estimate air temperature changes from changes in sea surface temperature

Question

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?

Answer 1

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.