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Adding vector arrows to tikz plot
I have code which outputs a number of parabolas
\begin{document}
\tdplotsetmaincoords{70}{20}
\begin{tikzpicture}[tdplot_main_coords,
declare function={pX=2;}]
 \begin{scope}[canvas is yz plane at x=0]
  \draw (-pX,-pX) coordinate (bl1) -- (pX,-pX) coordinate (br1)
   -- (pX,pX) coordinate (tr1)  -- (-pX,pX) coordinate (tl1) -- cycle;
   \draw[stealth-stealth] (0,pX) node[below left]{$x_2$}
   -- (0,0) --  (pX,0) node[below left]{$x_1$};
   \pgflowlevelsynccm
   %\draw[-{Latex[bend]}] (2,0) arc(0:380:1);
   %\draw[-{Latex[bend]}] (0,-1) arc(-90:290:1);
   %\draw[-{Latex[bend]}] (0,0) arc(-90:380:1);
 \end{scope}
 \foreach \X in {bl,tl,br,tr}
 {\draw[-latex] (\X1) -- ++ (1,0,0);}
 %
 \draw (0,0,0) -- (4,0,0);
 % plane
  \fill[gray,fill opacity=0.3] (-1,-4,0) -- (7,-4,0) --
  (7,4,0) -- (-1,4,0) -- cycle;
 \begin{scope}[canvas is yz plane at x=4]
  \draw (-pX,-pX) coordinate (bl2) -- (pX,-pX) coordinate (br2)
   -- (pX,pX) coordinate (tr2)  -- (-pX,pX) coordinate (tl2) -- cycle;
   \draw[stealth-stealth] (0,pX) -- (0,0) --  (pX,0);
   \pgflowlevelsynccm
   %\draw[-{Latex[bend]}] (1,0) arc(0:380:0.25);
   %\draw[-{Latex[bend]}] (0,-0.25) arc(-90:290:0.25);
   %\draw[-{Latex[bend]}] (0,0.25) arc(-90:380:0.25);
 \end{scope}
 \draw[-latex] (4,0,0) -- (6,0,0) node[below left]{$x_3$};
 \foreach \X in {bl,tl,br,tr}
 {\draw[-latex] (\X2) -- ++ (1,0,0);}
  \begin{scope}[canvas is xy plane at z=0]
   \begin{scope}
        \clip (0,-2) rectangle (4,3);
        \foreach \X in {-2,-1.6,...,2}
         {\draw (0,\X) parabola ++(5,1);}
       \end{scope} 
%       \draw[-stealth] (-1,0) -- (5,0) node[below right]{$x_3$};
%       \draw[-stealth] (0,-0.5) node[above left]{$0$} -- (0,3)
%       node[right=2em,yshift=-1em]{$\theta(x_3)$};
%       \draw[-stealth] (4,-0.5) node[above left]{$L$} -- (4,3) node[below right]{$x_1$};
  \end{scope}
\end{tikzpicture}
\end{document}

which looks like 
I am trying to replace the parabola curves by a series of lines 
like this

so as to make a series of layers with each layer having a random arrow direction. Can this be done in tikz? Thanks in advance.

This adds random arrows. 
\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz-3dplot}
\begin{document}
\tdplotsetmaincoords{70}{20}
\begin{tikzpicture}[tdplot_main_coords,
declare function={pX=2;}]
 \begin{scope}[canvas is yz plane at x=0]
  \draw (-pX,-pX) coordinate (bl1) -- (pX,-pX) coordinate (br1)
   -- (pX,pX) coordinate (tr1)  -- (-pX,pX) coordinate (tl1) -- cycle;
   \draw[stealth-stealth] (0,pX) node[below left]{$x_2$}
   -- (0,0) --  (pX,0) node[below left]{$x_1$};
   \pgflowlevelsynccm
 \end{scope}
 \foreach \X in {bl,tl,br,tr}
 {\draw[-latex] (\X1) -- ++ (1,0,0);}
 %
 \draw (0,0,0) -- (4,0,0);
 % plane
  \fill[gray,fill opacity=0.3] (-1,-4,0) -- (7,-4,0) --
  (7,4,0) -- (-1,4,0) -- cycle;
 \begin{scope}[canvas is yz plane at x=4]
  \draw (-pX,-pX) coordinate (bl2) -- (pX,-pX) coordinate (br2)
   -- (pX,pX) coordinate (tr2)  -- (-pX,pX) coordinate (tl2) -- cycle;
   \draw[stealth-stealth] (0,pX) -- (0,0) --  (pX,0);
 \end{scope}
 \draw[-latex] (4,0,0) -- (6,0,0) node[below left]{$x_3$};
 \foreach \X in {bl,tl,br,tr}
 {\draw[-latex] (\X2) -- ++ (1,0,0);}
  \begin{scope}[canvas is xy plane at z=0]
   \begin{scope}
      \clip (0,-2) rectangle (4,3);
      \foreach \X in {0.25,0.75,...,2.75}
       {\pgfmathsetmacro{\myrnd}{rnd*360}
       \foreach \Y in {-1.75,-1.25,...,1.75}
        {
        \draw[-stealth] ({\X-0.2*cos(\myrnd)},{\Y-0.2*sin(\myrnd)}) --
        ({\X+0.2*cos(\myrnd)},{\Y+0.2*sin(\myrnd)});
        }
        \draw[densely dashed] (\X+0.25,-2) -- (\X+0.25,2);}
%       \foreach \X in {-2,-1.6,...,2}
%        {\draw (0,\X) parabola ++(5,1);}
   \end{scope} 
  \end{scope}
\end{tikzpicture}
\end{document}


Or
\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz-3dplot}
\begin{document}
\tdplotsetmaincoords{70}{20}
\begin{tikzpicture}[tdplot_main_coords,
declare function={pX=2;}]
 \begin{scope}[canvas is yz plane at x=0]
  \draw (-pX,-pX) coordinate (bl1) -- (pX,-pX) coordinate (br1)
   -- (pX,pX) coordinate (tr1)  -- (-pX,pX) coordinate (tl1) -- cycle;
   \draw[stealth-stealth] (0,pX) node[below left]{$x_2$}
   -- (0,0) --  (pX,0) node[below left]{$x_1$};
   \pgflowlevelsynccm
 \end{scope}
 \foreach \X in {bl,tl,br,tr}
 {\draw[-latex] (\X1) -- ++ (1,0,0);}
 %
 \draw (0,0,0) -- (4,0,0);
 % plane
  \fill[gray,fill opacity=0.3] (-1,-4,0) -- (7,-4,0) --
  (7,4,0) -- (-1,4,0) -- cycle;
 \begin{scope}[canvas is yz plane at x=4]
  \draw (-pX,-pX) coordinate (bl2) -- (pX,-pX) coordinate (br2)
   -- (pX,pX) coordinate (tr2)  -- (-pX,pX) coordinate (tl2) -- cycle;
   \draw[stealth-stealth] (0,pX) -- (0,0) --  (pX,0);
 \end{scope}
 \draw[-latex] (4,0,0) -- (6,0,0) node[below left]{$x_3$};
 \foreach \X in {bl,tl,br,tr}
 {\draw[-latex] (\X2) -- ++ (1,0,0);}
 \foreach \Z in {0,0.5,...,2} 
  {\begin{scope}[canvas is xy plane at z=\Z]
   \begin{scope}
      \foreach \X in {0.25,0.75,...,2.75}
       {\pgfmathsetmacro{\myrnd}{rnd*360}
       \foreach \Y in {-1.75,-1.25,...,1.75}
        {
        \draw[-stealth] ({\X-0.2*cos(\myrnd)},{\Y-0.2*sin(\myrnd)}) --
        ({\X+0.2*cos(\myrnd)},{\Y+0.2*sin(\myrnd)});
        }
        \draw[densely dashed] (\X+0.25,-2) -- (\X+0.25,2);}
%       \foreach \X in {-2,-1.6,...,2}
%        {\draw (0,\X) parabola ++(5,1);}
   \end{scope} 
  \end{scope}}
\end{tikzpicture}
\end{document}



This is perfect almost! How can I change this so that between each row of arrows there is a dotted line, and also, in each row all the arrows point in the same direction? I just want each row of arrows in the x3 direction to be at random directions.

@rodger_kicks Done.

This is perfect. I can't thank you enough. Is it easy to extend this into the x2 axis, to make the "layers" extend upwards? So that the dashed lines go upwards and the arrows repeat?

@rodger_kicks I added a loop over \Z :\foreach \Z in {0,0.5,...,2}  . You can change the values to move the arrows closer or further away from each other.
