(Solved by Humans)-this about Differential Equations if you can solve any one of

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this about?Differential Equations

if you can solve any one of them is fine .

THANK YOU !

AMATH 383 Spring 2016
Homework 5 ? Due Thurs May 5th in class
Show work for full credit! The grader will subtract points for poor presentation.

1. The eleven year solar cycle
In class we derived the following differential equation for changes in the global mean temperature, ?T , due to climate
pertubrations ?Q:
?
B? ?T = (1 ? ?
? )?Q ? B?T /g,
(1)
?t
where ? is the characteristic timescale of the climate response, g is the climate gain, ?
? is the mean albedo across the
globe, and B is related to the black body radiation emitted by the Earth.
a. (20 points) The sun?s radiant output fluctuates on an 11-year periodic cycle that is modeled by ?Q(t) = a cos(?t), with
? = 2?/(11 years). Solve the time-dependent equation (1) for the periodic response of the atmosphere near the surface,
?T (t). Show that it can be written in the form
?T (t) =

?Q(t ? ?)(1 ? ?
? )g/B
?
,
1 ? 2

where  = g?? and ?? = tan?1 (). ? is the time lag of the response, and the factor in the denominator gives the reduction in amplitude from the equilibrium
value because of the periodic nature of the response. (Hint: The trigonometric
?
identity a cos(?t) + b sin(?t) = a2 + b2 cos(?t + tan?1 (b/a)) may be useful.)
b. (10 points) The variability of the sun?s radiation through the 11-year solar cycle has been measured since 1979 by
earth-orbiting satellites. We know that the solar constant varies by 0.06% from solar minimum to solar maximum. In
terms of the parameters of our model, we know that 2a/Q0 = 0.06%, so 2a = 0.2 watts/m2 , where Q0 = 343 watts/m2 is
the baseline solar constant. The atmosphere?s temperature response is found to lag only slightly (by about 1 year) and its
magnitude is measured near the surface to be about 0.2? C on a global average from minimum to maximum. Use these
values to deduce the climate gain factor g, and show that it should be about g ? 3. Recall that B has been measured to
be about 1.9 watts/m2 /? C.

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2. Latitude variations in temperature
In class we dodged the issue of variations in temperature with latitude by focusing on the global mean temperature.
In this problem we?ll look at the equilibrium distribution of global temperatures across latitudes for the two kinds of
models we discussed in class. Our model for the temperature of the Earth?s atmosphere is
R

?T
= Qs(y)(1 ? ?) ? A ? BT + D(y),
?t

(2)

where R is the heat capacity of the Earth,
Q is the total solar input to the Earth?s atmosphere, y = sin ? is the latitude

2
(0 ? y ? 1), s(y) = 1 ? 0.482 3y 2?1 describes how much solar radiation is absorbed at each latitude, A + BT is the
amount of black-body radiation emitted by the Earth. ? is the albedo, which we will assume to be constant in this model,
though it would really vary with latitude. D(y) describes the transport of heat from one latitude to another. It averages
to zero across the globe. We?ll consider two models for D(y).
a. (20 points) Let D(y) = C(T (t) ? T (t, y)), as in the model of Budyko. This model assumes that temperatures at each
latitude would relax to the global mean temperature, T , with rate C/R, if solar radiation were uniformly spread out
across the globe. We seek the equilibrium temperature distribution in this model. To do so,
i) Average equation (2) over the latitude to obtain an equation for T (by integrating over y from 0 to 1). Then, set
R1
?T /?t = 0 and solve for the equilibrium global mean temperature T eq . Use the fact that 0 dy s(y) = 1.
ii) Plug your result for T eq into equation (2) and set ?T /?t = 0 to solve for the equilibrium temperature distribution
Teq (y). Show that this may be written as
Teq (y) =

Q(1 ? ?)(s(y) ? 1)
+ T eq .
B+C

For what latitudes 0 ? y ? 1 is the temperature less than the global mean?
b. (15 points) A slightly more realistic modelh treats the itransport of heat through the atmosphere as a diffusion process.
?
This model, due to North, sets D(y) = ? ?y
(1 ? y 2 ) ?T
?y , where ? is the diffusion coefficient of heat through the atmosphere. Plug this expression for D(y) into equation (2). Then, set ?T /?t = 0. The result is now an ordinary differential
equation for Teq (y). Show that Teq (y) = a0 + a2 y 2 is the solution, determining the constants a0 and a2 in terms of the
other parameters of the model.

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3. Changes in the latitudinal temperature distribution in time
Finally, let?s look at how the latitudinal distribution of temperatures changes in time. We?ll focus on Budyko?s model.
For a climate perturbation Q ? Q + ?Q, we can derive a latitude- and time-dependent version of equation (1),
B?

?
?T (t, y) = (1 ? ?)?Q(t)s(y) ? B?T (t, y)/g + C(?T (t) ? ?T (t, y)),
?t

(3)

where ?T (t, y) is the perturbation to the atmospheric temperature at time t and latitude y. For simplicity, we?ll approximate the albedo as a latitude-independent constant. For a time-dependent perturbation (1 ? ?)?Q(t) = bt, we showed in
class that the response of the mean temperature obeyed the equation (1) and had solution
?T (t) =

g
(1 ? ?)?Q(t ? ?(t)),
B




t
is the time-dependent time lag of the climate?s response to the perturbation.
where ?(t) = g? 1 ? exp ? g?
a. (20 points) Let ?T (t, y) = ?T (t) + (t, y), where (t, y) is the deviation of the temperature at latitude y from the global
mean temperature. Plug this expression and (1 ? ?)?Q(t) = bt into equation (3) and, using equation (1) to cancel out
terms involving ?T , derive a differential equation for (t, y). Solve this differential equation using the initial condition
(0, y) = 0. Because there are no y-derivatives, you can just treat y as a constant parameter. Show that your result can
be written as
1
?
(1 ? ?)?Q(t ? ?(t)),
(t, y) = (s(y) ? 1)
B/g + C
?
where ?(t)
is a time-lag (different from the previously defined ?) that you should determine an exact expression for.
R1
R1
Note that because 0 dy s(y) = 1, 0 dy (t, y) = 0 for every time t.
b. (15 points) Show that for a certain range of y the temperature change (t, y) is not only negative but growing increasingly negative as time goes on. i.e., at some latitudes the temperature is getting colder (relative to the increasing mean
global temperature) in response to the increase in energy input ?Q! Last year, a certain United States senator infamously
threw a snowball onto the floor of the senate to ?prove? that global warming is not real because it was unseasonably cold
outside, rather than unseasonably hot. Based on the calculation we just performed, explain why decreasing temperatures
around the globe do not necessarily disprove global warming.

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