-
Notifications
You must be signed in to change notification settings - Fork 0
/
module3.Rmd
382 lines (290 loc) · 14.9 KB
/
module3.Rmd
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
# Module 3: Climate Modelling {#M3_1 -}
<h2>Energy Balance Models</h2>
### Objectives {#M3_2 -}
- To construct a simple energy balance climate model for the Earth in Excel
- To understand the role of greenhouse gases, and incorporate them into an energy balance climate model
### Before you start {#M3_before_start -}
- Listen to the energy balance lecture
### Module resources {#M2_14 -}
Download the Excel spreadsheet for this module by clicking the download button in the <i class="fa fa-download" aria-hidden="true"></i>.
<!--There are no module resources needed for these exercises! But please refer to the supporting information on LMS (Assignment 1/Week3) prior to commencing.-->
### Case Study 1: A “Zero-Dimensional” Energy Balance Model {#M3_3 -}
Here, we assume that the Earth is a sphere with uniform surface and no differences with latitude and longitude (i.e. everything receives the same amount of solar energy). We also assume that there is a balance, or equilibrium, between incoming short wavelength solar radiation absorbed by the ground and outgoing long wavelength radiation emitted by the ground.
To simplify the model further, we assume the Earth emits outgoing (longwave) radiation like a “blackbody”, i.e. it is a perfectly efficient radiator - it gives off the maximum amount of radiation that an object of its temperature can emit.
<center>
![](images/module3/energy_model.png){width=650px data-action="zoom"}
</center>
#### Outgoing Radiation {#M3_4 -}
The rate of the outgoing radiant energy (Joules per second = Watts) from a blackbody is given by the Stefan-Boltzmann equation:
<center>
\begin{equation}
E = \sigma T^{4}
(\#eq:module3-1)
\end{equation}
</center>
where, $\sigma$ = Stefan-Boltzmann constant (5.67x10^-8^), and $T$ = Temperature (in degrees Kelvin).
However, for the Earth, this is being emitted over the full surface area of the Earth:
<center>
\begin{equation}
E_{\text{out}} = \sigma 4\pi r^{2} T^{4}
(\#eq:module3-2)
\end{equation}
</center>
where, $r$ = Radius of the Earth (in m).
#### Incoming Radiation {#M3_5 -}
The energy absorbed by the earth is given by the amount of incoming solar radiation which is not reflected due to albedo. However, we have to take into account that solar energy is striking the Earth as a parallel beam of circular cross-section. So this incoming beam of radiation hits only a single “disc” of the Earth’s surface at any one time. This area has a surface area $\pi r^{2}$.
<center>
\begin{equation}
E_{\text{abs}} = S\pi r^{2} (1-a)
(\#eq:module3-3)
\end{equation}
</center>
where, $S$ = incoming solar radiation: the “solar constant” (1367 W/m^2^) .
#### Exercises {#M3_6 -}
1) Using the concept of an energy balance (i.e. energy in = energy out), rearrange the equations to find the Earth’s temperature, $T$.
2) In Excel, construct a simple spreadsheet and fill in the blank cells of Table \@ref(tab:tabM3-1) to calculate the mean temperature of the Earth using the equation from Q1, assuming an albedo of 0.33 (including in °C and °F):
a) The current mean temperature of the Earth is 288K (15°C). Does this simple model over or under predict the Earth’s temperature? What important factor is missing from this model?
<center>
```{r tabM3-1, echo=FALSE, message=FALSE, warning=FALSE}
library(knitr)
library(kableExtra)
library(hablar)
library(dplyr)
options(kableExtra.html.bsTable = F,
knitr.kable.NA = '')
module3_1 <-
read.csv("tables/module3/module3-1.csv", check.names = FALSE) %>%
convert(chr(Value, Units))
kable(
module3_1,
format = "html",
escape = F,
align = "c",
caption = "Fill in the blank cells to calculate the mean temperature of the Earth."
) %>%
kable_styling(full_width = F, position = "center") %>%
column_spec(1, width_min = "15.5em") %>%
column_spec(2, width_min = "15.5em") %>%
column_spec(3, width_min = "15.5em") %>%
row_spec(1:6, background = '#ffffff00') %>%
scroll_box(width = "100%",
fixed_thead = FALSE)
```
</center>
<br>
```{block2, M3-1, type='rmdtip2'}
Remember: $K = °C + 273.2$ and $°F = 1.8°C + 32$
```
3) Test the sensitivity of this model to the choice of albedo. Using values from 0.3 to 0.7 in Table \@ref(tab:tabM3-2), calculate and plot T (°C)
<center>
```{r tabM3-2, echo=FALSE, message=FALSE, warning=FALSE}
library(knitr)
library(kableExtra)
library(hablar)
library(dplyr)
options(kableExtra.html.bsTable = F,
knitr.kable.NA = '')
module3_2 <-
read.csv("tables/module3/module3-2.csv", check.names = FALSE) %>%
convert(chr("Stefan-Boltzmann constant"))
kable(
module3_2,
format = "html",
escape = F,
align = "c",
caption = "Test the sensitivity of this model."
) %>%
kable_styling(full_width = F, position = "center") %>%
column_spec(1, width_min = "6.5em") %>%
column_spec(2, width_min = "6.5em") %>%
column_spec(3, width_min = "6.5em") %>%
column_spec(4, width_min = "6.5em") %>%
column_spec(5, width_min = "6.5em") %>%
row_spec(1:5, background = '#ffffff00') %>%
scroll_box(width = "100%",
fixed_thead = FALSE)
```
</center>
### Case Study 2: A “Quick Fix” {#M3_7 -}
To try to reduce the error in the model estimates of Earth’s temperature, we can “parameterise” the model using observed or experimentally derived relationships between outgoing radiation and temperature. One such example is:
<center>
\begin{equation}
E_{\text{out}} = x + yT
(\#eq:module3-4)
\end{equation}
</center>
Where, $x = 204$ W/m^2^, and $y = 2.17$ W/m^2^/°C.
This gets inputted into the energy balance equation (energy in = energy out) to give:
<center>
\begin{equation}
\pi r^{2} S (1-a) = 4\pi r^{2}(x+yT)
(\#eq:module3-5)
\end{equation}
</center>
#### Exercises {#M3_8 -}
1) Solve the above equation for $T$ (note, this time, $T$ is in CELSIUS!)
2) Set up a spreadsheet to estimate $T$ for a range of albedo values. Different amounts of glacial ice cover on the Earth could produce albedos from 0.3 for minimal ice to 0.7 for a complete ice cover (this might seem unrealistic, but geologists have recently found evidence to suggest that the Earth was completely ice-covered during Neoproterozoic time 700 million years ago – a “snowball Earth”!)
a) Are your results reasonably close for the present-day Earth mean temperature? What is the sensitivity of this model to small changes in albedo?
<!-- b) Expanding a bit from your model (you might need to consider smaller steps in albedo), at what value of albedo do you think this simplified Earth would experience strong positive feedback involving ice cover, albedo, and mean temperature, and therefore move rapidly toward “snowball Earth” with albedo = 0.7? Could it ever be deglaciated? -->
<center>
```{r tabM3-3, echo=FALSE, message=FALSE, warning=FALSE}
library(knitr)
library(kableExtra)
library(hablar)
library(dplyr)
options(kableExtra.html.bsTable = F,
knitr.kable.NA = '')
module3_3 <-
read.csv("tables/module3/module3-3.csv", check.names = FALSE) %>%
convert(chr(Albedo, x, y))
kable(
module3_3,
format = "html",
escape = F,
align = "c",
caption = "Estimate $T$ for a range of albedo values."
) %>%
kable_styling(full_width = F, position = "center") %>%
column_spec(1, width_min = "11.5em") %>%
column_spec(2, width_min = "11.5em") %>%
column_spec(3, width_min = "11.5em") %>%
column_spec(4, width_min = "11.5em") %>%
row_spec(1:5, background = '#ffffff00') %>%
scroll_box(width = "100%",
fixed_thead = FALSE)
```
</center>
### Case Study 3: Incorporating Greenhouse Gases {#M3_9 -}
This model is a modification of the one described by @harte1988. He showed that it is appropriate, based on the physics of radiation (as water is the major absolver of terrestrial radiation), to divide the Earth’s atmosphere into a lower layer (from the surface to 1.8km altitude, containing 20% of the air and 50% of the water vapour) and an upper layer (containing 80% of the air and 50% of the water vapour).
This time we require three energy-balance equations:
1. For the Earth-atmosphere system as a whole;
2. For the upper layer of the atmosphere;
3. For the lower layer of the atmosphere.
In this model, all energy terms are expressed as long-term average fluxes, all temperatures are absolute. We assume that emissivity = 1 except at the surface, and (most importantly) that each of the three systems is in equilibrium.
<center>
![](images/module3/picture3.png){width=90% data-action="zoom"}
</center>
<center>
```{r tabM3-4, echo=FALSE, message=FALSE, warning=FALSE}
library(knitr)
library(kableExtra)
library(hablar)
library(dplyr)
#options(knitr.kable.NA = '')
module3_4 <-
read.csv("tables/module3/module3-4.csv", check.names = FALSE)
kable(
module3_4,
format = "html",
escape = F,
align = "c",
caption = "Energy balance model of Earth's atmosphere."
) %>%
kable_styling(full_width = F, position = "center") %>%
column_spec(1, width_min = "5em") %>%
column_spec(2, width_min = "15em") %>%
column_spec(3, width_min = "13em") %>%
row_spec(1:12, background = '#ffffff00') %>%
scroll_box(width = "100%",
height = "435px",
fixed_thead = FALSE)
```
</center>
<br>
#### Exercise 1 {#M3_10 -}
Using the chart above, draw the model yourself by hand, incorporating all fluxes.
We can now formulate the three energy-balance equations needed:
**Energy-balance for the Earth-atmosphere system as a whole:**
Energy enters the Earth–atmosphere system from above at the rate $S$ and from below at the rate $W$ (heat generated from nonrenewable energy sources: nuclear and fossil fuels).
Energy leaves these the system by three routes:
1. Reflected solar radiation: $a_{p}S_{av}$;
2. Thermal radiation from the top of the atmosphere: $\sigma T_{u}^{4}$; and
3. The portion of thermal radiation from the surface that is not absorbed in the atmosphere: $(1 − \varepsilon)\sigma T_{s}^{4}$.
Here:
- $\varepsilon$ is the fraction of surface radiation absorbed in the atmosphere (emissivity),
- $\sigma$ is the Stefan-Boltzmann constant, and,
- $T_{u}$ is the surface temperature (absolute). Thus, the energy balance for the whole system is:
<br>
<center>
\begin{equation}
S_{av} + W = a_{p}S_{av} + \sigma T_{u}^{4} + (1 − \varepsilon)\sigma T_{s}^{4}
(\#eq:module3-6)
\end{equation}
</center>
<br>
**Energy-balance for the upper layer of the atmosphere:**
The upper atmospheric layer absorbs a fraction, $k_{u}$, of the solar radiation that strikes it and also receives:
1) Energy radiated **upward** from the lower layer: $\sigma T_{l}^{4}$, and,
2) One-half of the latent heat that accompanies evaporation from the surface (because this layer holds one-half the atmospheric water vapour): $0.5L$.
The upper layer loses energy by thermal radiation upward to outer space **and** downward to the lower layer. Thus, the energy balance for the upper layer is:
<br>
<center>
\begin{equation}
k_{u}S_{av} + \sigma T_{l}^{4} + 0.5L = 2\sigma T_{u}^{4}
(\#eq:module3-7)
\end{equation}
</center>
<br>
**Energy-balance for the lower layer of the atmosphere:**
Energy enters the lower atmospheric layer from above as absorption of a fraction, $k_{l}$, of the solar radiation that enters it and as thermal radiation from the upper layer: $\sigma T_{u}^{4}$. From below, energy enters from:
1) The absorbed portion of thermal radiation from the surface: $\varepsilon \sigma T_{s}^{4}$;
2) One-half the latent-heat flux from the surface: $0.5L$;
3) The sensible-heat flux from the surface, $H$; and
4) The anthropogenic heat flux: $W$.
Energy is lost from this layer by upward and downward radiation. The energy balance for the lower layer is therefore:
<br>
<center>
\begin{equation}
k_{l}S_{av} + \sigma T_{u}^{4} + \varepsilon \sigma T_{s}^{4} + 0.5L + H + W = 2\sigma T_{l}^{4}
(\#eq:module3-8)
\end{equation}
</center>
<br>
#### Exercise 2 {#M3_11 -}
**Solve these equations to find a formula for surface temperature $T_{s}$:**
Equations \@ref(eq:module3-6) to \@ref(eq:module3-8) are a system of three equations in three unknowns: the temperatures $T_{l}$, $T_{u}$, and $T_{s}$. The other quantities are parameters whose values must be given. The values of temperatures can be found via the following steps:
1) Rearrange Equation \@ref(eq:module3-6) for $\sigma T_{u}^{4}$ in terms of $T_{s}$ and parameters.
2) Rearrange Equation \@ref(eq:module3-7) for $\sigma T_{l}^{4}$ in terms of $T_{u}$ and parameters.
3) Substitute the results of step 1 into the results of step 2 to give an equation for $\sigma T_{l}^{4}$ in terms of $T_{s}$ and parameters.
4) Rearrange Equation \@ref(eq:module3-8) for $\varepsilon\sigma T_{s}^{4}$ in terms $T_{l}$, $T_{u}$, and parameters.
5) Put the results of steps 1 and 3 into the results of step 4 and simplify to give the equation for $T_{s}$ as a function of parameters:
<br>
<center>
\begin{equation}
T_{s} = [\frac{(3-3a_{p}-2k_{u}-k_{l})S_{av}-1.5L-H+2W}{(3-2\varepsilon)\sigma}]^{\frac{1}{4}}
(\#eq:module3-9)
\end{equation}
</center>
<br>
```{block2, hintM3-1, type='rmdnote2'}
Show your workings!
```
<br>
Using the values given in Table \@ref(tab:tabM3-4), what value do you derive for $T_{s}$?
How close is this to the actual value of 290K?
#### Exercise 3 {#M3_12 -}
Using Excel, test the sensitivity of this model to albedo. Compare your findings to those found in [Case Study 1](#M3_3) (a plot of $T_{s}$ for values of albedo between 0.3 and 0.7 might help here).
1) For which model, and at what value of albedo, do you think the Earth would experience a strong positive feedback involving ice cover, albedo, and mean temperature, and therefore move rapidly toward “snowball Earth”? Could it ever be deglaciated?
### Submission {#M3_13 -}
:::: {.redbox2}
**Submit properly formatted graphs and tables of the following sections of the lab:**
1) Clear photograph or page scan of your **hand drawn conceptual model** of the *Harte 1-D Energy Balance Model*, including key explaining the terms.
2) Clear photograph or page scan of your handwritten workings of the *Harte Model* calculations, with a clear answer of $T_{s}$, the Earth’s temperature.
3) **Summary plot** from Case Study 1, 2 & 3 assessing albedo and temperature of the three models, with a brief explanation of less than 150 words of which model you prefer.
4) In 100 words or less, your answer to the Exercise 3 “snowball Earth” question.
These are to be uploaded as per the formatting specified in the [Style Guide](#StyleGuide). Marks will be deducted for incorrect formatting.
<!--
The submission is to be all together in a word doc or PDF format. No screenshots of figures from Excel/Excel spreadsheets to be uploaded.
**General professional formatting guidelines:**
- All figures are to have adequate captions explaining them
- For graphs, figure captions go below the plot
- For tables, the caption goes above the table
- Make sure figures and their text size is readable
**Excel hints:**
- When there is a caption for a plot, you remove the title
- Remove the plot border and gridlines
- Make sure both axes have visible lines and tick marks
- Units need to be noted properly with the axis label - 'Temperature (°C)'
- Round numbers to be reasonable
-->
::::