Appendix A

[59] This appendix documents the solution of ordinary differential equation (4) to obtain equation (5) described in this paper. Equation (4) is expressed as

Rearrange and integrate equation (4) so that

.

(A1) D

Let x = (T^e_w - T_w) and then dx = d(T^e_w - T_w). Substitute x and dx into equation (A1) where:

,

(A2) D

which equals

.

(A3) D

Take the exponent of both sides so that equation (A3) equals

;

(A4) D

note that C₂ equals .

[60] Substitute back for x so that equation (A4) equals

(A5) D

If we solve equation (A5) at the initial condition of t = 0, then

.

(A6) D

Introduce the assumption that a given air temperature change, T_a [T], eventually causes an equivalent water temperature change, where the water temperature change is expressed as the difference between the equilibrium and actual water temperatures, thus

.

(A7) D

Substitute T_a in equation (A5) so that

.

(A8) D

By rearranging equation (A8), the formula becomes equation (5), which is

[61] Acknowledgments. The U.S. Geological Survey (USGS) and South Florida Water Management District jointly funded this study. Eric Swain (USGS, Miami, Florida) helped derive equations. Mark Zucker (USGS, Miami, Florida) provided review comments and identified statistical methods appropriate for application in this study. Robert Renken (USGS, Miami, Florida), Amy Swancar (USGS, Tampa, Florida) and three anonymous Water Resources Research reviewers provided comments that improved the manuscript quality. Review comments from Water Resources Research Associate Editor Michiaki Sugita are gratefully acknowledged for substantially improving the manuscript quality.