35  Problem set 5 solutions

Q1. Evolution

Consider again the basic time-dependent EBM given in (Equation 27.1). Let $T^{\ast }$ be a steady-state solution and set $T=T^{\ast }+u(t)$ where $u(t)$ is a small perturbation from the steady state.

1. Show that the perturbation satisfies $$C\dot{u}=-Du+O(u^2).$$ and hence solve for the general solution of the leading-order perturbation (ignoring quadratic terms). What are the conditions on $T^{\ast }$ so that the steady state is linearly stable?

Solution

Graphical solution

2. Assuming $T^{\ast }$ is linearly stable, find the typical response time to a perturbation. For instance, what is the time it takes for the perturbation to reach the value $u(t)=0$ if $u(0)=1$? How does this response time change with $C$? What is the physical interpretation of this regarding the climate?

Solution

Graphical solution

Q2. Integral of energy over the planet

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Ignoring the effects of albedo, the total radiation absorbed over the surface of the planet (per unit time) is given by $${\iint }_{\text{planet}}Qs(y=\sin \phi )\mathrm{d}S.$$ This is what is known as a surface integral (Section 46.1). In the case of the spherical coordinate system, this is calculated by $${\int }_{\theta =0}^{2\pi }{\int }_{\phi =-\pi /2}^{\pi /2}Qs(y=\sin \phi )R_E^2\cos \phi \mathrm{d}\phi \mathrm{d}\theta .$$ Use the properties of $s(y)$ in (Equation 13.3) to conclude that the total radiation absorbed is $4\pi R_E^{2}Q$.

Solution

Graphical solution

Q3. Mean temperature in the latitude-dependent EBM

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Consider now the latitude-dependent EBM $$C\frac{\partial T}{\partial t}=Qs(y)[1-a(y)]-(A+BT)+k(\overline{T}-T).$$ Recall the albedo is given by $a=a_i$ for $y>y_s$ and $a=a_w$ for $y<y_s$.

1. By integrating the above equation over $y\in [0,1]$, show that the mean temperature is given by $$\begin{array}{}\text{(35.1)}& C\frac{\mathrm{d}\overline{T}}{\mathrm{d}t}=Q(1-\overline{a})-(A+B\overline{T}),\end{array}$$ where $$\overline{a}={\int }_0^{1}s(y)a(y)\mathrm{d}y={\alpha }_w{\int }_0^{y_s}s(y)\mathrm{d}y+a_i{\int }_{y_s}^{1}s(y)\mathrm{d}y.$$

Solution

Graphical solution

2. In the case that $s$ is given by (Equation 13.3), show that $$\begin{array}{}\text{(35.2)}& \overline{a}=a_i+(a_w-a_i)y_s[1-0.241(y_s^2-1)].\end{array}$$ What is $\overline{a}$ in the two situations of a completely ice-covered world and an ice-free world?

Solution

Graphical solution

Q4. Sensitivity of the climate

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Consider the equation for the global average temperature given in (Equation 35.1): $$C\frac{\mathrm{d}\overline{T}}{\mathrm{d}t}=Q(1-\overline{a})-(A+B\overline{T}).$$ We would like to understand how the climate behaves under a small perturbation in the solar forcing. In general, though, the ‘constants’, $A$, $B$, and $\overline{a}$ will depend on $T$. For example, their values may have been calibrated under a static situation. Therefore, let $A=A(T)$, $B=B(T)$, and $\overline{a}=\overline{a}(T)$.

Below, we drop all bars for convenience.

1. Consider a perturbation of the solar radiation, say $Q=Q_0+\delta$ where $\delta$ is small in comparison to $Q_0$. Expand now the temperature into a series: $$T=T_0+\delta T_1+\dots$$ Show that at $O(\delta )$, the perturbation is governed by $$C\frac{\mathrm{d}{T}_1}{\mathrm{d}t}=(1-a(T_0))-B(T_0)T_1-A^{\prime }(T_0)T_1-T_0{B}^{\prime }(T_0)T_1-Q_0{a}^{\prime }(T_0)T_1.$$

Solution

Each of the functions should be expanded in the usual way. For example, from Taylor series we have $$A(T)=A(T_0)+A^{\prime }(T_0)(T-T_0)+O(T-T_0{)}^2.$$ Then using $T=T_0+\delta T_1+\dots$, we have $$A(T)=A(T_0)+\delta A^{\prime }(T_0)T_1+O(\delta {)}^2.$$ Do the same for both $a(T)$ and $B(T)$. Then substituting into the equation gives (remember to drop bars): $$\text{LHS}=\delta C\frac{\mathrm{d}{T}_1}{\mathrm{d}t}+O({\delta }^2)$$ while for the right hand-side

$$\begin{array}{}\text{(4.1)}& \text{RHS}& =Q_0(1-a(T_0))-(A(T_0)+B(T_0)T_0)\text{(4.2)}& & +\delta [(1-a(T_0))-Q_0{a}^{\prime }(T_0)T_1-A^{\prime }(T_0)T_1-B^{\prime }(T_0)T_0{T}_1-B(T_0)T_1]+O(\delta {)}^2.\end{array}$$

Now the first, $O(1)$, grouping of terms above is zero by assumption that $T_0$ is steady-state. So considering only those terms that are multiplied by $\delta$, we have $$C\frac{\mathrm{d}{T}_1}{\mathrm{d}t}=(1-a(T_0))-B(T_0)T_1-A^{\prime }(T_0)T_1-T_0{B}^{\prime }(T_0)T_1-Q_0{a}^{\prime }(T_0)T_1.$$

2. Consequently, show that the temperature perturbation can be written as $$\begin{array}{}\text{(35.3)}& B(T_0)\tau \frac{\partial T_1}{\partial t}=[1-a(T_0)]-\frac{B(T_0)}{g}{T}_1,\end{array}$$ where

$$\begin{array}{rl}\tau & =\frac{C}{B(T_0)}\\ g& =\frac{1}{1-f}\\ f& =f_1+f_2\\ f_1& =\frac{1}{B(T_0)}(-T_0{B}^{\prime }(T_0)-A^{\prime }(T_0)),\\ f_2& =\frac{1}{B(T_0)}(-Q_0{a}^{\prime }(T_0))\end{array}$$

The parameter $\tau$ measures the time scale of the climate system’s thermal inertia. It involves the thermal inertia of the atmosphere and also the much larger inertia of the oceans. The factor $g$ is the climate gain; it amplifies any response to the radiative perturbation by a factor of $g$. Also $f_1$ incorporates the effect of water-vapor feedback and $f_2$ that of ice and snow albedo feedback.

Solution

There is nothing tricky about this, but it just requires keeping track of bookwork. On the right hand-side, separate those terms proportional to $T_1$ and write it in the desired form. Refer to the lecture videos where we completed this (modulo a possible sign!)

3. Consider (Equation 35.3) at steady state, so therefore the perturbed equilibrium temperature is equal to $$\delta T_1=\frac{[1-a(T_0)]\delta g}{B(T_0)}.$$ If the CO₂ level in the atmosphere doubles, then the radiative forcing might be adjusted as: $$(1-a(T_0))\delta =3.7\mathrm{W}\cdot {\mathrm{m}}^{-2}.$$ Assuming that the climate gain is $g=3$ and $B(T_0)=1.9\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot ({}^{\circ }\mathrm{C}{)}^{-1}$, what is the expected increase in temperature?

Solution

So we know that $(1-a)\delta =3.7W/m^2$. So according to the equation, we have the fact that $$\delta T_1=3.7\frac{3}{1.9}\approx {5.8}^{\circ }C.$$ So this is the expected increase in temperature. Note that if a climate gain factor of $g=1$ is used then this gives an expected increase of 1.9 degrees instead.. The range of current model predictions, in fact, is about 1.5 to 4.5 degrees so this is not bad.