# Using school physics to calculate oceans warming rate

I've been - er - *discussing *AGW with sceptics again. One intelligent chap is making the argument that "we are chosing timeframes MASSIVELY too short for this sort of science..." His point is that the paleo record suggests that CO2-forced warmings take millions of years, not decades. This is my attempt to answer:

OK. So we need to talk about rates. Physics is good at this. To approximate the rate at which the planet warms, we need to:

- Measure the rate at which the planet is likely to gain heat energy due to an increase in GHGs
- Measure the heat capacity of the oceans + atmosphere + land
- divide the power input by the heat capacity to calculate the time it will take to warm the planet

OK...**1) How fast will the planet gain heat given a doubling of atmospheric [CO**_{2}**]?**

- A doubling of CO
_{2}concentration will exert a positive forcing of 3.7 Watts/m^{2}(that's just for CO_{2 }ignoring any feedbacks). - The surface area of the planet is 510,072,000 km² = 5.1 x10
^{14}m^{2} - So the planet will gain heat at roughly 5.1 x10
^{14}m^{2}x 3.7 W/m^{2}= 1.9 x 10^{15}joules per second = 1.9 petawatts

**2) The heat capacity of the ocean + atmosphere + land.**

Well. Let's keep it simple and just consider the first 1000m of the ocean (the entire atmosphere has a heat capacity roughly equivalent to only a 3.2m slice of the ocean so it's not completely insane to ignore the atmosphere for this rough-and-ready calculation).

The surface area of the Earth's oceans are 361,132,000 km². The specific heat capacity of sea water is 3996 J/kg/°K. Let's say that 1 litre of sea water = 1kg. So the heat capacity of the oceans = 3996 J/kg/°K x 361,132,000 km^{2} x 1,000,000 m^{2}/km^{2} x 1000m depth x 1000 litres per m^{3} = 1.44x 10^{24} J/°K.**3) Given a heat input of 1.9 petawatts, how long would it take to heat the first 1000m of ocean by 1°K?**

The oceans require 1.44 x 10^{24} J to raise by 1 degree K. That amount of energy will be delivered in 1.44x10^{24} J / 1.9x10^{15} J.s-1 = 760,000,000 seconds = 24 years.*24 years per degree C of warming of the oceans.*

Yes: this is a very, very, VERY simplistic calculation (and I've probably ballsed up). But I hope it does demonstrate that the basic physics supports the notion that we can warm the planet within the timescales the IPCC talk about.

You've got to remember that - strictly speaking - humans aren't directly warming the planet. Instead we're modulating the effect that the Sun has on the planet. And the Sun is - um - powerful ;)

Some of the most obvious simplifications in this calculation:

- Given an instant pulse of a doubling of CO2, the earth will warm for a while and then reach a new equilibrium, it wont continue gaining heat at a constant rate
- No aerosols (negative forcing)
- No water-vapour (positive forcing) or other feedbacks
- The oceans mix in complex ways
- The oceans interact with the atmosphere in complex ways

But. Still. *If* I've got the maths right then the result of this simple exercise should be the correct order of magnitude. No, it doesn't require millions of years to heat the oceans.

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