Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2010 / A / 1

The heat capacity per unit horizontal area is $C=\rho c_{p}d$: mass density times specific heat capacity times atmospheric depth. Thus $C\dot{T}$ is the rate of stored thermal energy per unit area. The solar constant $Q$ is the incident radiative flux on a surface normal to sunlight; the ratio of the Earth's intercepting disk area to its spherical area is $1/4$. The absorbed fraction is $1-a$, where albedo $a$ is the reflected fraction. The outgoing radiative flux is $\sigma \gamma T^4$, a Stefan–Boltzmann law with effective radiation factor $\gamma$ incorporating the simplified greenhouse effect. Every term therefore has units of power per area.

The albedo curve decreases smoothly, passes through $(T_0,1/2)$, and approaches $1/4$ at high temperature. At $T=0$ it is $(2+\tanh \mu )/4$, close to $3/4$ when $\mu$ is large. Its derivative isAn ice-covered planet reflects more sunlight than exposed land or ocean. Warming removes reflective ice, lowers albedo, and increases absorbed radiative flux: this is ice-albedo feedback. The parameter $\mu$ controls the sharpness of this transition; the formula is an idealized global response, rather than a local melting law.

Set $x=T^4/T_0^4-1$. Since $\sigma \gamma T_0^4=Q/8$, the steady state energy balance is $2(1-a_0)=1+x$. Substitution givesThe two sides have slopes $\mu$ and $2$ at the origin. For $x>0$, $\tanh (\mu x)$ is strictly concave, its initial slope is $\mu$, and it remains below $1$. Consequently no positive root exists for $\mu \le 2$, while for $\mu >2$ there is exactly one positive root $x_{+}\in (0,1/2)$. Oddness supplies the corresponding negative root. At $\mu =2$ the only root is still zero. Hence multiple steady temperatures occur precisely when $\mu >{\mu }_c=2$.

The largest root corresponds monotonically to the largest positive absolute temperature:For large $\mu$, the positive root tends to $1/2$, so $a_{+}\simeq 1/4$ and $T_{+}/T_0\simeq (3/2{)}^{1/4}$. The negative branch has larger albedo and lower temperature.

Use dimensional quantities $T_d,t_d$ and set $T_d=T_{0}T$, $t_d=t_{0}t$. Dividing the energy balance by $\sigma \gamma T_0^4$ and the albedo equation by its time scale givesThus $\epsilon$ is the ratio of the atmospheric thermal response time to the much longer albedo response time. Converting kilometres to metres and years to seconds gives $C=10^7\mathrm{J}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$, $T_0\simeq 266.3\mathrm{K}$, $t_0\simeq 3.156\times 10^{10}\mathrm{s}$, andThe thermal time is about half a year, compared with a millennium for albedo adjustment. This is a singular perturbation: setting $\epsilon =0$ removes the independent initial condition for temperature.

Initially $a=1/2$ and $T=T_{+}/T_0>1$, so $\epsilon T_t=1-T^4<0$. On the fast time $s=t/\epsilon$, albedo changes by only $O(\epsilon )$ andThus temperature rapidly falls towards $1$. At the same time $a_t=a^{\ast }(T)-1/2<0$ while $T>1$, which seeds selection of the warm branch. Afterwards the approximate slow manifold is $T^4=2(1-a)$: decreasing albedo makes temperature rise slowly back towards $T_{+}/T_0$. The preceding figure separates these two time scales; a single time axis would compress the initial fall into a nearly vertical segment.

In the sharp-transition regime, after selection of the warm branch $a^{\ast }(T)\simeq 1/4$. Solving the linear ordinary differential equation with the leading matched value $a(0^{+})=1/2$ givesThese are outer solutions: their initial temperature is $1$, so the fast cooling layer is required to satisfy the actual initial temperature.

There is a useful qualification to the two-limit argument. For fixed finite $\mu >2$, put $x=1-2a$ on the slow manifold. Then $x_t=-x+\frac{1}{2}\tanh (\mu x)$, and $x=0$ is an unstable equilibrium with growth rate $\mu /2-1$. The small positive $O(\epsilon )$ seed from the fast layer can therefore produce a delay of order $\log (1/\epsilon )/(\mu /2-1)$. The simple exponential approximation assumes that the sharp albedo transition is sufficiently fast to make this delay negligible on the displayed slow scale. It is not a uniform approximation for every fixed $\mu >2$ as $\epsilon \to 0$.