Long \sqrt in two lines

Some suggestions

• Don't use eqnarray -- it's badly deprecated. If you need to align consecutive equations on some chosen marker (say, an = symbol), use the align environment of the amsmath package.

• The term \exp is a math operator and doesn't take an argument. Really.

• Don't over-use \left and \right to auto-size parentheses. Among other things, using \left and \right introduces extra white-space, which isn't desirable here, is it?

• Instead of \bigg(\!-\!\frac..., write \biggl(-\frac.... (And, use \biggr) instead of just \bigg).) TeX's spacing rules treat - as a unary operator if it's preceded by an object to type math-open; \biggl( is such an object, whereas \bigg( is not.

• The equation is simply too wide to fit inside the text block, it has to be broken up into two lines. I suggest you use a split environment inside an equation environment for this purpose. And, use \biggl\{ in the first row and \biggr\}^{1/2} in the second to denote the start and end of the material whose square root is being taken.

\documentclass{article}
\usepackage{amsmath} % for 'split' environment
\begin{document}
\begin{equation}
\begin{split}
y&=\pm\biggl\{
   \frac{Q}{2\pi \sigma_y\sigma_z u C(x,y,z) } 
   \exp\biggl(-\frac{y^2}{2\sigma_y^2}\biggr) \\
 &\qquad\quad \times\biggl[ 
        \exp\biggl(-\frac{(z-H)^2}{2\sigma_z^2}\biggr)+ 
        \exp\biggl(-\frac{(z+H)^2}{2\sigma_z^2}\biggr)
  \biggr] 
  \biggr\}^{\!1/2}
\end{split}
\end{equation}
\end{document}

Given the twocolumn constraint, multline is still an option:

\documentclass[twocolumn]{article}
\usepackage{amsmath} % for 'multline' environment
\begin{document}

And a multline equation ...
\begin{multline}
y=\pm\biggl\{
   \frac{Q}{2\pi \sigma_y\sigma_z u C(x,y,z) } 
   \exp\biggl(-\frac{y^2}{2\sigma_y^2}\biggr) \\
  \times\biggl[ 
        \exp\biggl(-\frac{(z-H)^2}{2\sigma_z^2}\biggr)+ 
        \exp\biggl(-\frac{(z+H)^2}{2\sigma_z^2}\biggr)
  \biggr] 
  \biggr\}^{\!1/2}
\end{multline}

\end{document}