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?