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Earth and Space Sciences · 术语表

什么是 Shortening？

Definition 20.7 High School Earth and Space Sciences · 第 20 章 — Building Mountains

The shortening of a layer is the fraction of its length that it has lost: if it was $L_0$ long and now spans $L$ ,

e = \frac{L_0 - L}{L_0}.

Left: a folded layer spans L; laid out flat, without being stretched, it is L_0 long (here the folds are as high as a quarter of their wavelength, and L is 68\,\% of L_0). Right: the shortening of a fold with straight limbs, 1 -, against the dip of its limbs. Gentle folds shorten a layer very little. — Left: a folded layer spans $L$ ; laid out flat, without being stretched, it is $L_0$ long (here the folds are as high as a quarter of their wavelength, and $L$ is $68\,\%$ of $L_0$ ). Right: the shortening of a fold with straight limbs, $1 - \cos\theta$ , against the dip of its limbs. Gentle folds shorten a layer very little.

示例

Example 20.10 (The Varden folds)

The limbs of the Varden folds dip $12{}^{\circ}$ , so the cover was shortened by $e = 1 - \cos12{}^{\circ} = 1 - 0.978 = 0.022$ , about $2.2\,\%$ : a stretch of the region $10\,\mathrm{km}$ wide today was $10/\cos12{}^{\circ} \approx 10.22\,\mathrm{km}$ wide before folding. Measured on section A–B as drawn, the limbs would look about $28{}^{\circ}$ steep, because the vertical scale is stretched 2.5 times: $\tan\theta$ is multiplied by 2.5, and $\arctan(2.5 \tan12{}^{\circ}) \approx 28{}^{\circ}$ . Read at face value, that would give a shortening of $12\,\%$ , more than five times too much.

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