How do I stack two figures on top of each other, rather than side to side?

Since you're looking to make the two graphs larger, you could (a) increase the widths of the two subfigure environments to, say, 0.75\textwidth and (b) set the widths of the graphs to 1\linewidth, i.e., to the full width of the enclosing subfigure environments. LaTeX will automatically insert a line break between the two subfigures.

\documentclass{article}
\usepackage{subcaption}
\usepackage[demo]{graphicx}  % remove 'demo' option for your real document

\begin{document}
\begin{figure}
\centering
\begin{subfigure}[b]{0.55\textwidth}
   \includegraphics[width=1\linewidth]{Nvariousg1}
   \caption{}
   \label{fig:Ng1} 
\end{subfigure}

\begin{subfigure}[b]{0.55\textwidth}
   \includegraphics[width=1\linewidth]{Nvariousg2}
   \caption{}
   \label{fig:Ng2}
\end{subfigure}

\caption[Two numerical solutions]{(a) Numerical solutions for the small-time system 
with a constant-curvature body shape showing the scaled leading-order veritcal 
reaction force $N_0$ versus the scaled body mass $M$ for various values of $g$. 
Again, $I=M$ for definiteness and $A=0.7$. (b) As for (a) but over a wider range of 
values of $M,I$.}
\end{figure}
\end{document}

An other option is provided here.

\documentclass[smallextended]{svjour3}
\usepackage{lipsum}% Just for this example
\usepackage{graphicx}
\begin{document}

See Figure~\ref{fig:myfig}(a) or~(b). \lipsum[1]

\begin{figure}
  \centering
  \begin{tabular}{@{}c@{}}
    \includegraphics[width=.7\linewidth,height=75pt]{example-image-a} \\[\abovecaptionskip]
    \small (a) An image
  \end{tabular}

  \vspace{\floatsep}

  \begin{tabular}{@{}c@{}}
    \includegraphics[width=.6\linewidth,height=100pt]{example-image-b} \\[\abovecaptionskip]
    \small (b) Another image
  \end{tabular}

  \caption{This is a figure caption}\label{fig:myfig}
\end{figure}

\lipsum[2]

\end{document}

Just add '\\' between the subfigures.

\begin{figure}[H]
\centering
   \begin{subfigure}[b]{0.3\textwidth}
   \includegraphics[width=0.8\textwidth]{Nvariousg1}
   \caption{}
   \label{fig:Ng1} 
\end{subfigure}

\\

\begin{subfigure}[b]{0.3\textwidth}
   \includegraphics[width=0.8\textwidth]{Nvariousg2}
   \caption{}
   \label{fig:Ng2}
\end{subfigure}
\caption{(a) Numerical solutions for the small-time system with a constant-curvature body shape showing the scaled leading-order veritcal reaction force $N_0$ versus the scaled body mass $M$ for various values of $g$. Again, $I=M$ for definiteness and $A=0.7$. (b) As for (a) but over a wider range of values of $M,I$.}
\end{figure}