-T_a)=\exp(\frac{\beta}{\alpha}t)+K$$

determine the integration constant with an initial condition, rewrite for $T_a$ and you can determine for what values of your parameters the value of $T_a$ should be.

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