Prof. Dr. Markus Meier
\n",
" Leibniz Institute for Baltic Sea Research Warnemünde (IOW)
\n",
" E-Mail: markus.meier@io-warnemuende.de
Figure 4: Thermal radiation of a blackbody (albedo=0) in an equilibrium energy state as function of the distance to the sun (solid line). The filled circles indicate the planetary equilibrium temeratures also considering the planetary albedos. The stars indicate the actually measured surface temperatures.
\n", "```\n", "\n", "- actually $T_{surf} \\approx 15°C$, have to consider the role of Earth's atmosphere, because greenhouse effect happens. most incoming solar radiation passes through atmosphere due too its short wavelength. outgoing thermal radiation with higher wavelengths is absorbed by the atmosphere and reemitted in all directions, also back to the surface. this implies that less energy escapes to space than what would be the case without an atmosphere. net effect is warming of the surface." ] }, { "cell_type": "markdown", "id": "62b37a65", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Greenhouse shield model\n", "\n", "```{figure} figures/L2/L2_4_atmospheric_transm.png\n", "---\n", "width: 50%\n", "---\n", "Figure 5: Sketch for the greenhouse shield model considering the short- and longwave transmissivity of the atmosphere.
\n", "```\n", "\n", "- can formulate radiation balances for top of atmosphere and Earth's surface\n", "```{math}\n", ":label: balances\n", "\\begin{align*}\n", "F_{solar} &= F_{atm} + \\tau_{lw}F_{thermal}\\\\\n", "\\tau_{sw}F_{solar} + F_{atm} &= F_{thermal}\n", "\n", "\\end{align*}\n", "```\n", "$\\tau_{sw} \\approx 0.67$ - atmospheric shortwave transmissivity \n", "$\\tau_{lw} \\approx 0.06$ - atmospheric longwave transmissivity \n", "\n", "- can quickly eliminate $F_{atm}$ from [](#balances) to obtain:\n", "```{math}\n", ":label: balance_both\n", "F_{solar} - \\tau_{lw}F_{thermal} = F_{thermal} - \\tau_{sw}F_{solar}\n", "```\n", "- which can be reframed to get:\n", "```{math}\n", "F_{thermal} = F_{solar} \\frac{1 + \\tau_{sw}}{1 + \\tau_{lw}}\n", "```\n", "- plugging in the definitions of $F_{solar}$ [](#def_fsolar) and $F_{thermal}$ [](#def_fthermal):\n", "```{math}\n", "\\sigma T_{surf}^4 = \\frac{1}{4}S_0(1-\\alpha_{surf}) \\frac{1 + \\tau_{sw}}{1 + \\tau_{lw}}\n", "```\n", "- now we can get an expression for the surface temperature including the greenhouse effect:\n", "```{math}\n", "T_{surf}^1 = \\sqrt[4]{\\frac{1}{4\\sigma}S_0(1-\\alpha_{surf}) \\frac{1 + \\tau_{sw}}{1 + \\tau_{lw}}} \\approx 285K = 12°C\n", "```" ] }, { "cell_type": "markdown", "id": "08531b9d", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "\n", "```{figure} figures/L2/L2_6_global_energy.png\n", "---\n", "width: 50%\n", "---\n", "Figure 6: Quantitative sketch of the global energy flows (2000-2005 global mean).
\n", "```\n", "- albedo = solar incoming/solar reflected = 100/340 = 0.29\n", "- imbalance = 0.7 $\\frac{W}{m^2}$\n", "- absorbed by surface = absorbed surface/solar incoming = 160/340 = 0.47\n", "- atmospheric shortwave transmissivity $\\tau_{sw}$ = absorbed surface/(solar incoming - solar reflected) = 160/(340-100) = 0.67\n", "- atmospheric longwave transmissivity $\\tau_{lw} \\approx$ 0.06, see atmospheric window " ] }, { "cell_type": "markdown", "id": "29af204f-e3da-4365-8b40-40a6a85b7320", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Greenhouse model with ice-albedo feedback\n", "\n", "- net absorbed energy of $0.7\\frac{W}{m^2}$ -> warming. warming means melting of ice and snow -> albedo decreases (see below values, difference from ice and snow to water): this is the ice-albedo feedback\n", "\n", "| Properties of the ground | Albedo [%] |\n", "| --- | --- |\n", "| Snow | 75-95 |\n", "| Old snow | 50-70 |\n", "| Ice | 30-40 |\n", "| Sand | 20-30 |\n", "| Grass | 15-20 |\n", "| Forest | 5-20 |\n", "| Water | 3-10 |\n", "| Water (with sun close to the horizon) | 10-100 |\n", "\n" ] }, { "cell_type": "markdown", "id": "fd747465", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "- idea of Budyko 1969 to implement ice-albedo feedback into the zero order model with thee greenhouse effect. start from zero order model [](#zom):\n", "```{math}\n", ":label: start_albedo_model\n", "\\gamma \\frac{dT_{surf}}{dt} = \\frac{S_0}{4}(1-\\alpha_{feedback}) - F_{thermal}\n", "```\n", "- adapt two things: 1. linearize earths thermal radiation $F_{thermal}$ and include the greenhouse effect in the parameters and 2. formulate earths albedo $\\alpha$ as a function of T_surf\n", "- 1: linearize thermal radiation as a function of surface temperature:\n", "```{math}\n", ":label: albedo_fthermal\n", "F_{thermal} = A + B*(T_{surf} - 273.15K)\n", "```\n", "$A = 203.3 \\frac{W}{m^2}$ \n", "$B = 2.09 \\frac{W}{m^2K}$\n", "- values for A and B taken from observations so they include the greenhouse effect\n", "\n", "```{figure} figures/L2/L2_7_plot_fthermal.png\n", "---\n", "width: 50%\n", "---\n", "Figure 7: Thermal radiation. left: blackbody radiation, right: blackbody radiation for a narrower temperature range and the linearization of the thermal radiation by Budyko.
\n", "```\n", "- see that linearization in red is a good approach and see that linearization parameter B includes greenhouse effect as red slope is smaller than black slope\n", "- 2: now we want to describe albedo $\\alpha$ as a function of $T_{surf}$ instead of setting a constant value" ] }, { "cell_type": "markdown", "id": "4a32142e-dd88-4ff9-b96d-759f1e0dd7ac", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "- there are two extremes: a totally ice-covered world with the albedo $\\alpha_{ice} = 0.62$, we assign that to be the case for temperatures lower than $T_{ice}=-10°C$. or a totally ice free world with an albedo of $\\alpha_{noice} = 0.30$ for temperatures over $T_{noice}=0°C$. for temperatures in between assume a linear decrease of the albedo:\n", "```{math}\n", ":label: albedo\n", "\\alpha_{feedback}(T_{surf})= \\left\\{\n", "\\begin{array}{ll}\n", " \\alpha_{ice} & T_{surf}Figure 9: Solar and thermal radiative forcing. The surface temperature is at an equilibrium point at the intersections.
\n", "```" ] }, { "cell_type": "markdown", "id": "bb8ae7ae-9b8d-406e-97c2-8e862fd79000", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Climate stability and sensitivity\n", "### Climate potential\n", "- what happens when deviation from these equilibrium temperatures? in general there are two types of equilibria: if adding a small perturbation to a stable equilibrium it returns to its initial state. if adding a small perturbation to an unstable equilibrium the perturbation starts growing and the system will not return to equilibrium state- it will drift away\n", "- for climate science a climate potential P describes these behaviours. P is defined as:\n", "```{math}\n", ":label: potential_def\n", "P(\\chi) = - \\int F_{total}(\\chi)~d\\chi,\n", "```\n", "- in our case of the greenhouse model with ice-albedo feedback: $\\chi$ the variable of interest that the potential depends on, $T_{surf}$; and $F_{total}$ the forcing on the system, for us $F_{solar}-F_{thermal}$. insert into [](#potential_def):\n", "```{math}\n", ":label: substitutions\n", "\\chi \\rightarrow T_{surf}\\\\\n", "F_{total}(\\chi) \\rightarrow F_{solar}(T_{surf})~-~F_{thermal}(T_{surf})\\\\\n", "P(T_{surf}) = -\\int F_{solar}-F_{thermal} ~dT_{surf}\n", "```\n" ] }, { "cell_type": "markdown", "id": "73f2db6d", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "- recalling the definitions of $F_{solar}$ and $F_{thermal}$ yields formula for the climate potential of the greenhouse model with ice-albedo feedback:\n", "```{math}\n", ":label: potential\n", "\\begin{align*}\n", "P(T_{surf}) &= -\\int \\left( \\frac{S_0}{4}(1-\\alpha_{feedback}(T_{surf})) - (A + B(T_{surf}-273.15K)) \\right)~ dT_{surf}\\\\\n", "&= AT_{surf} + B \\left( \\frac{1}{2}T_{surf}^2- T_{surf}*273.15K\\right)-\\frac{S_0}{4} \\int (1-\\alpha_{feedback}(T_{surf}))~dT_{surf}\n", "\\end{align*}\n", "```\n", "```{figure} figures/L2/L2_10_plot_potential_albedo.png\n", "---\n", "width: 50%\n", "---\n", "Figure 10: Climate potential for the greenhouse model with albedo-feedback and its equilibrium states.
\n", "```\n", "- recalling the definition of the potential [](#potential_def) we can describe the forcing of our climate system if the climate potential is known as:\n", "```{math}\n", ":label: forcing\n", "F_{total}(T_{surf}) = -\\frac{dP(T_{surf})}{dT_{surf}}\n", "```\n", "- apply this definition to climate potential shown in Figure 10. in point 1: perturbation to left/right cause force in opposite direction, back to equilibrium. same for point 3. but perturbing the system from point 2 causes a force in the same direction, away from the equilibrium.\n", " 1. Totally ice covered earth is stable.\n", " 2. Partially ice covered earth is unstable.\n", " 3. Earth without ice cover is stable." ] }, { "cell_type": "markdown", "id": "cfc452a1-9c82-4b45-b825-cdbbc20a4c27", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Tipping points of the climate system\n", "- tipping point is a criticial threshold beyond which a small change can cause strong and **often times irreversible** changes to the climate system. figure 11 shows the most important known tipping points of our climate system.\n", "\n", "```{figure} figures/L2/L2_11_tipping_points.png\n", "---\n", "width: 50%\n", "---\n", "Figure 11: Tipping points of the Earth's climate system Earth and under which warming scenarios they may be reached.
\n", "```" ] }, { "cell_type": "markdown", "id": "7a4b1342", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "- tipping point is described by a local maximum of the climate potential. if an additional constant forcing $C$ is added to the total forcing in equation [](#substitutions):\n", "```{math}\n", ":label: add_forcing\n", "P(T_{surf}) = -\\int (F_{solar} - F_{thermal} + C)~dT_{surf},\n", "```\n", "- we obtain\n", "```{math}\n", "P(T_{surf}) = P_0(T_{surf}) - CT_{surf},\n", "```\n", "- with the original potential\n", "```{math}\n", ":label: p0\n", "P_0(T_{surf}) = -\\int (F_{solar}-F_{thermal}) ~dT_{surf}.\n", "```" ] }, { "cell_type": "markdown", "id": "49d2245c", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "- constant additional forcing effectively rotates the potential clockwise\n", "- imagine ball in the modified potential rolling into new equilibrium\n", "```{figure} figures/L2/L2_12_plot_tipping_points.png\n", "---\n", "width: 70%\n", "---\n", "Figure 12: Climate potential of the greenhouse model with ice-albedo feedback for different additional external forcings C. After reaching the tipping point the ball rolls into a new equilibrium.
\n", "```\n", "- after removing the additional forcing to return to the potential $P_0$: system is still in new equilibrum state -> climate change is irreversible!\n", "```{figure} figures/L2/L2_13_plot_tipping_points_back.png\n", "---\n", "width: 70%\n", "---\n", "Figure 13: Decreasing the additional forcing for the climate potential of the greenhouse model with ice-albedo feedback after it reached the tipping point. The system stays in the new equilibrium.
\n", "```" ] }, { "cell_type": "markdown", "id": "e09b1a5b-f140-4779-a2e6-4f183c6bd301", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Climate sensitivity\n", "\n", "- describes the change of temperature or any other variable of interest per change in forcing. forcing can be different things, such as forcing through carbon emissions, a change in solar forcing, volcanic eruptions, many more. we look at\n", "```{math}\n", ":label: sensitivity_def\n", "\\lambda = \\frac{\\Delta T}{\\Delta Q}\n", "```\n", "- for our different climate models from *2.2*.\n", "- 1. IPCC AR6 from 2021 gives different climate change scenarios called SSP... (Shared Socioeconomic Pathways), shown in figure 14. last number of these pathways indicate the additional radiative forcing. for SSP1-2.6: $\\Delta Q = 2.6 \\frac{W}{m^2}$ with a temperature change relative to preindustrial time of $\\Delta T = 2K$. inserting these 2 values into [](#sensitivity_def):\n", "```{math}\n", ":label: sensitivity_ipcc\n", "\\lambda_1 = \\frac{2.0K}{2.6 \\frac{W}{m^2}} = 0.77 \\frac{m^2K}{W}\n", "```\n", "```{figure} figures/L2/L2_14_ipcc_sensitivity.png\n", "---\n", "width: 40%\n", "---\n", "Figure 14: Global temperature change for different warming scenarios based on an increase in radiative forcing.
\n", "```\n", "- 2. zero order model. use equation [](#eq5). Q is absorbed solar radiation $Q=F_{solar}$.\n", " ```{math}\n", " \\lambda_2 = \\frac{dT_{surf}}{dQ} = \\frac{d}{dQ} \\left[ \\left( \\frac{Q}{\\sigma} \\right)^{\\frac{1}{4}} \\right] = \\frac{1}{4\\sigma} \\left( \\frac{Q}{\\sigma} \\right)^{\\frac{-3}{4}} = \\frac{\\left( \\frac{Q}{\\sigma} \\right)^{\\frac{1}{4}}}{4Q} = \\frac{T_{surf}}{4Q} = \\frac{255K}{4*239.4\\frac{W}{m^2}} = 0.27 \\frac{m^2K}{W}\n", " ```\n", "- 3. Greenhouse shield model. using [](#equilibrium_albedo) and noting that the left side is just $F_{solar}$, which is our Q:\n", " ```{math}\n", " \\lambda_3 = \\frac{dT_{surf}}{dQ} = \\frac{d}{dQ} \\left( \\frac{Q -A}{B} -273.15K \\right) = \\frac{1}{B} = 0.48 \\frac{m^2K}{W}\n", " ```\n", "- 4. Greenhouse model with ice-albedo feedback. interesting part is where the world is partially ice covered and the feedback mechanism happens. but previous calculations revealed unstable equlibrium in this area. it doesnt make sense to analyse cliate sensititvity around unstable points. small perturbations cause high temperature changes which are not actually caused by the change in forcing but by the system falling into a new equilibrium. if we reduce the albedo gradient from $-0.032\\frac{1}{K}$ to $-0.003\\frac{1}{K}$. furthermore we assume $Q = \\frac{S_0}{4}$, so that the effects of the albedo feedback can be noticed. we can rewrite [](#equilibrium_albedo):\n", " ```{math}\n", " Q(1-\\alpha_{ice} + 0.003\\frac{1}{K}(T_{surf}-T_{ice}) = A + B(T_{surf} +273.15K)\n", " ```\n", "- rewriting this expression for $T_{surf}(Q)$ (can be done as homework) yields:\n", " ```{math}\n", " \\lambda_4 = \\frac{dT}{dQ} = 0.66 \\frac{m^2K}{W}\n", " ```\n", "- ice-albedo feedback increases climate sensitivity drastically.\n", "\n" ] }, { "cell_type": "markdown", "id": "8c8e62bf", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Feedback mechanisms\n", "- in general we have climate system Earth and look at the temperature. change in temperature can trigger two different kinds of feedback. first a positive feedback, which is a reaction of the system to the change in temperature, that further increases the trend. negative feedback is the opposite: the system reacts to a change in temperature in a way opposing the original trend thus decreasing it. lets go through a few examples:\n", "1. ice-albedo feedback: temperature increases which causes the melting of surface ice, which increases the planetary albedo which leads to an increase in absorbed solar radiation.\n", "\n", "```{figure} figures/L2/L2_15_icealbedo_feedback.png\n", "---\n", "width: 30%\n", "---\n", "Figure 15: Scheme of the ice-albedo feedback.
\n", "```\n", "```{figure} figures/L2/L2_16_blackbody_feedback.png\n", "---\n", "width: 30%\n", "---\n", "Figure 16: Scheme of the blackbody radiation feedback.
\n", "```\n", "```{figure} figures/L2/L2_17_watervapor_feedback.png\n", "---\n", "width: 30%\n", "---\n", "Figure 17: Scheme of the water vapor feedback.
\n", "```\n", "```{figure} figures/L2/L2_18_cloudssolar_feedback.png\n", "---\n", "width: 30%\n", "---\n", "Figure 18: Scheme of the solar cloud feedback.
\n", "```\n", "```{figure} figures/L2/L2_19_cloudsthermal_feedback.png\n", "---\n", "width: 30%\n", "---\n", "Figure 19: Scheme of the thermal cloud feedback.
\n", "```\n", "*Courtesy: D. Dommenget*" ] }, { "cell_type": "markdown", "id": "5cf311b8-fdcc-4c67-8189-81858d460278", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Questions:\n", "1. Explain what an Earth system approach is. Which are the five components of the Earth climate system? Provide brief characteristics.\n", "2. List three important greenhouse gases.\n", "3. Explain the greenhouse effect in detail.\n", "4. Define climate feedbacks and climate sensitivity and relate the two quantities to each other. Explain at least one example of a climate feedback.\n", "5. What is a tipping point of the climate system? (Hint: draw a sketch of a climate potential and explain in words why changing climate might be irreversible.)" ] }, { "cell_type": "markdown", "id": "82e5f812-bec9-4e95-a98f-231243c4b6b8", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [ "## Table of figures\n", "\n", "Figure 1: FAQ 1.2, Figure 1 in IPCC, 2007: Chapter 1. In Climate Change 2007: The Physical Science Basis. Contribution of Working group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change [Core Writing Team, Pachauri, R.K. and Reisinger, A. (eds.)]. IPCC, Geneva, Switzerland, 104pp.\n", "Figure 2: https://www.geomar.de/en/discover/ocean-and-climate/model-simulations/the-earths-climate-system, access on 22.02.2024\n", "Figure 3: Robert A. Rohde, https://commons.wikimedia.org/wiki/File:Atmospheric_Transmission.png, access on 24.02.2024, GNU Free Documentation License\n", "Figure 6: IPCC, AR6 Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change [Forster, P., T. Storelvmo, K. Armour, W. Collins, J.-L. Dufresne, D. Frame, D.J. Lunt, T. Mauritsen, M.D. Palmer, M. Watanabe, M. Wild, and H. Zhang, 2021: The Earth’s Energy Budget, Climate Feedbacks, and Climate Sensitivity. In Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change [Masson-Delmotte, V., P. Zhai, A. Pirani, S.L. Connors, C. Péan, S. Berger, N. Caud, Y. Chen, L. Goldfarb, M.I. Gomis, M. Huang, K. Leitzell, E. Lonnoy, J.B.R. Matthews, T.K. Maycock, T. Waterfield, O. Yelekçi, R. Yu, and B. Zhou (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, pp. 923–1054, doi: 10.1017/9781009157896.009 .] \n", "Figure 11: D. I. Armstrong McKay et. al.. Exceeding 1.5°C global warming could trigger multiple climate tipping points. Science, Vol. 377, Issue 6611, 2022. https://doi.org/10.1126/science.abn7950\n", "Figure 14: Figure 1.16 in IPCC, 2021: Chapter 1. In: Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change [Chen, D., M. Rojas, B.H. Samset, K. Cobb, A. Diongue Niang, P. Edwards, S. Emori, S.H. Faria, E. Hawkins, P. Hope, P. Huybrechts, M. Meinshausen, S.K. Mustafa, G.-K. Plattner, and A.-M. Tréguier, 2021: Framing, Context, and Methods. In Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change [Masson-Delmotte, V., P. Zhai, A. Pirani, S.L. Connors, C. Péan, S. Berger, N. Caud, Y. Chen, L. Goldfarb, M.I. Gomis, M. Huang, K. Leitzell, E. Lonnoy, J.B.R. Matthews, T.K. Maycock, T. Waterfield, O. Yelekçi, R. Yu, and B. Zhou (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, pp. 147–286, doi: 10.1017/9781009157896.003 .]" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" } }, "nbformat": 4, "nbformat_minor": 5 }