Atmospheric rivers as Lagrangian coherent structures Daniel Garaboa,1,a) Jorge Eiras-Barca,1 Florian Huhn,2 and Vicente P´erez-Mun˜uzuri1,b) 1)Group of Nonlinear Physics, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain. 2)Institute of Mechanical Systems, ETH Zu¨rich, Tannenstrasse 3, 8092 Zu¨rich, Switzerland. (Dated: January 6, 2015) We show that filamentous Atmospheric Rivers (ARs) over the Northern Atlantic Ocean are closely linked to attracting Lagrangian Coherent Structures (LCSs) in the large scale wind field. LCSs represent lines of attraction in the evolving flow with a significant impact on all passive tracers. Using Finite-Time Lyapunov Exponents(FTLE),weextractLCSsfromatwo-dimensionalflowderivedfromwatervaporfluxofatmospheric reanalysis data and compare them to the three-dimensional LCS obtained from the wind flow. We correlate 5 the typical filamentous water vapor patterns of ARs with LCSs and find that LCSs bound the filaments on 1 the back side. Passive advective transport of water vapor from tropical latitudes is potentially possible. 0 2 PACS numbers: 42.27.De,05.60.-k,92.60.-e,92.60.J- n a J Atmospheric Rivers (ARs) transport moisture Qian, 2009). The advection of moisture by ARs is a key 5 from the Tropics towards northern latitudes and process for the Earth’s sensible and latent heat redistri- they are responsible for extreme precipitation butionandhasastrongimpactonthewatercycleofthe ] h and flood events as they approach coastal areas. mid-latitudesbygeneratingextremeprecipitationevents. p Advection of Lagrangian tracers by geophysical The connection between extreme precipitation and flood - flows has shown the presence of persisting trans- events has been shown over the Western US Coast (Le- o port patterns of Lagrangian Coherent Structures ung and Qian, 2009; Stohl, Forster, and H.Sodemann, a . (LCS). In this paper, we argue that under some 2008) and over Europe (Lavers and Villarini, 2013). s c circumstances, ARs can be considered attract- In large scale geophysical flows, advection is often the i ing LCS in front of which moisture accumulates, primaryprocessshapingtracerpatterns,andeffectssuch s y while the rivers propagate northwards. This be- as turbulent diffusion or other sinks and sources of the h havior has been observed both in three dimen- tracer are secondary. Such tracer patterns have been p sional simulations of the wind flow and in terms observed in the atmosphere and in the ocean; examples [ of the two-dimensional vertically averaged water include volcanic ash clouds (Gudmundsson et al., 2012), 1 vapor field. planktonblooms(Toneretal.,2003; Lehahnetal.,2007; v Huhn et al., 2012), and oil spills (Olascoaga and Haller, 7 2012). If advection is dominant, Lagrangian Coherent 7 Structures (LCS) are the relevant finite-time structures 8 I. INTRODUCTION in the time-dependent flow that determine the deforma- 0 tionofthefluidandhencetheevolutionofanyadvective 0 . The transport of moisture from the tropics to mid- tracerfield. TheconceptofLCSshasitsoriginindynam- 1 latitudes is not continuous and uniform, but rather in- icalsystemstheory. IthasbeenintroducedbyHallerand 0 termittent. More than 90% of poleward water vapor is Yuan (2000) and is still being developed further (Faraz- 5 1 transported by narrow and elongated structures (longer mand, Blazevski, and Haller, 2014). Attracting hyper- : than2000kmandnarrowerthan1000km),mostlywithin bolic LCSs are lines evolving with the flow that maxi- v the Warm Conveyor Belt (WCB) ahead of cold fronts mallyattractfluid. InthevicinityofattractingLCSs,the i X and within the Low Level Jet (LLJ) of extratropical cy- fluid is stretched in one direction and compressed in the r clones commonly associated to the polar front (Newell orthogonal direction. Therefore attracting LCSs are the a etal.,1992; ZhuandNewell,1994; RalphandDettinger, cores of filamentous tracer patterns. A standard way of 2011). These structures, referred to as Tropospheric or estimating the position of attracting LCSs is the Finite- Atmospheric Rivers (ARs), are defined as elongated re- time Lyapunov Exponent (FTLE) (Haller, 2002; Shad- gionsofIntegratedWaterVaporcolumn(IWV)over2cm den, Lekien, and Marsden, 2005). The FTLE measures andwindsstrongerthan12m/s,thattransportmoisture the maximum stretching rate among trajectories over a inthelowertroposphereclosetothe850hPalevel(Ralph fixed time interval derived from neighbor fluid particles. and Dettinger, 2011; Neiman et al., 2008; Leung and Ridges of maximum FTLE values are a robust method to estimate hyperbolic LCSs (Haller, 2002; Shadden, Lekien, and Marsden, 2005), although some drawbacks have been reported (Haller, 2011). a)Electronicmail: [email protected] FTLE and similar versions of the Lyapunov exponent b)Electronicmail: [email protected] havebeenusedtoestimateLCSsindifferentmodelsofat- 2 mosphericflows,suchasazonalstratosphericjet(Beron- where Q is the vertically integrated water vapor, Φ = Vera et al., 2010), a jet-stream (Tang et al., 2010), a (Φ ,Φ ) are the eastern/northern water vapor flux, g is λ φ hurricane (Rutherford et al., 2010), transient baroclinic theaccelerationofgravity,andηisahybridverticalcoor- eddies(vonHardenbergandLunkeit,2002)andthepolar dinate (Kasahara, 1974; Simmons and Burridge, 1981). vortex (Koh and Legras, 2002). This coordinate uses the mean sea level as a bottom ref- Given that ARs over the Atlantic and Pacific Ocean erencelevelandpisthepressurelevelintheηcoordinate. appear as coherent filaments of water vapor with a per- uandvaretheeastwardandnorthwardwindcomponent sistencetimeofseveraldaysuptoaweek,andgiventhat and q is the specific humidity. Data are available with LCSshaveturnedouttoexplaintheformationofsimilar a spatial resolution of 0.7◦ and temporal resolution of 6 tracerpatternsingeophysicalflows,thispaperaddresses hours. The eastward and northward averaged velocities the questions: Are ARs associated to attracting LCSs in inalongitude-latitudesphericalcoordinatesystem(ϕ,θ) thelargescaletroposphericflow? TowhatextentdoARs are calculated as consist of humid air advected from tropic latitudes? (cid:104) (cid:105) Since ARs are defined and studied in terms of two- Vf = <λ˙(ϕ,θ,t)>,<φ˙(ϕ,θ,t)> dimensional maps of IWV, we use a two-dimensional (cid:20) (cid:21) Φ Φ height-averaged tropospheric flow to detect the main = λ, φ . (4) Q Q LCSs. We focus on selected ARs events that prop- agate over the Northern Atlantic Ocean and hit the When the velocity fields are vertically averaged in this Iberian Peninsula. Wind fields and water vapor flux way, the pressure levels are weighted according to their fields are retrieved from European Center for Medium- water vapor content, such that V is representative for f Range Weather Forecast reanalysis, ERA-Interim. We the dynamics of ARs. focus on spatiotemporal patterns in the two-dimensional Alongthispaper,wewillcomparetheresultsobtained watervaporfieldformedduringanARevent. Thesepat- with the previous vector field with those obtained from ternsarecomparedwithadvectivestructuresinthewind the wind velocity fields at individual pressure levels, field,namelyLCSs. Asmentionedearlier,ARsarethree- dimensional phenomena (Sodemann and Stohl, 2013), so V =[u(ϕ,θ,t),v(ϕ,θ,t),w(ϕ,θ,t)] (5) w p we have also considered a three-dimensional case study toaddressthelimitationsofatwo-dimensionalapproach Hence, two types of Lagrangian simulations have been (Stohl, 1998). performed. Either particle trajectories are computed at We find that ARs with a sharp filamentous shape are constantpressurelevels(2Dsimulation),orafull3Dsim- associated to a strong attracting LCS in the wind field. ulationisdone. Forthelastcase,thetopboundaryofthe Thissuggestsadominanceofhorizontaladvectioninthe domain is located at 300 hPa, while the bottom bound- formation process of the filamentous pattern. In these ary is at 1000 hPa. Particle trajectories are computed cases,theLCStypicallymarksthenorthernboundaryof by integrating (4) or (5) using a 4-th order Runge-Kutta theAR-alineofhighwatervaporgradientwherehumid scheme with a fixed time step of ∆t = 2 hours, and a and dry air converge. Filamentous LCSs appear mainly multilinear interpolation in time and space. A fine grid in winter, while wide, less coherent ARs structures with- ofparticleswithaninitialseparationof1/5◦ineachpres- out a clear attracting LCS develop in summer. sure level is advected to obtain fields of the Lagrangian quantities introduced below. For the two-dimensional simulations w =0 in Eq. (5). The finite integration time II. METHODS is chosen to be τ = 120 hours, a typical time scale for the formation and propagation of the ARs. In order to detect attracting Lagrangian coherent Atmospheric Rivers have been analyzed in terms of a structures, we use the Finite-Time Lyapunov Exponent two-dimensional vertically integrated flow based on data (FTLE)fieldσ(x)whichisameasureofstretchingabout retrieved from the European Center for Medium-Range fluid trajectories. It is computed from the trajectories of Weather Forecast reanalysis, ERA-Interim (Dee et al., Lagrangiantracersintheflow(PeacockandDabiri,2010) 2011), as 1 1(cid:90) ∂p 1 (cid:112) Q= g q∂ηdη (1) σ(r0,t0,τ)= |τ| log µmax(C(r0)), (6) 0 where µ is the maximum eigenvalue of the right 1(cid:90)1 ∂p Cauchy-Gmraexen deformation tensor C = FTF. F is de- Φ = uq dη (2) λ g ∂η fined by F(r ) = ∇(r(t +τ)), and r(t +τ) is the final 0 0 0 0 position of the tracers. The time variation of the FTLE 1(cid:90)1 ∂p field has been computed by following the same steps ex- Φ = vq dη (3) plainedpreviously,butvaryingtheinitialtimet infixed φ g ∂η 0 steps∆t =6hoursinordertoreleaseanewinitialtracer 0 0 3 grid. FTLEarecomputedinforwardandbackwardtime latitudes transporting water vapor to medium latitudes. directions. However,potentialARssituationswithascatteredshape Finite-Time Lyapunov Exponents have been used ex- donotcorrespondtowelldefinedwatervaporjets. Large tensively to quantify mixing and especially to extract areas of water vapor surround the jet. The last situa- persisting transport patterns in the flow, referred to as tionistypicalofsummercases,whilethefirstoneoccurs Lagrangian Coherent Structures (LCS). The FTLE at a mainly in winter or beginning of spring. For the winter given location measures the maximum stretching rate of ARs cases, the threshold criteria used to extract LCSs an infinitesimal fluid parcel over the interval [t ,t +τ] from the FTLE fields is insensitive to variations. How- 0 0 starting at the point r at time t . Ridges of the FTLE ever, small variations in the thresholds for the summer 0 0 field are used to estimate finite time invariant manifolds cases lead to very different LCSs patterns, an indication in the flow that separate dynamically different regions. that the LCSs patterns associated to ARs are more co- Repelling (attracting) LCSs for τ > 0 (τ < 0) can be herent in winter than in summer. thoughtofasfinite-timegeneralizationsofthestable(un- In all analyzed winter ARs events, LCSs are located stable)manifoldsofthesystem. Thesestructuresgovern behind the river (in the direction of propagation) and do the stretching and folding mechanism that control flow not coincide with it. Here the shape of the river is de- mixing. LCSs as FTLE ridges are extracted using the finedastheridgeoftheQfield. FromaLagrangianpoint Hessianmatrixandthegradientofσ(r0,t0,τ)(Shadden, of view, the attracting LCS accumulates water vapor in Lekien, and Marsden, 2005; Sadlo and Peikert, 2007). front of the pattern moving towards the east, and a cer- To discard weak structures, a threshold criterion associ- tain gap in between both maxima, LCSs and Q field, is ated to the 90th percentile is used for the FTLE values observed. and for the eigenvalues of the Hessian matrix, a fixed These results are obtained using the vertically aver- threshold is used for the minimal connected ridge area. aged flow V where the averaging is weighted with wa- f For the 3D simulations, the LCSs are estimated using a ter vapor content. Another natural choice is to define high isosurface of the FTLE field. a two-dimensional flow on isobaric levels. We compute the FTLE fields at different individual isobars using the windV (5)andneglectingtheverticalwindcomponent, w III. RESULTS w =0. Figure3showsanexampleoftheFTLEfieldsob- tained for the pressure levels 850hPa and 1000hPa. At A. 2D results 850 hPa, ridges of the FTLE field show a close relation- ship with those shown in Fig. 1(b). For other pressure Atmospheric Rivers are observed as filamentous struc- levels, we find that the main LCS preserve its continu- tures in fields of the integrated water vapor column Q, ity, but loses the smoothness of the curvature, showing a Eq. (1). Figure 1(a) shows an example of an AR extend- different shape. ing north-westwards from ∼ (20◦N,75◦W) and trans- However, the AR has a vertical extent and it is portingwatervaporovertheAtlanticOceantowardsthe not clear which isobar to choose, such that the two- Iberian Peninsula. The inset shows the daily precipita- dimensional flow best represents the dynamics of the Q tion rates well above the mean values for this area that tracer,i.e.,suchthatisismostsimilartoV . Inorderto f are caused by this event. The long ridge of large val- comparetrajectoriesforbothtwo-dimensionalflowfields, ues of Q connects the tropics with the North Atlantic V andV ,weintroduceasimilaritymeasure,themean f w Ocean and represents a continuous AR structure. The distance error δ. It measures the mean arc length dis- backward FTLE field is shown in Fig. 1(b). The FTLE tance between the final position of the tracers obtained ridge reaching from the Iberian Peninsula to the Gulf for both flows, of Mexico shows a clear relationship to the Atmospheric River depicted previously. From a Lagrangian point of N (cid:88) view, the AR can be considered as an attracting LCS or δ =N−1 |r −r |. (7) vw vf unstable manifold of the flow dynamical system V (4). f i=1 Similar results have been obtained for different dates. Lagrangian Coherent Structures calculated from the Thesubindexofraccountsfortheusedflowfield,andN backwardFTLEfieldsarecomparedtotheQfield(1)in is the number of passive tracers released. The mean dis- Fig. (2) for four ARs events. Note that regions of max- tanceerrorδisthencalculatedfor20pressurelevelsfrom imum Q values match LCSs locations. The good agree- 300hPa to 1000hPa for different ARs events, as shown ment of LCSscomputed from theflow field V with pat- inFig.4. Aminimumofδ isobservedforapressurelevel f ternsinthetracerfieldQindicatesthatV capturesthe near850hPaforallARseventsanalyzed. Theminimum f main dynamic processes shaping the water vapor field. suggests that the flow field at 850hPa is closest to the In three cases, Figs. (2)(a) to (2)(c), LCSs and Q water vapor flow V . It is consistent with observations f ridges show a filamentous structure, while for the event that the core of ARs is typically located at that height in Fig. (2)(d), LCSs have a scattered shape. Filamen- (RalphandDettinger,2011; Neimanet al.,2008; Leung tous ARs have one or two narrow jets growing from low and Qian, 2009). 4 Figure1. (a)AtmosphericRivereventon15October1987intermsoftheintegratedwatervaporcolumn,Eq.(1). TheΦflow is shown superimposed. The inset shows the accumulated precipitation rates over the Iberian Peninsula [mm]. (b) Backward FTLE field for the same date and the flow given by Eq. (4) . Figure 2. Lagrangian Coherent Structures (blue dots) and integrated vapor column Q for four different events. Note the coincidence between LCSs and the maximal Q values in the first three cases (winter and spring). B. 3D results ing a fixed isosurface from the FTLE field for a winter AR.NotetheribbonextendingfromtheTropicstowards So far, LCSs have been calculated using the two- the Iberian Peninsula. The surface of the LCS is contin- dimensional flow given by Eq. (4). However, including uous,independentofthepressurelevel,althoughslightly theverticalwindcomponentmightsignificantlyalterthe changes with increasing height. Note that the 3D FTLE LCSs obtained from a purely two-dimensional analysis. isosurface follows the Atmospheric River depicted below To verify that, we compute the FTLE backward fields in the figure. This fact suggests that the AR dynamics from a full 3D simulation using the wind field V (5). can be considered two-dimensional and the vector field w Figure 5 shows the three-dimensional LCSs obtained us- defined by Eq. (4) accounts for the global dynamics of 5 Fig.6b,c. The 3D LCS seems to act as a lateral barrier for the vertical transformation processes of water vapor that develop in front of it as the AR moves eastward, showing that the LCS is a dynamical barrier to precip- itable water. One of the issues that emerges from these findings is that high vertical flux also means that Q is not a con- Figure 3. FTLE backward fields computed using the wind servedtracer,andifwewanttocharacterizeverticalmo- field at two different pressure levels, 850 hPa (a) and 1000 tionsofprecipitablewateractivetracersareneeded. But hPa (b), for the ARs event on October 23, 1987. an implication of this is the possibility that the LCSs can identify regions which are not sensitive to the use ofactivetracersforhorizontalmovementsofprecipitable water. C. Origin of air Since we have observed stronger attracting LCSs for filamentous winter ARs than for scattered summer ARs, wealsoinvestigatetheoriginofairparcelsinbothcases. Wequantifythedisplacementofpassiveparticlesfortwo ARs, once the AR hits the Iberian Peninsula. To that end, an initial grid of particles is advected backward in time for τ = 5days. Figure 7 shows the origin of the particlesinlatitudeforafilamentous(a)andascattered (b) river. Note in panel (a) that particles coming from low latitudes coincide with high values of the integrated Figure 4. Mean distance error defined by Eq. (7) for three water vapor column. This potentially gives the possibil- ARs events at different pressure layers. December 1994 (red ity that moisture from the Tropics is passively advected crosses),August1988(bluedots),andNovember1995(green triangles). northwards,eventuallyformingtheAR.Thecontribution ofevaporationandprecipitationcan,however,notbeas- sessed with this simple simulation of advection. Both the ARs. The vertical FTLE slice shows the presence processes tend to represent significant sinks and sources of the atmospheric river at approximately 40◦. For this in the water vapor budget. For the unstructured river latitude, Qdiminishesdrastically, asthewatervaporac- (b), the contours of the original latitude do not enclose cumulatesinfrontoftheriverasitdisplacestowardseast. the ARs structure. Only a very thin zone from low lati- Note the accumulation of numerous filaments with large tudeswithnearlyzeroareaiscoupledtotheARevent. In FTLEvaluesbehind theARduetoflowmixingprobably this case, two-dimensional passive advection of moisture induced by the interaction between the jet stream and from low latitudes as a main source can be excluded. the AR (Sodemann and Stohl, 2013). Even if the horizontal shape of an AR turns out to be dominated by the horizontal flow, vertical motions in IV. CONCLUSIONS the atmosphere are needed to account for precipitation and evaporation processes. These processes may be rep- ThepropagationofAtmosphericRiversovertheNorth resented by the vertical flux of precipitable water on a AtlanticOcean,finallyhittingtheIberianPeninsula,has given level, been studied in terms of Lagrangian tools. Based on an integrated water vapor flux obtained from the ERA- φ =ρw (q +q +q ) (8) z v l s Interim database, AR events have been identified with where q , q and q are the amount of water in the attracting LCSs, both in 3D and 2D simulations. Two v l s gaseous, liquid, and solid states, respectively, retrieved different AR events have been characterized. from the ERA-Interim database, ρ is the density of the On the one hand, narrow filamentous ARs with a fast air and w is the vertical component of wind. Figure 6 and persistent eastward transport typically develop in shows the vertical flux of precipitable water for two days winter. An attracting LCSs with the same shape as the where ARs are present. The vertical flux (8) reaches ARs exists in the flow. It acts as a lateral boundary to maximum values all along the main attractive LCS bar- the ARs. This boundary is more clearly visible for the rier, but rather directly in front of the LCS than on top 3D simulations where the AR is identified as a vertical of it. This location of maximal vertical moisture flux ribbon displacing eastward. As an advection experiment corresponds to ridge of maximal water vapor content, cf. shows, for this kind of ARs, the air originates from low 6 Figure 5. Backward three-dimensional FTLE for the Atmospheric River event show in Fig. (1). The isosurface of the FTLE fieldof0.04h−1isshowningrayscaleshadeswhiletheQfieldcorrespondingtothisARisdepictedbelow. Averticalcutat304◦ longitude shows the presence of the AR at 40◦ latitude. For comparison, Q values at this longitude are shown superimposed over the FTLE field. ance of these ARs must be dominated by local sources, sincethepassivetransportofmoisturefromlowlatitudes is practically excluded. Our Lagrangian analysis assumes that water vapor behaves as a passive tracer. This assumption works to assess the deformation in the flow field and obtain LCSsthatshapetracerpatternsinthewatervaporfield, namely ARs. For a quantitative analysis of the water vapor balance in ARs, inertial tracers which may ex- Figure 6. Vertical flux of precipitable water at 850 hPa for change matter and thermodynamic properties with the two Atmospheric Rivers shown in Fig. (2). For comparison surrounding flow should be considered. the LCSs detected in both cases are shown as black lines. Finally,weconcludethatthecloseconnectionbetween attracting LCSs and ARs should be taken into account for future studies and may help to characterize this kind of event. Most probably, our results are not limited to the Atlantic Ocean, but can be extended to ARs over the Pacific Ocean. An extensive analysis with more ARs events, including other regions, could help to set up a definition of ARs in terms of Lagrangian analysis. Figure 7. Contours of latitudes of origin of passive tracers advected backward in time for two ARs events. The red, green and blue contours enclose particles coming from 25◦N, ACKNOWLEDGMENTS 21◦N, and 17◦N. Shaded gray images correspond to water vapor concentration Q. ERA-Interim data was supported by the ECMWF. This work was financially supported by European Com- mission FP7 programme (EARTH2OBSERVE) through latitudes and passive transport of water vapor from the the project Global Earth Observation for Integrated Wa- Tropics northwards is potentially possible. terResourceAssessment,andbyMinisteriodeEconom´ıa Ontheotherhand,forunstructuredorscatteredrivers, y Competitividad (CGL2013-45932-R). 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