# Q&A: Erosion and Rising Sea Levels

June 18, 2018**Question:** I read in the news that Representative Mo Brooks (R–AL) claimed sea levels are rising because of “rocks falling into the ocean.” Makes total sense to me. When I get into the bathtub the water level rises. Why does everyone assume it’s melting ice that’s causing the rise in sea level? — SP, Washington DC

**Answer: **You are correct that *anything* falling into the ocean must cause the sea level to rise. And that includes eroded sediments carried by rivers, rocks falling from seaside cliffs, new boats being launched, and even stones skipped into the water by beachgoers. But let’s take a close look at the scale of each of those contributions.

Here’s the relevant data:

- Total surface area of the world’s oceans = 3.60 × 10
^{8}km^{2} - Total volume of the world’s oceans = 1.34 × 10
^{9}km^{3} - Total annual sediment delivered by the world’s rivers = 6.98 km
^{3}(ref: AGU100) - Total amount of ice on Earth =2.4 × 10
^{7}km^{3}(ref: Earth’s Water Resources)

If you followed the link to AGU100 you’ll see they estimate 18.5 × 10^{9} metric tons of sediment. Assuming most of that sediment is sand (SiO_{2}) with a density of 2650 kg/m^{3}, the total volume of sediment would be:

V = m/ρ = 1.85 × 10^{13} kg / 2650 kg/m^{3} = 6.98 × 10^{9} m^{3} = 6.98 km^{3}

That’s equivalent to a solid sphere of sediment measuring 2.4 km in diameter. In comparison, the other contributions (falling rocks, launched boats, skipped stones) are negligible. The landslide at Big Sur, CA in May 2017 was one of the largest in recent history. It dumped enough rock into the ocean to create 0.05 km^{2} (13 acres) of prime coastal real estate (which, if you’re curious, became land owned by the state of California). But that was a tiny volume compared to river sediments — USGS estimates the Big Sur landslide at “a million tons of rock.” Suffice it to say river sediments (at 20 *billion* tons) are the major *erosive* contributor “filling up” the ocean.

Historical records show that over the 20th century, sea level rise averaged 1.8 mm/year (about 1/8 inch). More recent records using satellite data measure the current rate at 3.3 mm/yr. The question is: Could river sediment delivered to the ocean account for the current rate of sea level rise? We can use science to answer that question and validate (or not) Brooks’ claim.

We start by constructing a mathematical model of Earth’s ocean using two simplifying assumptions: The ocean is continuous and cylindrical (top graphic). We can now write an equation relating the variables:

V = πR_{o}^{2}H_{o }⇒ H_{o} = V/πR_{o}^{2}

R_{o} is calculated as √(A/π), and V is given, so we can calculate H_{o} as:

H_{o} = (1.34 × 10^{9} km^{3}) / π(1.07 x 10^{4} km)^{2} = 3.7255240__4__ km

This model assumes a constant depth (H_{o}) for the ocean. In reality, depth ranges from 11.9 km in the Challenger Deep, to around 0.15 km along the 80 km-wide continental shelf, to zero at the beach. My cylindrical ocean model is not to scale (NTS). If it was, H_{o} in the drawing would be only a few pixels tall and I needed some space for the labels.

Still, my cylindrical model is valid for a first-order approximation. So let’s drop that 6.98 km^{3} of river sediment into the ocean and see how much the sea level rises:

H_{o} = (1.34 × 10^{9} km^{3} + 6.98 km^{3}) / π(1.07 x 10^{4} km)^{2} = 3.7255240__6__ km

Compared to 3.7255240__4__ km, that’s an increase of 2 x 10^{-8} km or 0.02 mm in one year. Sediment can account for only 0.6% of the observed 3.3 mm annual rise in sea level. There must be something else filling up the ocean.

[Note to fellow math geeks: Given that we started with data to 3 significant figures, it may seem unjustified to extract our “answer” from the 8th decimal place of the two H_{o} calculations. But if we use calculus to find the *rate of change*:

dH_{o}/dt = dV/dt × 1/πR_{o}^{2} = 6.98 km^{3}/yr × 2.78 × 10^{-9} km^{-2} = 1.94 × 10^{-8} km/yr ≈ 0.0200 mm/yr

You can see we get the same result when following significant figure rules. This is because we’re using an ideal model where significant figures propagate to whatever decimal place we need. Calculus is cool, but not really needed here.]

To create a 3.3 mm rise in sea level we’d need to drop 1187 km^{3} of *something* into the ocean. That would be a sphere measuring 13 km in diameter. It dwarfs the sediment sphere by a factor of 170. What is this *something* and where is it coming from?

After eliminating the sedimentary sources, it must be additional water from melting ice. No other source could provide enough volume to account for the observed rise in sea level. Scientists have long been aware of shrinking glaciers, melting ice at the poles (and Greenland), and breakup of Antarctic ice shelves. The expected result was rising sea levels, so no surprises there.

In fact, using topographic data from NASA satellites, more sophisticated models than mine allow predictions of changing shorelines for coastal areas around the planet. You can zoom to your favorite oceanfront property, input the sea level rise in meters, and see how the coastline changes at: http://flood.firetree.net/.

For comparison, the graphic below shows spheres for the total amount of ice on Earth (358 km diameter), the amount that melted last year (13 km diameter), the amount of sediment deposited by rivers in one year (2.4 km diameter), and the Big Sur landslide (0.0002 km diameter estimated). All are drawn to scale.

Thanks for the question SP. I hope it’s now clear that “falling rocks” could not possibly explain the observed rate of sea level rise. This was a great opportunity to show how science (and a little math) can often provide answers in the face of differing opinions or uninformed speculation.

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