phyHydroWiSe2324.pdf

Physical Hydrology WS 2023/24, Thursday 10:0-11:30h, 1002/B Prof. Dr. Harald Kunstmann Email: [email protected] Prof. Dr. Harald Kunstmann Lehrstuhl für Regionales Klima und Hydrologie Forschungsschwerpunkte der Arbeitsgruppe Auswirkungen des regionalen Klimawandels auf den Wasserkreislauf Dynamisches und statistisches Downscaling von meteorologischen Feldern Vollgekoppelte regionale atmosphärische und hydrologische Modellierung @ KIT Campus Alpin, Anwendung kommerzieller Mikrowellen-Links (CMLs) zur Garmisch-Partenkirchen Niederschlagsbestimmung Geostatistisches Merging von hydrometeorologischen Variablen Saisonale Vorhersagen Research Focus Africa Lehre am Institut für Geographie Physikalische Hydrologie, Exp. Methoden in der Hydrologie, hydr. Modellierung Mathematik für Geographen, Geostatistik Python Basics @ KIT Campus Alpin, Garmisch-Partenkirchen @ KIT Campus Alpin, Garmisch-Partenkirchen Anbindung an den KIT Campus Alpin von der Uni Augsburg abgeordnet zu Forschungszwecken an den Campus Alpin des Karlsruher Instituts für Technologie Institut für Meteorologie und Klimaforschung in Garmisch-Partenkirchen Klima- und Wasserforschung weltweit Forschungsbeispiele How to investigate the hydrological cycle? Observational networks Hydrological models Usage of observations to estimate Usage of models to simulate the amounts of e.g. precipitation, runoff, quantities of the hydrological cycle groundwater flow Model equations based on the physics of the environment Observed annual mean precipitation 1970 - 2000 (CRU TS 2.0) in [mm/day]; Scheme of 3d atmosphere - land surface - [https://data.giss.nasa.gov, last access: 08/2018] subsurface model; [B. Fresch, KIT/IMK-IFU] 8 Hydrological cycle 9 Residence times 10 Lecture content Solar forcing & energy balance Evapotranspiration & turbulent fluxes Atmospheric water & water transport Condensation & clouds Precipitation Soil water processes Subsurface flow: Darcy’s law & general unsaturated flow equation Channel flow Snow & snow melt Stable water isotopes 11 23.11.2023: Lecture is Part of PCS 12 Literature Dingman L.S., Physical Hydrology, 3nd edition, Prentice Hall, 2015 Hornberger G. et al., Elements of Physical Hydrology, John Hopkins University Press, 1998 Chow, Maidment, Mays, Applied Hydrology, McGraw Hill, 1998 Tindall, J., Kunkel, J., Unsaturated Zone Hydrology for Scientists and Engineers, Prentice Hall, 1999 Bras R., Hydrology – An Introduction to Hydrologic Sciences, Addison Wesley, 1990 Dyck, Peschke, Grundlagen der Hydrologie, Verlag für Bauwesen, Berlin, 1989 Eagleson P., Dynamic Hydrology, McGraw Hill, 1970 13 Recommended Further Lectures Obligatory for Module „Hydrology“ - Hydrological Modeling (Dr. Thomas Rummler) - Seminar “Experimental Methods in Hydrology“ (SS2023) Recommended - Mathematik für Geographen (Dr. Jan Bliefernicht, SS 2023) - Geodatenverarbeitung mit Python (Dr. Thomas Rummler, SS 2023) 14 15 Solar forcing Stefan-Boltzmann law Energy balance Greenhouse effect Global distribution of insolation Global circulation system Variations of solar forcing (Milanković cycle) 16 Solar forcing Solar radiation is energy, released as electromagnetic waves; radiation can be seen on the electromagnetic spectrum: Increased energy 0,4 - 0,7 µm 17 Solar forcing Solar radiation is energy, released as electromagnetic waves; radiation can be seen on the electromagnetic spectrum: Increased energy Sun: 0,2-4 µm 0.2 to 3 μm shortwave radiation 0,4 - 0,7 µm 18 Solar forcing Solar radiation is energy, released as electromagnetic waves; radiation can be seen on the electromagnetic spectrum: Increased energy Sun: 0,2-4 µm 0.2 to 3 μm shortwave radiation λ.ν=c 0,4 - 0,7 µm λ: Wavelength ν: Frequency c: Speed of light (3.108 ms-1) 19 Solar forcing Sun’s energy arrives at an average rate of 1.74×1017 W Solar constant S = 1,364 Wm-2 is the averaged arrival rate divided by the area of the planar projection of earth (1.28×1014 m2) Interactions of matter and radiant energy at a given wavelength (λ): Absorptivity (αλ) is the fraction of the incident energy at a given λ that is absorbed by surface Reflectivity (ρλ) is the fraction of the incident energy at a given λ that is reflected by the surface Transmissivity (τλ) is the fraction of the incident energy at wavelength λ that is transmitted through the matter For 0 ≤ (αλ, ρλ, τλ) ≤ 1 is: For many materials on Earth τλ= 0 20 Solar forcing: Stefan-Boltzmann law Incoming Reflected All matter at a temperature above zero radiation radiation radiates energy in form of electromagnetic waves traveling at the speed of light This energy is emitted with a rate which is given by the Stefan–Boltzmann law: FEM 𝑭𝑬𝑴 = 𝝐 # 𝝈 # 𝑻𝟒 FEM: Energy flux in Wm-2 σ: universal constant : 5.67×10-8 Wm-2K-4 (Stefan-Boltzmann constant) Surface with ε=1 is called blackbody; for most materials ε: Emissivity of surface [.], 0 ≤ ε ≤ 1 on earth is ε≈1 (e.g. fresh T: Absolute temperature in K snow: ε=0.99, dry sand: ε=0.95) 21 Solar forcing: Energy balance (average) Average global energy balance of the earth Average insolation over spherical atmosphere system surface at the top of the atmosphere is 341 Wm-2 After absorption and reflection by atmosphere, 184 Wm-2 of energy reach the surface; most of this energy is in the visible range [0.4 to 0.7 μm] Fraction of insolation that is reflected by surface is called albedo: Dingman, 2015 Solar (shortwave) radiation On average 161 Wm-2 absorbed by Terrestrial (longwave) radiation Earth’s surface 22 Solar forcing: Energy balance (average) Average global energy balance of the earth atmosphere system Dingman, 2015 Solar (shortwave) radiation Terrestrial (longwave) radiation 23 Solar forcing: Energy balance (average) Average global energy balance of the earth Energy for evaporation or turbulent atmosphere system processes is provided by surface warming Following Stefan–Boltzmann law the average surface temperature (16°C) emits terrestrial radiation (396 Wm-2) Gases and clouds absorb most of radiation (356 Wm-2) emitted by surface (Greenhouse effect) Radiation resulting from correspon- ding heating produces a upward flux Dingman, 2015 to outer space (199 Wm-2) and a backward flux to the surface Solar (shortwave) radiation (333 Wm-2) Terrestrial (longwave) radiation 24 Solar forcing: Greenhouse effect Absorption spectrum for atmosphere Absorptivity Different gases and clouds strongly absorb only specific wavelengths of the longwave radiation Without greenhouse the average surface temperature would be about -18°C Energy flux [cal cm-2 min-1 µm-1] Often discussed “Greenhouse gases” (GHGs) are : CO2, N2O, CH4, O3, and CFCs Only 25% of the greenhouse effect is directly due to GHGs; the warming induced by increasing GHGs allows the air to hold more water vapor and clouds, creating a positive feedback that produces the remaining effect Wavelength [µm] Outgoing radiation Incoming radiation Dingman, 2015 26 Solar forcing: Greenhouse effect Greenhouse gases: absorb and emit radiant energy within the thermal infrared range Carbon Dioxide Methane Solar forcing: Greenhouse effect Greenhouse gases: absorb and emit radiant energy within the thermal infrared range Electromagnetic wave: absorbtion at resonance frequency Solar forcing: Greenhouse effect Greenhouse gases: absorb and emit radiant energy within the thermal infrared range CO2 https://www.esrl.noaa.gov/gmd/ccgg/trends/ CO2 Solar forcing: Distribution of insolation Columns represent energy hitting the surface, which is the same for each column Area which is hitting by the same amount of energy is larger near the poles as near Dingman, 2015 equator As a consequence at the poles the incoming energy flux is lower 33 Solar forcing: Global circulation 1. Hadley cell: Near the Equator warm, moist air is lifted, creating equatorial low-pressure areas (Intertropical Convergence Zone, ITCZ), to the tropopause About 30°N or S some of the air fuels the tropical jet, but most of the air descend to form high-pressure areas Some of the descending air travels equatorially along surface Polar front © Sonjia Leyva 2018 H L H L 34 Solar forcing: Global circulation 2. Polar cell: Cool air masses at the poles descend creating a cold, dry high-pressure area From there air moves along the surface towards the sub-tropics, until approx. 60°N or S; here the air warms and raises, and moves back poleward Outflow from the polar cell creates waves which play important role in determining the path of the Polar jet Polar front © Sonjia Leyva 2018 H L H L 35 Solar forcing: Global circulation 3. Ferrel cell: Surface winds flow poleward form a high- to a low-pressure area Approx. 60°N or S air converges to ascend along the boundary between cool and warm air; the return flow at high altitudes towards the tropics join the sinking air from Hadley cell Ferrel cell is not temperature driven, acting like a gear between two other cells Polar front © Sonjia Leyva 2018 H L H L 36 Solar forcing: Global circulation - Coriolis force Why the Earth hasn’t only one circulation cell Only if the Earth would not rotate and was a simple land mass with no oceans, the Earth would have only one circulation cell! Since the Earth rotates, the Coriolis force appears perpendicular to the Earths axis Winds in northern hemisphere are deflected "to the right“ Winds in southern hemisphere are deflected "to the left“ 37 Arbogast, 2007 Solar forcing: Influence of global circulation Distribution of precipitation Glawion et al., 2009; Dingman, 2002 Source: http://www.geocities.com /CapeCanaveral/Hall/6104/atmosphe.html 38 Solar forcing: Distribution of insolation AU := astronomical unit = 149.597.870.700 m ± 3 m 39 Solar forcing: Orbital variations (Milanković cycle) 40 Solar forcing: Orbital variations (Milanković cycle) Stages of Glaciation forced by the Milanković cycle [http://www.erdwissen.ch/2017/06/milankovic-zyklen-und-das-klima, last access: 07/2018] 41 Solar forcing Simple 0-dim energy balance model of planetary temperature (Tp): To maintain energy balance, average emitted energy (Stefan-Boltzmann Law) must equal i: 𝑖 = 𝐹!" = 𝜀 % 𝜎 % 𝑇 # % 𝐴 Ɛ≈1 Tp Calculate Tp with: S = 1.74×1017 W σ = 5.67×10-8 Wm-2K-4 Tp≈ 254 K ≈ -18°C A = 5.10×1014 m2 ‼ A includes atmosphere 42 Summary Greenhouse Gas Impact 43 Globale Abschätzung für die Zukunft STIEG RAN LIG A TU INMA R PE VTL E EM ER T ZEIT E RTIG ZEN Ä ENW R KUR G GE IESE IN D... entsprechen den Unterschieden zur letzten Kaltzeit! Ist Klimaänderung neu? Höhepunkt der letzten Vereisung: vor 20 000 Jahren Isar/Loisach-, Inn-, Salzach- Vorlandgletscher: Eisdicke in Garmisch rund 1000 m Freie Gipfel von Zugspitze & Watzmann Quelle: Krenmayer and Hofmann 2002 Quelle: Bayerisches Landesamt für Wasserwirtschaft Warum machen wir uns Sorgen? Basierend auf Osman, M.B., Tierney, J.E., Zhu, J. et al. Globally resolved surface temperatures since the Last Glacial Maximum. Nature 599, 239–244 (2021). https://doi.org/10.1038/s41586-021-03984-4 Warum machen wir uns Sorgen? °C gl ob a l kälter -6 u h eute: l ei c h z im Verg erer 120 m tief iegel Meeressp Quelle: https://www.spiegel.de/wissenschaft/mensch/was-die-eiszeit-ueber- den-klimawandel-lehrt-a-cfce2e0b-564d-4887-949d-ab3bafec4363 Warum machen wir uns Sorgen? l kälter -6°C globa u heute: r gl ei c h z im Ve ocke n e A dria! tr https://de.wikipedia.org/wiki/Weichsel-Kaltzeit#/media/Datei:Weichsel-W%C3%BCrm-Glaciation.png Water Balances Air moisture and it’s measures Atmospheric moisture flux Balance equations for atmospheric moisture Discussion of long term mean datasets regarding the water balance 50 cle? y alc l ogic o dr h y bal ets g l o s he ata d owt bal kn g lo l l y d r ea s an e l o w ode ell d ba lm oww - G lo H Modelling Modelling Precipitation Soil Moisture Evapotranspiration State of the Art Reanalyses & Observation Data Sets GPCC: Global Precipitation Climatology Centre GPCP: Global Precipitation Climatology Project CRU: Climate Research Unit CPC: Unified gauge based analysis of Global Daily Precipitation from Climate Prediction Centre DEL: University of Delaware Air Temperature & Precipitation Rainfall Gauges per Gridcell Rainfall Gauges per Gridcell Signiﬁcant decrease in the number of gauges! Africa Example West Africa: Both a scientific and a data challenge © Dr. Jan Bliefernicht, UniAugsburg Number of monthly measurements used for interpolation of the GPCC rainfall reanalysis version 7 (West Africa) Africa Example West Africa: Both a scientific and a data challenge Illustration of Problem: GPCC Precipitation August 2007 © Dr. Jan Bliefernicht, UniAugsburg Africa Example West Africa: Both a scientific and a data challenge Illustration of Problem: GPCC Precipitation August 2007 N i ge r i a?.g f or t st i m at es e dp r oduc PC C e gr idde G eh i n d t he le ga uge b si n g is not a T h er e © Dr. Jan Bliefernicht, UniAugsburg How about Germany? How about Germany? o r er… po eh ow et som ic hg o the r ls …a Trends from Gridded Observation Data Sets? Trend des Jahresniederschlags zwischen 1979 und 2010 Trend des Jahresniederschlags zwischen 1979 und 2009 Datengrundlage: GPCC Version 6.0 Datengrundlage: CRU Version 3 P - d at a d r g r i dde o s o u r ce f c t i o n of n d s! s e l e : n t r e d u e to c e s sar y ci p i t at i o ty e re n c e r t ai n a u t i on n ns of p C g U f e r e n t si i f , e v en d ly e g i o n al R [mm/year] [mm/year] 15 10 5 0 5 10 15 15 10 5 0 5 10 15 Karlsruher Institut für Technologie Karlsruher Institut für Technologie Institut für Meteorologie und Umweltforschung (IMK-IFU) Institut für Meteorologie und Umweltforschung (IMK-IFU) Trends from Gridded Observation Data Sets? Europe Mittlerer Jahresniederschlag zwischen 1979 and 2010 Trend des Jahresniederschlags zwischen 1979 and 2009 Datengrundlage: GPCC Version 6.0 Datengrundlage: CRU Version 3 a n ean e n d s: d i t er r r e d s! c a le t h e M e n r g es e r in t i o n tr l a a t a t m i l ar s sw e c i pi t S i l e r r o p e, s of p u n e r nE n t si g e N or t h d i f f er n a t er i y e ven re w ll Mo g i ona e a i n, r u t : Ag B [mm/year] [mm/year] 5 4 3 2 1 0 1 2 3 4 5 5 4 3 2 1 0 1 2 3 4 5 Karlsruher Institut für Technologie Karlsruher Institut für Technologie Institut für Meteorologie und Umweltforschung (IMK-IFU) Institut für Meteorologie und Umweltforschung (IMK-IFU) Precipitation and Temperature Dataset Variability Variability in each pixel: Vi,j = max(ens)i,j - min(ens)i,j The Hydrological Cycle dS Large-scale terrestrial = P− E− R dt Long-term terrestrial R≈P −E Oceanic-continental P land − E land = − ( P ocean − E ocean ) € dW ⃗ ⃗ Atmospheric-terrestrial (a) + ∇⋅Q = E− P dt dS dW ⃗ ⃗ Atmospheric-terrestrial (b) =− − ∇⋅Q− R dt dt Long-term atmospheric-terrestrial (a) ⃗ ⋅Q= ∇ ⃗ E− P Long-term atmospheric-terrestrial (b) ⃗ ⋅Q= −∇ ⃗ R Water balance equations Balance equations - long term mean data sets Balance between oceans and land not closed! 66 Balance equations - long term mean data sets Table shows long term mean of the ﬂuxes, in contrast to the >me series depicted below Issues within this analysis: (P-E)CFSR is overestimated (P-E)CFSR,land is overes>mated compared to the other data sets (seen in both, long term mean and >me series) mean 1998 - 2006 Due to the assimila>on of new satellite data huge gaps in (P- E)ocean for MERRA and CFSR (only seen in >me series) 1998: assimilation of mean 1989 - 1998 new satellite data 67 Revision – Basic Maths Derivations and its corresponding difference quotients: ! 𝑓 𝑡 = Application: Calculation of next time step (t+Δt): Δf (Total) differential and partial differential for multi- dimensional functions: total differential t Δt partial differentials 68 Revision – Basic Maths Velocity in cartesian coordinate system u ⃗ = '⃗𝑢 + *⃗𝑣 + 𝒌 𝒗 ⃗𝑤 ⃗ = v 𝒗 𝟐 𝟐 𝟐 ⃗ = 𝒗 𝒖 +𝒗 +𝒘 w '⃗, *,⃗ 𝒌 ⃗ unit vectors: 1 0 0 ⃗1 = 0 *⃗ = 1 ⃗ = 0 𝒌 0 0 1 𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑢 = 𝑣 = 𝑤 = 𝑑𝑡 𝑑𝑡 𝑑𝑡 69 Revision – Basic Maths ! ! ! ! a = (ax ,ay ,az ) = iax + jay + kaz ! ! ! a ⋅ b = ax bx + ayby + azbz = a b cosθ ! ! ! c = a × b = (aybz − azby ,azbx − ax bz ,ax by − aybx ) ! ! c = a b sin θ Revision – Basic Maths Nabla operator 𝜕 𝜕 𝜕 𝛻⃗ = 𝚤⃗ + 𝚥⃗ + 𝑘⃗ … is an operator; result is a vector 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑓 𝜕𝑓 𝜕𝑓 … operator applied to function f; result is a vector. 𝛻𝑓 = 𝚤⃗ + 𝚥⃗ + 𝑘⃗ 𝜕𝑥 𝜕𝑦 𝜕𝑧 $ $ $ $( $) $* 𝛻 B 𝑣⃗ = 𝚤⃗ + 𝚥⃗ + 𝑘⃗ B 𝚤⃗𝑢 + 𝚥⃗𝑣 + 𝑘⃗𝑤 = + + $% $& $' $% $& $' … result is a scalar. $ $ $ $ $ $ 𝑣⃗ B 𝛻⃗ = 𝚤⃗𝑢 + 𝚥⃗𝑣 + 𝑘⃗𝑤 B 𝚤⃗ + 𝚥⃗ + 𝑘⃗ =𝑢 +𝑣 +𝑤 $% $& $' $% $& $' … is an operator; result is a scalar. 71 Revision – Basic Maths G ra d i w it h e n t : e x p N ab r la - O e s s e d p e ra to r → 𝟐𝒙 𝟐 𝟐 𝒇 𝒙, 𝒚 = 𝒙 + 𝒚 𝛁𝒇 𝒙, 𝒚 = 𝟐𝒚 72 Revision - mathematical basis Divergence: $( $) $* 𝛻 B 𝑣⃗= + + $% $& $' Interpretation: It is a local measure of its "outgoingness" – the extent to which there is more of some quantity exiting an infinitesimal region of space than entering it. 73 Atmospheric Water - Air Moisture Example of daily global distribution of vertical integrated water vapor: Air moisture or air humidity is the fraction of water vapor in the air Transport of water vapor is always associated with transport of energy (latent heat) Latent heat flux is one driver for many atmospheric processes 74 Atmospheric water - moisture flux Definition vertical integrated moisture fliux p=0 p : air pressure [kg/m/s²] Q q(p) : specific humidity  vh ( p ) q(p) [kg H2O/kg moist air] € dp  vh ( p ) : horizontal wind speed [m/s] € g : Earth acceleration [m/s²] € p = psurface soil  p= psurfac e  € vh ( p) q( p) Q= ò → Column integrated moisture flux Q [kg/m/s]: dp p=0 g Atmospheric water - moisture flux Exchange atmosphere & land surface ∂S % % ∂W R+ =P ! $ −"ET $#a = −∇⋅ Q − !"#∂t !$$"$$ ∂t # Terrestrial Atmospheric Q ΔW water budget % % water budget Q’ ≈ −∇⋅ Q P ET with div Q: R ! ! 1 ! p = p sfc ! ΔS € ∇⋅ Q = ∇⋅ g ∫ p =0 v h ( p) q( p)dp Vertically integrated moisture€ divergence div Q Atmospheric water - moisture flux [Q] = kg/m/s [P] = kg/m²/s ΔW Q P Q’ [ET] = kg/m²/s [ΔW] = kg ET [ΔS] = kg [R] = kg/s R ΔS ! Atmosphere ΔW = − ∫∫ (P − ET)dAsurface dt − ∫∫ ∇⋅ QdAlateral dt ΔW ≈ 0 für ausreichend langes Δt (1 Monat oder länger) Soil ΔS = + ∫∫ (P − ET) dA surface dt − ∫ Rdt € The Hydrological Cycle (II) dS Large-scale terrestrial = P− E− R dt Long-term terrestrial R≈P −E Oceanic-continental P land − E land = − ( P ocean − E ocean ) € dW ⃗ ⃗ Atmospheric-terrestrial (a) + ∇⋅Q = E− P dt dS dW ⃗ ⃗ Atmospheric-terrestrial (b) =− − ∇⋅Q− R dt dt Long-term atmospheric-terrestrial (a) ⃗ ⋅Q= ∇ ⃗ E− P Long-term atmospheric-terrestrial (b) ⃗ ⋅Q= −∇ ⃗ R Water balance equa0ons Atmospheric water - measures Water vapor density ρv [kg・m-3] Density of dry air ρd [kg・m-3] Density of moist air ρa 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 ⇢a = ⇢d + ⇢v [kg・m-3] ⇢v Specific humidity qv qv = [kg H2O・kg-1 moist air] 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 ⇢a pv = ⇢v Rv T , Water vapor pressure pv ⇥ 1 1 ⇤ [mbar] Rv = 461.40 J · kg 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 K pd = ⇢d Rd T , Partial pressure due to dry air pd ⇥ 1 1 ⇤ [mbar] 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 Rd = 287.04 J · kg K Gas law for moist air 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 pa = ⇢a Ra T 79 Atmospheric water - measures ✓ ◆ 17.3 · T Saturation vapor pressure pv,s pv,s ⇡ 0.611 611 · exp [Pa]=[Nm-2], and T in [°C] 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 T + 237.3 pv Relative Humidity Rh Rh = [.] 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 pv,s pv,s(15°C)=1709,9 N/m2 pv,s(16°C)=1818.7 N/m2 1°C Temperaturerhöhung (von 15°C ausgehend)

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