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Response of snowfall extremes to climate change: theory and simulations Paul O’Gorman, MIT Workshop on Water in the Climate System, 2014 Simulations of climate warming: declines in annual-mean snowfall in many regions 709 710 711 712 713 714 715 716 Figure 2. Multi-model ensemble trends in snowfall (2006 to 2100, units cm year-1 decade-1). Annual (A), SON (B), DJF (C), and MAM (D). Contours of 2-m temperature at intervals of 10 ºC are also shown from the multi-model ensemble for the period 1986-2005. Hatching denotes regions of statistically significant trends (p <= 0.01). CMIP5 multimodel mean trends in snowfall depth (2006-2100 based on rcp4.5) Contours of 2m temperature shown in degrees Celsius for 1986-2005 35 Krasting et al, J. Climate, 2013 What about snowfall extremes? (heavy daily snowfall events) • Important because of disruption of transportation (roads, air, rail), business, schools • May not respond to climate change like mean snowfall e.g., heavy snowfall events in both anomalously cold and warm years (Kunkel et al, 2013; Changnon et al 2006) 1928 Regional studies of observed snowfall extremes: Decadal variability but inconsistent long-term trends JOURNAL OF CLIMATE VOLUME 14 FIG. 6. Nationally averaged 20-yr return values (relative to the values for 1971–90) of annual maximum daily precipitation, rainfall, and snowfall. The 20-yr return values are first estimated using 20-yr running windows for every station, and then normalized by the values estimated for the period 1971–90. Values are plotted in the center of the 20-yr window. Zhang et al, J. Climate, 2001 (Canadian observations) ally fewer events after the mid-1970s during both spring ing. The observed 18C warming trend over Canada Effect of climate change on daily snowfall extremes in global simulations • High percentiles of daily snowfall in liquid water equivalent • CMIP5 (use 20 models) under RCP8.5 • Compare warm climate (2081-2100) to control climate (1981-2000) Ratio 0.5 Analyze according Extremes to climatological temperature in Mean control climate 0 Ratio (warm/control) 1.5 1 0.5 0 −20 −15 −10 −5 0 5 10 Climatological temperature (°C) in control climate Grid boxes and days binned by climatological monthly surface air temperature in control climate 99.99th 99.99th percentile (mm day−1) 40 30 20 Observations Simulations (control) 10 Simulated snowfall extremes compare well with estimates from observations 0 99.9th percentile (mm day−1) 40 30 99.9th 20 10 0 99th percentile (mm day−1) 40 99th 30 20 Northern hemisphere land only 10 Observational estimates based on daily precipitation rates (GPCP 1DD), snowfall fraction (Feiccabrino et al 2013), and daily surface temperatures (NCEP2) 0 Mean (mm day−1) Mean 1 0.5 0 −20 −15 −10 −5 0 5 Climatological temperature (°C) in control climate 10 Ratio 0.5 Extremes Response of mean snowfall to climate warming: Mean ratio of warm over control-climate values 0 Ratio (warm/control) 1.5 1 0.5 0 −20 −15 −10 −5 0 5 10 Climatological temperature (°C) in control climate (Northern Hemisphere land below 500m) Rati 0.5 Extremes Mean Weaker response of daily snowfall extremes as compared to mean snowfall 0 Ratio (warm/control) (warm/control) Ratio (b) (a) 1.5 1.5 11 Increasing Percentile 0.5 0.5 99.99th Extremes Theory Mean Theory Simple 00 /control) (b) −20 1.5 1 −15 −10 −5 0 5 10 Climatological temperature (°C) in control climate 99th, 99.9th and 99.99th percentiles of daily snowfall Features of the response of snowfall extremes that would like to understand: • Climatological temperature at which snowfall extremes response goes from positive to negative • Weaker fractional changes at higher percentiles Simple theory (based on known physics/ observations) for the response of snowfall extremes to changes in climate ds on the vertical temperature ple theory is next developed that accounts for the main features of the resp Theory assumptions 1: 22 s to climate change. Surface precipitation on the Relate daily snowfall rate (s) to (p)vertica d to surface air temperature precipitation .type dependsrate and surface air temperature (T) e lower troposphere23 , but to first order it may be related to surface air t pitation rate p by s = f (T )p, nowfall rate s in the theory is related to the daily precipitation rate p b the daily air temperature, andfalls f (T ) is the fraction of precipita Snowfall fraction: tion ofsurface precipitation that 100 Snowfall fraction (%) Control snowfall fraction) 80at a given temperature T . The snowfall fraction in th fall fraction in the simulations Warm 60 p decline near freezing in the multimodel median (Fig. 2) and in most of 40 .)S2). need not occur precisely at a surface t andThis in rain-snow most oftransition the individual 20 0 −9 −6 −3 0 3 6 9 Daily surface air temperature (°C) Figure 2 do not immediately e hydrometeors change phase as they cross the mel Fig. 2. Importantly for the theory, and as expected given mod Theory 2: ow is almost exactly sametheory in the control and wa ilytransition precipitation rateassumptions inthethe is assume Relate precipitation rate (p) to temperature (T) and simulated snowfall in Fig. 2. Importantly for the theory, and as expected gi ure according to p = e p̂, where = 0.0 in lapse rates, the rain-snow transition almostaexactly the dependence same in the contro n rate in the theory is assumed toishave simple on T p (Fig. (T 2). Tc ) =e p̂, and simulated snowfall1in Fig. 2. Importantly for the the where = 0.06 C is a representative ther in lapse rates, rain-snow transition is almost ase ofTheextratropical extremes wi daily precipitation rateprecipitation in thethe theory is assumed to have a simpleexactly depen opical extremes respect= to surface (T with Tc ) p̂, where 1 is a tempera (Fig. 2).evariable temperature according to p = 0.06 C velocity; representa Normalized precipitation behaves like upward • precipitation follows gamma distribution on wet days 9 rate daily precipitation inwith the theory rate of increase of extratropical precipitation extremes respectistoassumed surface specific humidity (9). The The normalized precipitation variabl aturation specific humidity . The normalize (T Tc ) p̂, where temperature according to p The = e normalized = 0. Temperature is normally distributed and(9). independent of to •increases in saturation specific humidity precipitation variable closely related to upward motion in the atmosphere; it rate of increase ofrelated extratropical precipitation thought of as a dynamic variable closely to upward motion inextremes the atmosw amic variable closely related to upward mo bution on wet days with scale parameter 1/ and shape param to increases in saturation humidity1/(9).and The n to follow a gamma distribution on wet days withspecific scale parameter shap With these assumptions, the qth percentile of snowfall, sq , is exceeded if 1 normalized precipitation rate p̂ here.) The and) the s that p̂ > hsq where h(Ttemperature ) = e T fT (T . Assuming sq is non-zero, t Integral expression for eded may be written as T th p̂ e f (T ) >of sq ,snowfall q percentile (sq) With these assumptions, the qth percentile of snowfall, sq , i T 1 which requires thatZp̂ > hs where h(T ) = e f (T ) . Assuming sq is non-z q Z 1 1 q w k k 1 = dT dp̂ p̂ that sq is exceeded may 100 1 be written hsq as (k) (T T )2 1 1 e p̂ p̂pe T f (Te ) >2sq2, . 2⇡ Z 1 Z 1 1 T )2 k h(T ) = e T f (T ) (T which the requires thatofp̂ wet > h(T )sis . Ass where days q fraction w and 1 q where k 1 p̂ p 1 = dT dp̂ p̂ e e 22 . 100 (k) 2⇡ 1 hsq otic methods are next to evaluate thewritten double bility that used sq is exceeded may be as integral in (S3) i → Evaluate using asymptotic methods for large sq Z Z 1 1 k q w of large s . The integral in p̂ is first evaluated using a standard 1 q 1 used = dT the double dp̂ p̂kasympto e inp̂ (p Asymptotic methods are next to evaluate integral 100 (k) 1 h(T )sq S1 plete gamma function, snowfall limit of large sq . The integral in p̂ is first evaluated using a standard asy Z methods are next used to evaluate the double S1 1 Asymptotic for the incomplete gamma function, ⇥ ⇤ k 1 t k 1 z 1 dt t e =z e 1 + O(z ) , Asymptotics gives expression for snowfall extremes of snowfall sqthat is given by involves optimal temperature Tm m ( sq hm ) 3 2 k e sq hm = w 1 q 100 asymptotically for large sq , and where (k) s hm e 00 hm (T Tm ) 2 2 2 , is the gamma function, h is the second derivative of h at Tm . For a change in mean tempe igible changes in all other parameters, the change in snowfall extrem expression f(T) exp( T) Temperature dependence of snowfall reaches a maximum at Tm (roughly -2°C) −6 −4 −2 0 2 Surface air temperature (°C) Competition between increasing precipitation and decreasing snowfall fraction with increasing temperature ✓ ◆s ✓ ◆ ble in all gives other in snowfall ext srearranging Ts T parameters, k the 2 (Tchange s 2 terms q changes q Tm ) rearranging terms gives w h = T + T + log 1 + changes . (S m c k sq hmSimple result if only mean temperature 2 2 2 2 sq ew ✓ = sq hm (Tq◆ 2T2m ) s e , sq q ✓ ◆ 00 ✓ ◆ e ◆2 ,(k) 2 h✓m (1) 1 q T T k s 00 q 100 k ression T100 T(k) T 2 s h q m +is beingTctaken, Tc ++(S11) log log1 +1that + sq./sq. ! 0,(S11) (S11) T + of s ! 1 implies and the sec 2 q 2 ssq ✓ ◆ 2 s s 2 sq q sq q q t hand side of (S11) may be neglected. (TheTsecond term on the right h T oruldlarge s , and where is the gamma function, h qthe m is h eval s = T + T , where is gamma function, h is h evaluated q m m not betaken, neglected if implies either sq that !s /s sqq/s or sq /s ! 1, but neithe 2 hthat q and 1isisbeing being taken,(S11) (S11) implies s ! 0, the second 1 ! 0, and the second q 2 q q m with (S11) if k < 2.) The final result is that eofof(S11) (S11)may maybe beneglected. neglected. (The second term on the econsistent (The second term on the rightright handhand Trivative change inmmean temperature of ◆ T andtemperature of T h at T . For a change in mean m . For aof ✓ sqsq!! sq sor 1,1, but but neither of of eneglected neglectedififeither either sqq/s! neither q orsq /s q ! upplementary Information. T T th with final result is that + T , with(S11) (S11)ififksk <2.) 2.)The The final result iscsnowfall that Change in qT percentile of with climate change (S q<= 2 ers,other the change in snowfall extremes, 2 inssnowfall q , is givenextremes, s , is all parameters, the change q ✓✓ ◆◆ TT ChangeTinTmean temperature me as main According to (S12), the cha ssq(2) = in the T T+body (S12) cTc,paper. +2 ofTthe , (S12) q = 22 o the approximation (2), changes in snowfall extremes transit Variance of daily temperature 2 es is independent of q, , f , and k, although > 0 is required for ✓ ◆ T ✓ ◆ lt (S8) to be valid. If it is found that sq to < (S12), sq when applying (S inin main body of the paper. According the change Tthe + T , (2) Optimal temperature m T T the main body of the paper. According to0,(S12), the change 2 point (S3) is invalid because it assumes s > and we must instead npendent temperature in the control climate of T T /2. Changes i s + T and k, although is required for the q = of 2q, , f , T m , > q0 m pendent of q,hm , fInverse , andprecipitation k, although scale > 0 is required for the 2 be valid. If it is found that sq < sq when applying (S12), be valid. If it is found that sq < sq when applying (S12), 2 tesimal, the slightly simpler result ) is invalid because it assumes sq >as0,measured and we must set mak sely on temperature variability by instead , which Ratio 0.5 Extremes Theory matches simulations Mean (and dynamic changes don’t matter very much) 0 Ratio (warm/control) (b) 1.5 1 0.5 99.99th Theory Theory Simple 0 −20 −15 −10 −5 0 5 10 Climatological temperature (°C) in control climate Shading shows interquartile range of model ratios le changes in all other parameters, the change in snowfall extr Changes in snowfall extremes don’t depend on percentile q! ession ✓ ◆ T T sq = T+ Tm , 2 h 2 m upplementary Information. Example: same change for 99th percentile as 99.99th percentile of the percentile considered, such that the fractiona o the approximation (2), changes in snowfall extremes transiti percentiles ( sq /sq ! 0 as sq ! 1). This is the temperature in the control climate of Tm T /2. Changes in dependencies of precipitation extremes and the rain- Rati 0.5 Extremes Mean Fractional changes in snowfall extremes tend to zero for high percentiles 0 Ratio (warm/control) (warm/control) Ratio (b) (a) 1.5 1.5 11 Increasing Percentile 0.5 0.5 99.99th Extremes Theory Mean Theory Simple 00 /control) (b) −20 1.5 1 −15 −10 −5 0 5 10 Climatological temperature (°C) in control climate 99th to 99.99th percentiles of daily snowfall Joint PDF Intuition: Probability of optimal temperature (Tm) for snowfall extremes does change as climate warms ← Snow Rain → Joint PDF Control Warm T m 0◦ C (b) Tc (b) Joint PDF (a) T Also mean snowfall decrease substantially 0 Control Warm s Results in changes Tm in 0◦ Csnowfall percentiles T (b) Joint PDF Control Warm 0 sq sq p̂/hm Results in changes Tm in 0◦ Csnowfall percentiles T (b) Joint PDF Control Warm 0 sq sq p̂/hm ...but fractional change in sq is fairly small and is similar for all high percentiles Ratio (warm/control) (a) 1.5 1 Increasing Percentile 0.5 Extremes Mean 0 Ratio (warm/control) (b) 1.5 Similar results above 500m elevation (but models have issues there) 1 0.5 99.99th Theory Theory Simple 0 −20 −15 −10 −5 0 5 10 Climatological temperature (°C) in control climate cf. Kapnick & Delworth, 2013 mptotically for large sq , and where Conclusions is the gamma function, he second derivative of h at Tm . For a change in mean temp • Simulations: Smaller fractional changes in snowfall extremes le changes all other parameters, the change in snowfall extr than ininmean snowfall in many cases • Simple asymptotic theory: captures main features of response ession ✓ ◆ T T sq = T + T , m 2 h 2 m pplementary Information. Implications: detection and perception of climate change, • changes in snowfall extremes still likely to have impacts o the approximation (2), changes in snowfall extremes transiti By contrast: Probability of snowfall and mean snowfall decrease substantially Snow ←← Snow Rain Rain →→ Joint PDF Joint PDF (a) (a) Tm 0◦ C Big decrease in area under curve to left of rain-snow transition temperature (b) (b) Control TT 19504 AND BROWN: SPRING SNOW COVER EXTENT REDUCTIONS L19504 RapidlyDERKSEN changing snow cover in Northern Hemisphere L19504 DERKSEN AND BROWN: SPRING SNOW COVER EXTENT REDUCTIONS L19504 igure 2. Time series Northern June snow (NOAA snow chart seaice iceextent extent (NASA TEAM) Figure 2. of Time series ofHemisphere Northern Hemisphere June cover snow cover (NOAA snow chartCDR) CDR)and and sea (NASA TEAM) or 1979–2012for(1979–2011 for sea ice). denotes 5-yr running mean. 1979–2012 (1979–2011 for Thick sea ice).line Thick line denotes 5-yr running mean. andtoBrown, 2012 declines are being drivenbylargely by pervasive warming rcp8.5 [Riahi 2011]Derksen (2006 to create a GRL, 150-year CE declines SCE are being driven largely pervasive warming rcp8.5 [Riahi et et al.,al.,2011] (2006 to2100) 2100) to create a 150-year pan-Arctic temperatures (as described in Screen et al. [2012]), time series of model simulated and projected snow cover. an-Arctic temperatures (as described in Screen et al. [2012]), timeOne series of model simulated and projected snow cover. independent of these low-frequency climate variables. model run (typically the first member) was selected from Probability of snowfall given that precipitation occurs 3-hourly synoptic weather reports over land aggregated globally Dai, GRL, 2008 Conditional probability of snowfall (%) Rain-snow transition in climate models (CMIP5) versus observations 100 Multimodel Median Observations 80 60 40 20 0 −9 −6 −3 0 3 Surface air temperature (°C) 6 9 - 3-hourly observed (Dai 2008): 2 curves depending on whether mixed counted as snow - Daily accumulations in multimodel median (black) and individual models (gray); snowfall Conditional probability of snowfall over land as in Fig. 2 but showing taken to occur if precipitation is 50% solid 2. the l median only in the control climate (the black solid line) and showing the prob r individual models in the control climate (gray lines). Intuition: snowfall extremes occur when temperatures close to freezing Temperature (°C) (otherwise too cold to snow heavily) 10 0 −10 −20 Snow Extremes (Control) Snow Extremes (Warm) Mean (Control) Mean (Warm) −20 −15 −10 −5 0 5 10 Climatological temperature (°C) in control climate Note snowfall extremes remain at roughly same temperature (with same humidities) as climate changes - unlike rainfall extremes Quiz: World record daily snowfall Where? How much (inches of depth)? World record (probably): 75.8 inches in 24 hours MONTHLY WEATHER REVIEW 38 FEBBUABY 1963 RECORD SNOWFALL OF APRIL 14-15,1921, AT SILVER LAKE, COLORADO J. L. H. PAULHUS U. S. Weather Bureau, Washlngton. D. C. [Manuscript received January27, 19531 ABSTRACT A snowfall of 87 inches in 27% hours on April 14-15, 1921, was reported at Silver Lake, Colo. This snowfall, if INTRODUCTION Although the meteorologist and hydrologist are generally interested in the water equivalentof a snowfall rather than in the snow depth, there are many, including the general public, highway and street maintenance engineers, etc., who are very much interested in the depth. Consequently, Changes in precipitation extremes: some aspects are well understood! • Theory • Climate models • Observed trends matological mean temperatures; for example, at latitud here precipitation is related to poleward movement of Simple scaling captures changes in simulated asses, they may be tied more closely to mean temperatu precipitation extremes Theory: factors controlling intensity of precipitation rther equatorward, and mean temperatures change differen extremes different latitudes in global warming simulations (25). Taking into account these factors, we can express the intens Saturation specific humidity precipitation extremes at a given latitude as Precipitation rate $ ! % dq s Pe # " ! e dp . [ " *,T e Vertical velocity ere, Pe is a high percentile of precipitation, !e the correspon Moist adiabatic integral o g upward vertical velocity, {!} is a mass-weighted derivative e troposphere, and the moist-adiabatic derivative of saturat O’Gorman and Schneider, PNAS, 2009 temp ecific humidity is evaluated at the conditional mean O’Gorman and Schneider, 2009 O’Gorman and Schneider,J. Climate, PNAS, 2009 occurs. A large-scale av ure Te when extreme precipitationO’Gorman and Schneider, J. Climate, 2009 e over precipitation systems is implied, so that !e is a Growth of precipitation extremes with warming in simulations with climate models Fig. 1. The 99.9th percentile of daily precipitation (millimeters per day) for CMIP3 multi-model mean 99.9th percentile of daily precipitation the periods 1981–2000 (blue) and 2081–2100 (red) in the SRES A1B scenario (20c3m A1B) and observations (GPCP)Project (multimodel median),toand based oncurrent Global Precipitation Climatology (GPCP) data for the period 1997–2006 (black). Model scatter (shading) the O’Gorman and Schneider, for PNAS, 2009 period 1981–2000 is shown using the interquartile range (50% of models lie Fig. 2. (blue), zo water va extreme itation e Problem: Response of tropical precipitation NATURE GEOSCIENCE DOI: 10.1038/NGEO1568 extremes varies widely between models Climate change (% K¬1) 20 Inferred 10 Observed 0 0 20 40 Variability (% K¬1) 60 BCCR BCM2.0 CGCM3.1 T47 CGCM3.1 T63 CNRM¬CM3 CSIRO¬Mk3.0 CSIRO¬Mk3.5 GFDL¬CM2.0 GFDL¬CM2.1 FGOALS¬g1.0 ECHAM4/INGV ECHAM5/MPI INM¬CM3.0 IPSL¬CM4 MIROC3.2¬med MIROC3.2¬hi MRI¬CGCM2.32 NCAR¬PCM1 NCAR¬CCSM3.0 Figure 2 | Sensitivities (% K 1 ) of the 99.9th percentile of precipitation Problem: Response of tropical precipitation NATURE GEOSCIENCE DOI: 10.1038/NGEO1568 extremes varies widely between models Climate change (% K¬1) 20 Use observed variability to constrain: 10±4%/K Inferred 10 Observed 0 0 BCCR BCM2.0 CGCM3.1 T47 CGCM3.1 T63 CNRM¬CM3 CSIRO¬Mk3.0 CSIRO¬Mk3.5 GFDL¬CM2.0 GFDL¬CM2.1 FGOALS¬g1.0 ECHAM4/INGV ECHAM5/MPI INM¬CM3.0 (O’Gorman, Nature IPSL¬CM4 MIROC3.2¬med MIROC3.2¬hi MRI¬CGCM2.32 NCAR¬PCM1 NCAR¬CCSM3.0 20 40 Variability (% K¬1) 60 Geo. 2012) (see also Muller, O’Gorman, Back 2011) Figure 2 | Sensitivities (% K 1 ) of the 99.9th percentile of precipitation Observed trends in annual-maximum precipitation over land FIG. 12. Variation estimated sensitivity of annual maximum to a 1 K Sensitivity (%)in of the annual maximum precipitation per kelvin warmingprecipitation of global nearincreasesurface in global mean temperature byrecords latitude. (top) Thewith number stations within each 658 temperature (1900-2009; > 30 years), lightof blue shading indicating latitudethe band. (middle) The fraction of stations exhibiting significant positive association, with upper 97.5% confidence bound light blue shading indicating the upper 97.5% confidence bound and dark blue shading indicating the median of the confidence interval. (bottom) Sensitivity (%) of annual maximum precipitation per kelvin warming of global near-surface temperature, with light blue shading Westra et al, J. Climate, 2013 indicating the upper 97.5% confidence bound. Intuition: snowfall extremes occur when temperatures close to freezing Temperature (°C) (otherwise too cold to snow heavily) 10 0 −10 −20 Snow Extremes (Control) Snow Extremes (Warm) Mean (Control) Mean (Warm) −20 −15 −10 −5 0 5 10 Climatological temperature (°C) in control climate