Should We See Sea Level Rise to be Accelerating? (No)

One of the questions you sometimes hear is that, if more of the GHG-trapped heat is lately going into the ocean, why isn't sea level rise itself rising -- accelerating?

Here I'll show that, no, we should not be seeing more sea level rise, not now, via a back-of-the-envelope physics calculation. Really basic physics, like freshman year physics.

Alas, I don't expect anyone to follow this, or work through it and look for errors, or comment on it. Because Willie Soon. Because sexual harassment. Because climate McCarthyism. But I don't know how else to think about these questions. I find these kind of little calculations fun, and even sometimes wise. I wish they were done more often.

Sorry to get technical. But there's no other way to answer this question.

Everything starts, of course, with the definition of specific heat

dQ = mc dT = ρVc dT =ρAhc dT

where ρ is the density of sea water, V its volume, c its specific heat, and T its temperature, A is the area of the ocean, and h the ocean's height. (This is a simple model -- same temperature throughout, and I'll ignore any sea level rise from melting land ice, the so-called "eustatic" change. I'm doing a calculation of the "steric" changes.) (For some reason I have a mental block and can never remember which of these is which.) Then

dQ/Adt = ρch dT/dt

The left-hand side is how much heat is going into the ocean per unit area. That's f_OF(t) where F(t) is the energy imbalance for the planet (which I'll take as the forcing due to GHGs, ignoring cooling forcings like aerosols from fossil fuel pollution, etc. Note that this is a significantly larger number than reality, which is more like the energy imbalance of 0.5 W/m2 of Loeb et al Nature Geo 2012), and f_O is the fraction of that energy that goes into the ocean. f_O is about 93E%.

For the right-hand side, use the thermal expansion relation for water to give, assuming the area of the ocean stays the same (viz. the ocean is only expanding upward)

dT/dt = (1/αh) dh/dt

where h is the height of the ocean and α the thermal expansion coefficient of water. Then

F(t) = β dh/dt

where β = ρc/α = 1.6 x 10^-11 m³/J, or 1/β = 6.1 x 10¹⁰ J/m³, an energy density.

The global radiative forcing is increasing linearly. In 2013 it was 2.916 W/m2, according to NOAA, and its slope is, if you calculate it, ε = 0.034 W/m²/yr. (The Loeb et al 2012 result noted above would likely a signifcantly lower epsilion. That will increase the numers below correspondingly.) Earth's actual energy balance is less than this, because of manmade pollutants -- chemicals like sulphuric acid that reflect away sunlight. But I'll go this for now, as an upper bound.

Put this all together, taking F=F₀ = 2.916 W/m2 at t = t₀ = now, and you get simply

d²h/dt² = βεf_O ≡ a = 0.017 mm/yr2

This is the acceleration of sea level rise, due to thermal expansion. It's not so very big, but it's happening year-after-year.

Here βε sets the scale for this problem -- a quantity that, by units alone, is the acceleration, up to a constant that is a pure number, with no units.

It's an amazing aspect of physics, which no one has ever really explained, not even Dirac, that physical systems never deviate much from their fundamental constants. That is, in this case, d²h/dt² = βε*f_O, but it doesn't equal 100 times βε or a billion times it, or 10^^-21 times it. It's right in the sweet spot of the algebra -- a few times it, or the inverse.

Why? I don't know, and I don't know if anyone knows. But it does mean you can often make a good guess for the answer to a physics problem simply by doing dimensional analysis -- by deciding what constants are a necessary part of the answer, and seeing what combination of them works out to the units of the thing you're trying to calculate. It's trivial once you understand it, but powerful.

John Wheeler, who taught Richard Feynman, said you should never start a physics calculation before you know the answer. By this he means, do dimensional analysis first. That's how the system is going to respond to any external force.

Here beta is the fundamental unit of the system, and ε the cale of the input. It's not surprising the beta*epsilon shows up for the acceleration.

Now the equations are no different from basic kinematics of an accelerating car. With a = 0.017 mm/yr2, we find the present rate of sea level rise to be

S0 = S(t₀) = (dh/dt)|t=t₀ = βf_Oε = 1.4 mm/yr

where S(t) is the rate of sea level rise (= dh/dt), which is about what you'd expect, since SLR is now 3.2 mm/yr and about half of this is steric. As is expected, we get

S(t) - S0 = a*(t-t0)

How long will it take sea level rise to double, from today's S0 to 2*S0? Call that time D. Then

D - t₀ = S₀/a = F₀/ε = 85 years

How much sea level acceleration should we expect to see, for the thermal expansion of seawater, 20 years in the future?

S(20 years in the future) - S0 = 0.3 mm/yr

But note that CU's uncertainty on the rate of sea level rise is +/- 0/4 mm/yr. So what we'd expect is within the error bars.

--

Suppose though that what caused the hiatus is more heat going into the ocean. Should sea level rise be noticeably jumping up in that case?

No. We have

S0 = β*F₀*f_O

So if f_O goes from, say, 93% to 95%, current sea level should rise only by a factor of only 0.95/0.93 = 1.02 so S₀ -> (1.4 mm/yr)*1.02 = 1.43 mm/yr, which just isn't going to be noticeable with today's error bars.