Data Analysis Tutorial : Two-dimensional vector

2.2 Two-dimensional vector

In contrats to scalars such as temperature, pressure, some variables consist of two or three components. The horizontal current vector $u,\: v$ is a good examle of this. Consider time series of $u,\: v$ measured by an ADCP at some bin, and think of ilustrating the velocities by plotting a single symbol on a $u,\: v$ plot for each time that is available. The resulting figure could look line any of the six possibilities illustrated in Fig.6.

$\includegraphics[width=12 cm]{figs/scatteruv.eps}$

Figure 6: Scatter plot of $u$ and $v$ . The red arrows represent $u,\: v$ at the initial time and the black arrows represent the vector at the next to initial time.

Most real observations will look like the upper left hand frame, where

$\begin{equation} \label{eq:syntheticuv} u=0.1\cos {2\pi t}+0.1\: rand;\quad v=0.05\sin {2\pi t}+0.02\cos {2\pi t}+0.1\: rand \end{equation}$

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whre $rand$ represents a gaussian random nmber generator, representing either noise or currents that are uncorrelated between $u$ and $v$ . In this case the data form a shotgun pattern seemigly included within an ellipse whose major axis is slightly tilted from the horizontal.

The upper middle panel illustrates the case when the noise is set to zero, and the two velocity components are exactly out of phase:

$\begin{equation} \label{eq:syntheticuv} u=0.1\cos {2\pi t}\quad v=0.05\sin {2\pi t} \end{equation}$

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This pair of equations are called parametric equations and the figure is a Lissajous figure ¹. If we take thesquare of each velocity component and multiply by a constant we can write:

$\begin{equation} \label{eq:ellipseuv} \frac{u^2}{a^2}+\frac{v^2}{b^2}=1\quad a=0.1;\: b=0.05; \end{equation}$

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which is the equation for an ellipse (http://www.analyzemath.com/EllipseEq/EllipseEq.html). The red arrow represents the vector $u,\: v$ at $t=1$ , and the black arrow represents the vector at the second, so that in this case the vector is said to rotate in a counterclockwise (CCW) fashion. In this case $a=0.1$ is called the major axis, aligned with the $x$ axis; $b=0.05$ is the minor axis.

In the upper right panel, the $u$ velocity is unchanged but the sign of the $v$ velocity is opposite:

$\begin{equation} \label{eq:CWuv} u=0.1\cos {2\pi t}\quad v=-0.05\sin {2\pi t} \end{equation}$

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The ellipse is the same as before, but the sense of rotation is now clockwise (CW). The convention is to report the minor axis as negative for CW rotation and positive for CCW rotation, so $a=0.1;\: b=-0.05$ .

In the lower left panel, the $u$ velocity is still unchanged but the $v$ velocity is in phase with $u$ , although with different amplitude:

$\begin{equation} \label{eq:CWuv} u=0.1\cos {2\pi t}\quad v=0.2\: u \end{equation}$

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The ellipse shrinks to a straight line, tilted with respect to the $x$ axis by an inclination angle. In this case $a=\sqrt {(0.1)^2+(0.02)^2};\: b=0;\: \tan {\theta }=0.02/0.1$ .

The lower middle panel illustrates the combination of the second and fourth plots:

$\begin{equation} \label{eq:syntheticuv} u=0.1\cos {2\pi t}\quad v=0.05\sin {2\pi t}+0.2\: u \end{equation}$

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In this case the figure represents a tilted ellipse.

Finally the lower right panel represents random noise for both $u$ and $v$ , but with different amplitude:

$\begin{equation} \label{eq:randomuv} u=0.05\: rand;\quad v=0.025\: rand \end{equation}$

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The $u,\: v$ pairs fill out a rectangle whose size is determined by the amplitude of the components.

The upper left hand panel is the sum of the expressions illustrated in the last two frames.