Problem — The Adriatic Sea

Problem

Estimate the volume of water contained in the Adriatic Sea, treating it as a rectangular parallelepiped of plausible dimensions. Then compare the result with the volume of a mountain lake of $1\;\text{km}^2$ surface area and average depth $30\;\text{m}$.

Solution

Step 1 — Plausible dimensions. The Adriatic is an elongated basin. We assume dimensions of the order of:

• length $L \approx 800\;\text{km} = 8\cdot 10^5\;\text{m}$;
• average width $\ell \approx 150\;\text{km} = 1{,}5\cdot 10^5\;\text{m}$;
• average depth $H \approx 250\;\text{m}$ (shallow in the north, deeper in the south).

Step 2 — Volume as a parallelepiped.

$V = L\,\ell\,H \approx 8\cdot 10^5 \cdot 1{,}5\cdot 10^5 \cdot 2{,}5\cdot 10^2 = 3\cdot 10^{13}\;\text{m}^3$

Taking into account that the basin is deeper towards the centre and allowing a margin of uncertainty, the order of magnitude is:

$\ev{V \sim 10^{14}\;\text{m}^3}$

Step 3 — Comparison with the mountain lake. The lake has volume:

$V_\text{lago} = A\,h = 1\;\text{km}^2 \cdot 30\;\text{m} = 10^6\;\text{m}^2 \cdot 30\;\text{m} = 3\cdot 10^7\;\text{m}^3$

Step 4 — Ratio.

$\frac{V}{V_\text{lago}} \approx \frac{10^{14}}{3\cdot 10^7} \approx 3\cdot 10^6$

The Adriatic contains about a few million times the water of a mountain lake: a gap of about six to seven orders of magnitude. The comparison shows how quickly volumes grow when moving from the local to the geographic scale.

Linked atoms

Argomenti: Physical quantities and measurement Concetti: Scientific notation and orders of magnitude Competenze: Fermi estimates