Modeling the Continental-Scale Dynamics of the Coupled Land-Surface and Atmospheric Water Balances with a Stochastic Differential Equation

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1 Hydrology Day 003, 07-8 Modeling he Coninenal-Scale Dynamic of he Coupled Land-Surface and Amopheric Waer Balance wih a Sochaic Differenial Equaion John P. Kochendorfer and Jorge A. Ramírez Waer Reource, Hydrologic and Environmenal Science Diviion, Civil Engineering Deparmen, Colorado Sae Univeriy. Abrac. Uing a ochaic differenial equaion (SDE) approach, we examine he dynamic of he coninenal-cale waer balance in he cenral Unied Sae. Whie-noie and colorednoie verion of he model are developed baed on analyi of amopheric daa from he Naional Cener for Environmenal Predicion (NCEP) re-analyi projec and long-erm record of precipiaion and oil moiure. Improved correpondence o obervaion i achieved by incluion of erm in he SDE repreening boh hydrodynamic and hermodynamic oil-moiure feedback. We how ha emporally correlaed (i.e., colored) amopheric moiure flux remove he bimodaliy of he oil moiure probabiliy diribuion and ha our colored-noie formulaion uccefully capure he waer balance dynamic in he udy region depie he abence of muliple oil moiure ae. Specifically, he improved model reproduce boh he auocorrelaion rucure and he iner-annual variabiliy in he obervaion.. Inroducion Since he eminal work of Charney (Charney e al. 977), land-urface feedback have increaingly been recognized a a ignifican ource of amopheric variabiliy. Soil moiure i perhap he mo imporan mediaor of hee feedback. In hi paper, we are pecifically inereed in he role oil moiure play in he variabiliy and perience of precipiaion a large paial and emporal cale. The majoriy of previou modeling udie a hi cale have involved he ue of general circulaion model (GCM). Mo GCM experimen are of he eniiviy variey in which urface condiion are alered in ome ignifican way compared o a e of conrol condiion. While hee udie are ueful for underanding he phyical procee involved, hey generally are no uiable for fully quanifying he effec of he given feedback mechanim on he variabiliy and perience of precipiaion a he eaonal o inerannual cale. Thi i due in par o he neceary lengh of model run and in par o queion a o he abiliy of GCM o reproduce accuraely variabiliy a long ime cale (Mearn e al 990). In conra o GCM, modeling mehodologie are available which rade phyical realim and deail for analyical racabiliy and compuaional eae (e.g., Brubaker and Enekhabi 995). A he very end of hi ide of he climae-modeling pecrum are lumped-parameer, coninenal-cale waer balance model. In he preen work, we employ one uch model ha i baed Waer Reource, Hydrologic and Environmenal Science Diviion, Civil Engineering Deparmen, Colorado Sae Univeriy, F. Collin, CO ramirez@engr.coloae.edu jk@lamar.coloae.edu Hydrology Day 003

2 Kochendorfer and Ramírez on he work of Rodriguez-Iurbe e al. (99) and oher (Enekhabi e al. 99, Wang e al. 997). Becaue he model i fir formulaed a a ingle ordinary differenial equaion, iue of variabiliy and perience can be addreed in an analyical and probabiliic manner by reformulaing he ordinary differenial equaion a a ochaic differenial equaion (SDE).. Deerminiic Differenial Equaion Governing he Waer Balance We begin wih he ame ordinary differenial equaion governing he large-cale waer balance ued by Rodriguez-Iurbe e al. (99): d nzr = P( ) φ( ) E( ) () d relaive oil auraion (dimenionle); n oil poroiy (dimenionle); Z r hydrologically acive deph of oil (L); P() rainfall rae (L/T); φ() infilraion funcion (dimenionle); E() evaporaion rae (L/T). The evaporaion and infilraion funcion are (Rodriguez-Iurbe e al): E( ) = E c () p φ( ) = ε r (3) E p poenial evaporaion rae (L/T); c, ε, r nonnegaive conan. In he remainder of hi paper i will be aumed ha ε alway ake on a value of one, enuring ha he runoff raio i uniy for compleely auraed oil (i.e., =). Where we differ wih Rodriguez-Iurbe e al. i in he formulaion of he equaion governing he rainfall rae. Wherea hey aume ha he porion of precipiaion originaing from amopheric moiure ha i adveced o he region i conan, we aume ha oal precipiaion i proporional o he average amopheric moiure flux over he region. Following he reamube analogy (e.g., Enekhabi e al. 99), we wrie E P P( ) = h( )( Α + ) (4) A moiure flux per uni area adveced o he region, (L/T) h(.) fracion of moiure flux which fall a precipiaion. Solving for P() give h( ) P ( ) = ( Α + E ) (5) + h ( ) Hydrology Day

3 Coninenal Scale Dynamic of Coupled Land Surface and Amophere Becaue he energy balance a he urface conrol he amoun of amopheric convecion, and he energy balance i parially conrolled by oil moiure hrough he Bowen raio, we ake h o be a funcion of oil moiure. For he udy region, Α~5 m/yr, P~0.9 m/yr and P-E~0. m/yr. Therefore, h~0. and h h/(+ ½ h). Ou of a deire for impliciy and flexibiliy, raher han from any heoreical bai, we ue a power law relaionhip for h(), a wa done for E() and φ(). Our equaion for he precipiaion rae a a funcion of oil moiure i hu P( ) = ( a + k d )( Α + E c ) (6) a, k and d are nonnegaive conan (auming a ne poiive feedback). Noice ha appear in wo erm of (6). Thee erm repreen hermodynamic feedback and direc precipiaion recycling, repecively. Seing k or d equal o zero will remove he hermodynamic erm, leaving only precipiaion recycling a a feedback. We ubiue (), (3) and (6) ino () and divide by nz r o arrive a a form of he differenial equaion for he waer balance which involve only he ime dimenion: d d c r c = ( a + k )( α + b )( ε ) b d (7) p α = Α/nZ r (T - ); b = E p /nz r (T - ). 3. Whie-Noie Formulaion of he Sochaic Differenial Equaion The ranformaion of (7) from a deerminiic o a ochaic differenial equaion ake place wih he reamen of α a a ochaic variable. α i a funcion of he inpu moiure flux and herefore can be viewed a he driver of he yem (e.g., Rodriguez-Iurbe e al. 99). However, our formulaion of he precipiaion funcion ha reuled in an α which i proporional o he inpu moiure flux (a oppoed o he α of Rodriguez-Iurbe e al., which i inverely proporional.) We can view α a he um of i mean plu a ochaic noie erm: α = α + σ γ (8) α he mean of α; σ he andard deviaion of α; γ ochaic noie. In addiion, E[γ] = 0 and Var[γ] =. I i alo ypical o aume ha he noie in an SDE i Gauian. Wang e al. (997) how ha he Gauian aumpion i unneceary for oluion of he Fokker-Planck equaion. Non-Gauian noie i paricularly ueful for applicaion, uch a he preen, in which he driver ake on only poiive value. For he purpoe of making he SDE olvable uing Io calculu, he noie i alo ypically aumed o be whie. Hydrology Day

4 Kochendorfer and Ramírez 985): Subiuing (8) ino (7) give a SDE of he Langevin form (Gardiner, G( ) drif erm; σ g( ) diffuion erm; and G( ) d = G( ) ) +σ g( d γ (9) d c r = ( a + k )( α + b )( c ) b (0a) d r g( ) = ( a + k )( ) (0b) The ue of he erm drif and diffuion dae from he original applicaion of ochaic calculu o he udy of Brownian moion and oher diffuion procee. The ubcrip ha been ued in he uual ene o indicae he ochaic ime-dependence of a quaniy. A in (0), we will drop he ubcrip when ime i no explicily par of he equaion. If we aume γ i a whie noie proce: E[ γ γ ] = δ ( ) () Becaue whie noie doe no race a differeniable pah in ime, i i no inegrable by he normal rule of deerminiic calculu. The ochaic inegral of whie noie i defined a he Weiner proce W uch ha dw = γ d () dw he derivaive of he Weiner proce. Uing (), we can wrie (9) for he whie-noie cae a d = G( ) + σ g( ) dw (3) The ime evoluion of he probabiliy diribuion of i governed by he Io Fokker-Planck equaion (Gardiner, 985): f (, ) = [G( ) f (, )] + σ [ g ( ) f (, )] (4) f(,) i he probabiliy diribuion of a ime. Under eady ae condiion, he LHS of () equal zero, f(,)=f S () wih he oluion, f G( u ) ( ) = C exp + du ln g( ) σ g ( u ) S (5) C i a normalizaion conan uch ha 0 fs ( ) d = (6) An analyical oluion exi only if a = 0. Thi alo urn ou o be a condiion for = 0 o be a reflecing boundary. Wih a = 0, he oluion of (5) i, Hydrology Day 003 0

5 Coninenal Scale Dynamic of Coupled Land Surface and Amophere f S 5 nr C rv ( ) = exp U + i i ( Vi ) (7a) d r k ( ) i= σ r n= 0Vi + n U α α b b b T =,,,, kr kr kr kr k r (7b) = d d c d c d c d V, +,, +, T (7c) r r r r r 4. Colored-Noie Formulaion of he Sochaic Differenial Equaion The general approach o adding emporal correlaion o α i o rea γ, i noie componen in (5), a being generaed by he Ornein-Uhlenbeck proce (e.g., Wang e al., 997). The origin of he Ornein-Uhlenbeck proce i in Langevin model of Brownian moion (Gardiner, 985) in which he velociy (raher han he poiion) of he paricle i he principal ochaic variable ubjec o a whie-noie forcing. The reul i ha he velociy i no longer non-differeniable in ime (a in earlier model) and hu poee emporal correlaion. Concepualizing γ a velociy in he Ornein-Uhlenbeck proce, we can wrie an SDE for γ of he form d γ = γ d + D dw (8) τ τ correlaion ime cale of γ; D diffuion coefficien. Gardiner (985) how ha τ E[ γ ] = γ = 0 e (9) Var[ ] Dτ τ γ = [ e ] (0) τ E[ γ γ ] = Dτ e () In order ha (0) converge o a and ha () equal when =, D = () τ Solving for γ in (9) and ubiuing ino (8) along wih () yield Hydrology Day 003

6 Kochendorfer and Ramírez d d g( ) d d dw G( ) G( ) + σ d = τ g( ) d (3) τ d Applying he chain rule of ordinary calculu o he LHS of (34) give d g( ) d G( ) d G( ) dw = g( ) σ g( ) d g( ) d g( ) τ d τ τ d (4) Dicarding he quared and econd-derivaive erm, G( )d + τ g( )dw d = d G( ) τ g( ) d g( ) (5) We define new drif and diffuion erm, G( ) G ( ) = (6) d G( ) τ g( ) d g( ) τ g( ) g ( ) = (7) d G( ) τ g( ) d g( ) in order o wrie (5) in andard Langevin whie-noie form: d = G ( )d + g( ) dw (8) A in he whie-noie cae, chooing a=0 allow for an analyical oluion. Thi oluion, which conain 37 erie erm of he form in (7), i oo long o reproduce in hi paper. 5. Model Parameerizaion and Soluion for he Sudy Region The udy region i he graicule bounded by 3.5 N, 4.5 N, 87.5 W and 0.5 W. The boundarie of he region are overlain ono a map of mean annual precipiaion in Figure. Long-erm, large-cale daa e of wind peed, humidiy, precipiaion, poenial evaporanpiraion, runoff and oil moiure were ued o develop a parameerizaion of he model for he udy region. A daily verion of reul from he NCEP/NCAR re-analyi projec (Kalnay e al. 996) wa he ource of he wind peed and humidiy daa. Thee daa are reolved verically by 4 preure level and horizonal a 0.5 degree laiude and longiude. The precipiaion daa were exraced from a monhly daa e generaed by he PRISM model (Daly e al. 994) for he enire Unied Sae a a 0-km reoluion. The wo daa e overlap over a Hydrology Day 003

7 Coninenal Scale Dynamic of Coupled Land Surface and Amophere 37-year period ( ). A long-erm mean value of poenial evaporaion over he udy region wa derived from eimae made by Hobbin e al (000) for he enire U.S. a a 0-km reoluion baed on modified verion of he Penman-Moneih equaion. Figure : Sudy region and mean annual precipiaion Unforunaely, reliable oil moiure daa a imilar paial and emporal cale do no exi. To our knowledge, he be e of long-erm, large-cale obervaion of oil moiure in he U.S. i he oil moiure climaology developed by he Illinoi Waer Reource Survey (Hollinger and Iard 994). We ued daa from 6 of he ISWR ie o develop monhly ae-wide average of relaive oil auraion for he year Daa from a imilar number of USGS ream gauge were ued o eimae annual runoff over he ame period. A hown in Table, he hydroclimaic characeriic of Illinoi and he larger udy region are fairly imilar. Therefore, i hould be reaonable o exrapolae value of he oil parameer nzr, c and r, a derived for Illinoi, o he larger udy region. The eimaed value of hee and he remaining parameer are given in Table. The exponen c, r and d have been rounded o whole and half-ineger value o increae he analyical racabiliy of he oluion of he Fokker-Plank equaion. The analyi of he precipiaion daa and he amopheric moiure flux eimae uppor incluion of he hermodynamic feedback facor. Hydrology Day 003 3

8 Kochendorfer and Ramírez Table : Hydroclimaic Characeriic of Daa Locaion LOCATION Sudy Region Illinoi R5 Waerhed Daa Year Area (q. km).86e+06.45e Mean Annual: Precipiaion, P (m) Poenial Evap., E p (m).5..8 Runoff, Q (m) E p /P Q/P Effecive Soil Sauraion? Table : Parameer value Parameer Value Mean rae of adveced moiure inpu, A (m/yr) 4.7 Sandard Deviaion of A, σ A (m/yr).7 Correlaion ime cale of A, τ (yr) Poenial evaporaion rae, E p (m/yr).5 Sorage capaciy of acive oil deph, nz r (m) 0.6 Evaporaion funcion exponen, c.5 Runoff funcion exponen, r 4 Precipiaion facor conan, a 0 Precipiaion facor coefficien, k 0.9 Precipiaion facor exponen, d For he parameer value in Table, he analyical oluion o he whie- and colored-noie formulaion of he Fokker-Planck equaion are compared in Figure. A doe he model of Rodriguez-Iurbe e al, our model produce a bimodal diribuion of oil moiure under a whie-noie oluion. However, in marked conra o boh our whie-noie oluion and he colored-noie oluion of Wang e al., our colored-noie oluion ha a claical Gauian hape. Clearly, whie-noie and colored-noie procee are fundamenally differen. Given ha he amopheric moiure flux i correlaed in ime, one hope ha he colored-noie model beer repreen he underlying dynamic. 6. Comparion of Model Reul o Obervaion For model validaion, a wih he parameerizaion, we would ideally like a muli-decadal ime erie of obervaional eimae of area-averaged oil moiure over he udy region. In he abence of uch a daa e, we again urn o he Illinoi daa. The area and climae of Illinoi are cloe enough o hoe of he larger udy region o make comparion of he aiical properie of he model reul and he obervaional daa worhwhile. Hydrology Day 003 4

9 Coninenal Scale Dynamic of Coupled Land Surface and Amophere Probabiliy deniy, f S () Whie noie oluion Colored noie oluion Relaive oil auraion, Figure. Comparion of whie noie and colored noie oluion The mo baic aiic ha we can compare are he expeced value of f S () wih he long-erm mean of he Illinoi daa, which, a lied in Table, i 0.7. The expeced value of he whie-noie diribuion hown in Figure i The expeced value of he colored-noie diribuion i Given ha he climae of he udy region i only lighly drier han ha of Illinoi, he colored-noie formulaion appear o produce he beer eimae of mean oil moiure. There are alo enough obervaion o make reaonably accurae variance and auo-covariance calculaion wih he Illinoi daa. Baed on he SDE model, he auo-covariance funcion for oil moiure can be derived from a emporally dependen oluion of he Fokker-Plank equaion. Becaue here i no general analyical oluion (unlike he eady-ae oluion), we have reored o a numerical oluion, he deail of which are beyond he cope of hi paper. The variance of he emporal average, and he auocovariance a a funcion of ime, are ploed in Figure 3 for boh he colorednoie and whie-noie model. Alo ploed are he variance and i 95% confidence limi for he annual-average oil moiure for he 3 complee year of he Illinoi daa. The whie-noie formulaion overeimae he variance of he annual-average oil moiure by an order of magniude. The colored-noie formulaion appear o produce le variabiliy hen found in he Illinoi daa. Thi i o be expeced given ha he Illinoi daa repreen a maller paial cale and alo likely conain ignifican meauremen and ampling error. The amoun of perience in he wo model and he Illinoi daa can be compared direcly wih he auocorrelaion plo of Figure 4. Auocorrelaion coefficien for he Illinoi daa were calculaed from eaonally andardized monhly average. Given hi fac and hoe noed above, i i expeced ha he Illinoi coefficien would fall below he Hydrology Day 003 5

10 Kochendorfer and Ramírez auocorrelaion funcion for he udy region. Neverhele, he whie-noie formulaion appear o over-predic grealy he perience of oil moiure. Variance/Auocovariance.E+0.E+00.E-0.E-0.E-03.E-04.E-05 Variance of emporal average - whie noie Auocovariance - whie noie Variance of emporal average - colored noie Auocovariance - colored noie Sample variance of annual average - Illinoi Time (year) Figure 3. Auocovariance and variance of emporal-average oil moiure a calculaed uing colored and whie noie Auocorrelaion Whie Noie Colored Noie Illinoi Time/Lag (monh) Figure 4. Comparion of whie-noie and colored-noie auocorrelaion funcion wih correlogram of andardized monhly oil moiure from Illinoi By way of (6) and probabiliy diribuion of A and, an analyical form for he probabiliy diribuion of precipiaion can be derived. However, becaue of he lack of eaonaliy in he model and he highly epiodic naure of precipiaion a any cale, hi diribuion ha no obervable counerpar. We have developed a quai-analyical mehod for deriving he probabiliy diribuion of boh emporally averaged precipiaion and oil moiure. I i, however, le compuaionally efficien han generaing frequency diribuion from numerical imulaion of he SDE. Accordingly, frequency Hydrology Day 003 6

11 Coninenal Scale Dynamic of Coupled Land Surface and Amophere diribuion of annual precipiaion were produced from 50,000-year imulaion ha were driven by amopheric moiure flux generaed uing he fir-order gamma auoregreive model of Fernandez and Sala (990). Ploed on a normal probabiliy cale in Figure 5 are modeled frequency diribuion of annual average precipiaion for he parameerizaion in Table and a parameerizaion, which remove he hermodynamic feedback erm. Alo ploed are he 37 PRISM-baed obervaional value ha were ued in he parameerizaion. The ample variance of he obervaion i 6 cm a compared o 0 cm and 43 cm for he hermodynamic-feedback and precipiaion-recycling-only cae, repecively. Thee reul ugge no only ha he colored-noie formulaion of he model capure mo of he iner-annual variabiliy of precipiaion bu ha hermodynamic feedback are a ignifican ource of ha variabiliy. Annual precipiaion (cm) Percenile Sudy Region ( ) a=0, k=0.9, d=.0 (hermodynamic feedback) a=0, k=0.8, d=0 (precipiaion recycling only) Sandard normal, Z Figure 5. Modeled frequency diribuion of annual precipiaion compared o he obervaion ued in he parameerizaion and obervaion from a longer ime erie. 7. Summary Baed on a lumped-parameer waer balance model, we have developed a ochaic differenial equaion ha decribe he dynamic of coninenal-cale amopheric moiure fluxe and oil moiure. Depie i impliciy, i reproduced quie well he amoun of variabiliy and perience in large-cale, long-erm obervaion of oil moiure and precipiaion in he cenral Unied Sae. I wa hown ha a colored-noie formulaion of he SDE i eenial for agreemen wih he obervaion. While a whie-noie formulaion produce he very inereing and implicaion-ridden cae of a bimodal/muli-ae probabiliy diribuion for oil moiure, we believe i doe no repreen he acual yem in he cenral Unied Sae. In conra, he colored-noie formulaion eem o capure he poiive oil-moiure Hydrology Day 003 7

12 Kochendorfer and Ramírez feedback ha are likely a ignifican conribuor o iner-annual precipiaion variabiliy in he udy region. Acknowledgemen. Suppor for hi work wa provided by a gran of he Souh Cenral Regional Cener of Naional Iniue for Global Environmenal Change of he Deparmen of Energy. Reference Brubaker, K.L. and D. Enekhabi, 995: An analyical approach o modeling land-amophere ineracion,. Conruc and equilibrium behavior. Waer Reource Reearch, 3(3), Charney, J., J.Q. Quirk, S-H. Chow and J. Kornfield, 977: A comparaive udy of effec of albedo change on drough in emi-arid region. Journal of he Amopheric Science, 34, Daly, C., R.P. Neilon and D.L. Phillip,994: A aiical-opographic model for mapping climaological precipiaion over mounainou errain. Journal of Applied Meeorology, 33, Enekhabi, D., I. Rodriguez-Iurbe, and RL. Bra, 99: Variabiliy in largecale waer balance wih land urface-amophere ineracion. Journal of Climae, 5, Fernandez, B. and J. Sala, 990: Gamma auoregreive model for ream flow imulaion. Journal of Hydraulic Engineering, 6(), Gardiner, C.W., 985: Handbook of Sochaic Mehod for Phyic, Chemiry and he Naural Science. Springer-Verlag, Berlin, nd ediion, pp. 44. Hobbin, M.T., J. A. Ramirez, And T. Brown, 000: The Complemenary Relaionhip In The Eimaion Of Regional Evaporanpiraion I: The CRAE And Advecion-Aridiy Model. Waer Reource Reearch. (In review). Hollinger, S.E. and S.A. Iard, 994: A oil moiure climaology of Illinoi. Journal of Climae, 7, Kalnay, E. and co-auhor, 996: The NCEP/NCAR 40-year reanalyi projec. Bullein of he American Meeorological Sociey, 77(3), Mearn, L.O., S.H. Schneider, S.L. Thompon and L.R. McDaniel, 990: Analyi of climae variabiliy in general circulaion model: comparion wih obervaion and change in variabiliy in xco experimen. Journal of Geophyical Reearch, 95(D), Rodriguez-Iurbe, I., D. Enekhabi and R.L. Bra, 99: Nonlinear dynamic of oil moiure a climae cale,. ochaic analyi. Waer Reource Reearch, 7(8), Wang, J., R.L. Bra and D. Enekhabi, 997: Srucure in flucuaion of largecale oil moiure climae due o exernal random forcing and inernal feedback. Sochaic Hydrology and Hydraulic,, Hydrology Day 003 8