Evolution of Dipole-Type Blocking Life Cycles: Analytical Diagnoses and Observations

Ev olution of Dip ole-T yp e Blo c king Life Cycle s: Analytical Diagnoses and Observ ations F ei Huang 1 , 2 , 3 , Xiao yan T ang 1 , 3 , S. Y. Lou 1 , 3 and Cuih ua Lu 4 1 Dep artment of P hysics, Shanghai Jiao T ong University,Shanghai, 20 0030, China 2 Physic al Oc e ano gr aphy L ab or atory, D ep artment of Marine Mete or olo gy, Oc e anUniv ersity o f China, Qingdao, 266003 , China 3 Center of N online ar Scienc e, Ningb o University, Ningb o, 315211, China 4 We ather Bur e au of Zaozhuang, Zao zhuang, 277148, China Abstract A v ariable co efficient Kortewe g d e V ries (VCKdV) system is derived b y considering the time- dep end en t basic flo w and b ound ary conditions from a nonlinear, in viscid, nondissipativ e, and equiv alen t barotropic vo r ticit y equation in a b eta-plane. One analytical solution obtained f rom the V CKd V equation can b e su ccessfully used to explain the ev olution of atmospheric dip ole- t yp e blocking (D B) life cycles. Analytical diagnoses sho w that bac kground mean w esterlies hav e great influence on ev olution of DB during its life c ycle. A w eak weste r ly is necessary for blo c king dev elopment an d the blo c kin g life p erio d shortens, accompanied with the enhanced westerlie s . The shear of the bac kgroun d westerlies also p la ys an imp ortant role in the ev olution of blo cking. The cyclonic shear is p referable for the dev elopment of blo c kin g b ut wh en the cyclonic shear increases, the intensit y of blo c king decreases and the life p erio d of DB b ecomes sh orter. W eak an ticyclonic shear b elo w a critical thresh old is also fav orable f or DB formation. Time-dep endent bac kground weste r ly (TD W) in the life cycle of DB has s ome m o dulations on the blo c king life p erio d and in tensity due to the b eha vior of the mean w esterlies. Statistical analysis regarding the climatolo gical features of observ ed DB is also inv estigat ed . Th e P acific is a p r eferred region for DB, esp ecially at high latitudes. These features may asso ciate with the w eak est w esterlies and the particular w esterly shear structure o v er the north western Pac ifi c. P ACS n umbers: 47 .3 2.-y; 47.35.- i; 92.60.-e ; 02.30 .Ik 1 I. INTR O DUC T ION A tmospheric blo c king is an imp ort a n t large-scale w eather phenomena at mid-high lat- itudes in the atmosphere that has a profound effect on lo cal and regiona l climates in the immediate blo ck ing do ma in (R ex 1 950a,b; Illari 1984) as w ell as in regions upstream and/or do wnstream of the blo ck ing ev en t (Quiroz 19 8 4; White and Clark 1 9 75). Therefore, it has long been of in terest to synoptic and dynamical meteorologists. The formatio n, main tenance and collapse of a tmospheric blo c king alw ay s cause larg e-scale weather or short-term climate anomalies. Hence, the prediction fo r atmospheric blo cking pla ys a significan t role in regional midterm we at her forecast and short- term climate trend prediction. Because of the needs of midterm w eather fo r ecast, there are has b een a lot of researc h on dynamics of blo ck ing . Ho w eve r , the phy sical mec hanism leading to blo c king formation remains unclear, whic h ma y result in the difficult y of blo c king onset prediction using general circulation mo dels (T racton et al. 1989 ; Tibaldi and Molteni 1990). Theoretical (Long 1964; Charney and DeV ore 1979; T ung and Linzen 1979; McWilliams 1980; Malguzzi and Malanotte-Rizzoli 1 984; Haines and Marshall 1987 ; Butc hart et al. 1989; Mic helangeli and V autard 1998; Haines and Holland 1 998; Luo 2000; Luo et al. 2 0 01), observ ational (Rex 19 50a,b; Colucci and Alb erta 199 6; Berggren et al. 1949; Illari 198 4; Quiroz 1984; White and Clark 1975 ; D ole a nd Gordon 1983; Dole 1986, 1989; Lejen¨ and ∅ land 19 83; Lup o and Smith 1 995a), and nu merical (T anak a 1991, 199 8; Luo et al. 2 002; DAndrea et al. 199 8 ; Ji and Tibaldi 1983 ; Chen and Juang 1992; Colucci and Baumhefner 1998) studies on the atmospheric blo c king systems hav e undergone substan tia l and extensiv e dev elopmen t ov er the past sev eral decades. Usually an atmospheric blo c king anticyclone has three type o f patterns: monop o le-t yp e blo c king (or Ω-type blo cking), dip ole-t yp e blo ck ing (McWilliams 1980; Malguzzi and Malanotte-Rizzoli 1984; Luo and Ji 1991), and m ultip ole- t yp e blo c king (Bergg ren et al. 1949; Luo 200 0). The setup o f the tw o later patterns are usually due to the strong nonlinear in teraction b etw een upstream high frequency synoptic- scale tr ansien t eddies and planetary-scale w av es in exciting blo c king circulation (Hansen and Chen 1982; Sh utt s 1983, 1986; Ji and Tibaldi 1983; Illari and Marshall 198 3; Illari 1984; Hoskins et al. 1985; Egger et al. 1986; Mullen 1 9 87; Holopainen and F ortelius 198 7 ; Haines and Marshall 1 987; V autard and Legras 198 8; V autard et al. 19 88; Chen and Juang 1 9 92; T anak a 1991; Lup o and Smith 1995b; Lup o 1997; Lup o and Smith 1998; Haines and Holland 2 1998; Luo 2000, 2005a,b). How ev er, the nature of the planetary-synoptic scale in teraction has y et to b e clarified theoretically (Colucci 1985, 1987). A tmospheric blo c king ev ents, mainly lo cated in the mid-high latitudes and usually o ver the o cean, w ere first disco vere d as a dip ole pattern by Rex (19 50a,b). La t er, Malg uzzi and Malanotte-Rizzoli (1984) first fo und the dip ole-type blo c king f rom the Kortew eg- de V ries (KdV) Rossb y soliton theory . Ho w eve r, their analytical results cannot explain the onset, dev eloping and deca y of a blo c king system . Recen tly , Luo et al. (200 1) prop osed the en v elop e Rossb y soliton theory based on t he deduced nonlinear Sch r ¨ odinger (NLS) t yp e equations to explain the blo c king life cycle. Ho w ev er, they o nly obta ined their solutions of the NLS t yp e equations nume rically instead o f analytically . In 1895, the KdV equation w as firstly deriv ed in the classic pap er o f Kortew eg and de V ries (1895 ) as a fundamental equation g o ve r ning pr o pagation of w a ves in shallow w at er. After that, m uc h prog ress has b een ma de on this equation in b oth mathematics and phy sics. No w, phys ical applications of the KdV equation hav e b een seen in a n um b er of problems suc h as plasmas (Z a busky and Krusk al 19 65; Rao et al. 1990), quasi-one-dimensional solid- state ph ysics (Flytzanis et al. 1985), nonlinear transmission lines ( Y oshinaga and Kakutani 1984), and so on. In all these applications, the KdV equation a r ises as an appro ximate equation v alid in a certain a symptotic sense. T aking accoun t of additional ph ysical factors, one ma y a lso obtain v arious KdV-t yp e equations. F or instance, a v ariable co efficien t KdV- type (VC K dV) equation with conditions give n at the inflow site w as deriv ed as gov erning the spatial dy- namics in a simplified one-dimensional mo del for pulse wa v e propagation thro ugh fluid-filled tub es with elastic w alls (Flytzanis et al. 1985), nonlinear transmission lines (Y oshinaga and Kakutani 198 4 ), and so on. F rom the discussion b elow of a problem on the dynamics of atmospheric blo ckings, it will b e seen that V CKdV equation is fairly univ ersal and useful to explain t he lifetime of the atmospheric blo cking system s. One imp orta n t fact is that in the usual treatmen t of deriving either KdV- or NLS-t yp e equations in the study of atmospheric blo c king dynamics, one alw ays separates the bac k- ground flo w, assumed to linearly dep end on y only , a nd takes ze ro b o undar y v alues. Nonethe- less, in realit y , the bac kground w esterly c ha ng es with time as w ell as the shear type, whic h is something that has not ye t b een considered. Therefore, in this pap er, w e are motiv ated to in tro duce time in to the bac kground flo w and b oundary conditions to r einv estigate the 3 Rossb y KdV theory for a tmospheric blo c king systems. The term “blo c king” denotes a breakdo wn in the prev ailing trop ospheric we sterly flo w at midlatitudes, often asso ciated with a split in t he zonal jet and with p ersisten t ridging at higher latitudes (Rex 1950a,b; Illari 1984). Therefore, the basic pattern of blo c king is dip ole t yp e and it alw ay s o ccurs under w eak bac kground w esterlies (Sh utts 1983; Luo and Ji 1991; Luo 1994; Luo et al. 2001 ; Luo 2 005b). W eak bac kground w esterlies as a precondition for blo c k onset w ere first noted b y Sh utts (1983), who found that the blo ck do es no t o ccur fo r more rapid flows in whic h a stationary free state cannot b e excited. How ev er, Tsou and Smith (1990) and Colucci and Alb erta (1996 ) suggested that the preconditioned pla netary- scale ridge (incipien t blo cking ridge) ma y b e another necessary condition for blo c k onset in addition to t he w eak pla netary-scale we sterly flow . Although pr evious theoretical analysis hav e sho wn that the w eaker ba c kground w esterlies w ere prev ailing for the onset of DB than that of Ω-ty p e blo c king (Luo 1994), the role of the w eak w esterlies on the ev olution of DB life cycle is still unclear. F urther, ho w v ariations of the time-dep end back gr o und w esterlies influence the ev olution of DB during its life p e- rio d is also an interes ting problem, since the realtime wes terlies v ary with time. Shear in the bac kground w esterlies was alw a ys in tro duced in to theoretical mo dels (T ung and Linzen 1979; McWilliams 1 980; Malguzzi and Malanotte-Rizzoli 1984; Haines and Marshall 198 7; Butc hart et al. 1989 ; Luo 19 9 4; Haines and Holland 1998; Luo et al. 20 01) in the study o f blo c king, while Luo (1995) rev ealed that dip ole structure o f solitary Rossb y w av es excited b y the β parameter could b e obtained excluding the effect of shearing basic-state flo w. In addition, the role of an ticyclonic shear of basic-state flow on DB w as emphasized (Luo 1994 ). In contrast, Luo et al. (2001) found that the cyclonic shear of the bac kground we sterly wind at the b eginning of blo ck onset w as a fav ora ble preblo c k environmen t, whic h increased the strength of the precursor blo c king ridge, but the anticy clonic shear w eake ned the precursor blo c king ridge considerably . Recen tly , Luo (2 0 05b) also found that the asymmetry , in tensity , and p ersistence of dip ole blo c k depend strongly up o n the horizontal shear of the basicstate flo w prior to blo ck onset. Then there arises another question: what do es the role o f cy- clonic/an ticyclonic shear of the bac kground w esterlies play in the evolution of DB during differen t stages of its life cycle? In this pap er, w e will fo cus on the tw o questions men tioned ab o ve by diagnosing the an- alytical solutio n obtained from a VCKdV equation deriv ed from the atmospheric no nlinear, 4 in viscid, nondissipativ e, and equiv alen t barotropic vorticit y equation in a b eta plane consid- ering the time-dep enden t background flo w and b oundary conditions. The pap er is org anized as follows. Sec t io n 2 in t r o duces the deriv ation of the V CKdV equation and its ana lytical solution. Analytical diagnosis of bac kground w esterly v a riations (including the v ariation of mean flow, w esterly shears, and the time-dep enden t bac kgro und flo w) on the ev olution of DB during its life cycle are inv estigated in Section 3. In the following section, some sta- tistical observ a tional results ab out D B in the Northern Hemisphere are giv en and p o ssible in terpretations from ab ov e theoretical mo del are demonstrated. Section 5 will outline our conclusions. I I. THEO RETICAL MO DEL A. Deriv ation of the VCKdV e qua t ion Our starting mo del is the atmospheric nonlinear, invisc id, nondissipativ e, and equiv alen t barotropic v ort icity equation in a b eta-pla ne c hannel (P edlosky 1979; Luo 200 1, 2005a): ( ∂ t + u ∂ x + v ∂ y ) ( v x − u y − F ψ ) + β ψ x = 0 , (1) whic h is w ell kno wn and o ne of the most imp ortant mo dels in the study of atmospheric and o ceanic dynamical systems . In equation (1), u = − ψ y , v = ψ x , ψ is t he dimensionless stream function, u and v a r e comp onen ts of the dimensionless v elo cit y , F = L 2 /R 2 0 is the square of the ratio of the c haracteristic horizontal length scale L to the R o ssb y deformation radius R 0 , β = β 0 ( L 2 /U ), β 0 = (2 ω 0 /a 0 ) cos φ 0 , in which a 0 is the Earth’s radius, ω 0 is the angular frequency of the earth’s rot a tion, φ 0 is the latit ude, and U is the ch a racteristic v elo city scale. The characteristic horizontal length scale can b e L = 10 6 m and the c ha r acteristic horizon tal v elo city scale is ta k en as U = 1 ms − 1 . Similar to the usual treatmen ts, w e rewrite the stream function as ψ = ψ 0 ( y , t ) + ψ ′ where ψ 0 means the bac kground flow term, and in t r o duce the stretc hed v ariables ξ = ǫ ( x − c 0 t ) , τ = ǫ 3 t , ( c 0 is a constan t) . In the previous studies, the background field ψ 0 is often tak en only as a linear function of y , and in most cases one simply makes the expansion ψ ′ = P ∞ n =1 ǫ n ψ ′ n ( ξ , y , τ ). Ho wev er, here w e also hav e the expansion for t he background flo w term as ψ 0 = U 0 ( y ) + P ∞ n =1 ǫ n U n ( y , τ ). Then, again similarly , substitute the expansions 5 in to (1) and set the co efficien t of each order of ǫ equal to zero. F or notat io n simplicit y , the primes are dropp ed out in the remaining o f this pap er. In the first order of ǫ , w e obtain ψ 1 = A ( ξ , τ ) G ( y , τ ), where G is determined b y ( U 0 y + c 0 ) G y y − ( F 2 c 0 + β + U 0 yy y ) G = 0 . In the second order, w e ha v e ψ 2 = A ( ξ , τ ) G 1 ( y , τ ) + A 2 ( ξ , τ ) G 2 ( y , τ ), where G 1 satisfies ( U 0 y + c 0 ) 2 G 1 yy − ( U 0 y + c 0 )( F 2 c 0 + U 0 yy y + β ) G 1 − [( U 0 y + c 0 ) U 1 yy y − ( F 2 c 0 + U 0 yy y + β ) U 1 y ] G = 0 , and G 2 satisfies 2( U 0 y + c 0 ) 2 [( U 0 y + c 0 ) G 2 yy − ( F 2 c 0 + U 0 yy y + β ) G 2 ] + [( F 2 c 0 + U 0 yy y + β ) U 0 yy − ( U 0 y + c 0 ) U 0 yy y y ] G 2 = 0 . Substituting the solutions obtained from the previous t w o orders a nd ψ 3 = 0 into the co efficien t of the third order of ǫ , then in t egr a ting the result with resp ect to y f rom 0 to y 0 as in the usual treatmen ts of the o ceanic and atmospheric dynamics, yield the mo dified V CKdV equation ( e 2 6 = 0) and/or V CKdV equation ( e 2 = 0) e 1 A ξ ξ ξ + ( e 2 A 2 + e 3 A + e 4 ) A ξ + ( e 5 A ) τ + e 6 = 0 , (2) with ( e i ≡ e i ( τ )), e 1 = R y 0 0 − G ( U 0 y + c 0 )d y , e 2 = R y 0 0 2( G y y y G 2 − G y G 2 yy ) + GG 2 yy y − G y y G 2 y d y , e 3 = R y 0 0 2 U 1 yy y G 2 − 2 U 1 y G 2 yy + G y y y G 1 − ( G y G 1 y ) y + GG 1 yy y d y , e 4 = R y 0 0 U 1 yy y G 1 − G y y U 2 y + GU 2 yy y − U 1 y G 1 yy d y , e 5 = R y 0 0 G y y − F 2 G d y , e 6 = R y 0 0 ( U 1 yy − F 2 U 1 ) τ d y . B. Exact solution of the V CKdV equat ion Ob viously , in the ab ov e deriv ation of the VCKdV equation, the ba ckground flow is left a s an arbitrary f unction of { y , t } . F or the sak e of obtaining meaningful analytical solutions to the V CKdV equation, w e hav e to fix its v a r iable co efficien ts further. In the expansion of the bac kground flow term ψ 0 , if U n = 0 for n ≥ 1, it b ecomes a function of y as usual. Therefore, 6 the higher or der items U n can b e view ed as higher-or der corrections to the low est one, U 0 . Because of this, w e also tak e U 0 as a linear function of y ; that is, U 0 = a 1 y + a 0 with a 0 , a 1 constan t, a nd for the next higher order, as men tio ned b efore, it is chose n as a quadratic function o f y with time dep enden t coefficien ts; that is, U 1 = a 2 y 2 + a 3 y + a 4 with a 2 , a 3 , a 4 all time dep enden t functions. The higher orders are all tak en as zero, namely U n = 0 for n ≥ 2 so as to simplify the problem. In this case, the back g round flo w is similar to the cases in Gottw ald and Grimshaw (1999) except that the mean flow configuratio ns are time- v arying instead of stable. It is noted that the quan tity ǫa 2 pla ys the role of t he shear of the bac kground flow , a nd t he bac kground wes terly wind is determined b y a 1 + ǫa 3 . Apparen tly , b oth the shear and the back g round w esterlies v ary with time. After these selections, G, G 1 and G 2 can b e easily solv ed as G = F 1 sin( M y + F 2 ), G 1 = F 5 sin( M y + F 6 ) + ( F 2 + M 2 ) F 1 4 M ( a 1 F 2 − β ) [ M (2 a 2 y + a 3 ) sin( M y + F 2 ) − (2 a 2 M 2 y 2 + 2 a 3 M 2 y − a 2 ) cos( M y + F 2 )], G 2 = F 3 sin( M y + F 4 ), where c 0 = − a 1 M 2 + β F 2 + M 2 . F 1 ∼ F 6 , a nd a ll functions with resp ect to time should b e determined b y the b oundary conditions. No constrain t s are made on these functions b ecause in the follo wing, w e will fo cus on the b ehav io r of the VCKdV equation with differen t v alues for the parameters app eared in the background flo w so a s to inv estigat e what effect of the bac kground can mak e on the p ossible coheren t structure; that is, the soliton solution. With the ab o ve selections and results, we can finally write dow n an exact solution to the V CKdV equation as A = C 2 ξ − H 2 C 3 e 6 √ − C 1 + C 2 τ + ( F 2 a 1 − β ) 1 / 3 e 6 ( F 2 + M 2 ) 2 / 3 √ − C 1 + C 2 τ ( C 4 C 2 3 C 1 / 3 7 +12 C 3 K 2 C 2 / 3 7 sec h 2  8 K 3 C 3 3 C 7 − 2 K C 4 C 2 √ − C 1 + C 2 τ + K C 5 − 4 K 3 C 6 + K C 3 C 1 / 3 7 ( F 2 + M 2 ) 2 / 3 2( F 2 a 1 − β ) 1 / 3  2 ξ √ − C 1 + C 2 τ + Z H ( − C 1 + C 2 τ ) 3 / 2 d τ  − e 4 √ − C 1 + C 2 τ C 3 e 2 6 , (3) where H = R 2( C 1 − C 2 τ )( e 4 e 6 τ − e 4 τ e 6 ) e 2 6 + 2 e 7 C 3 √ C 2 − C 1 τ − e 4 C 2 e 6 d τ , e 4 = [2 a 2 sin( M y 0 + F 6 ) − 2 a 2 sin( F 6 ) − M (2 a 2 y 0 + a 3 ) cos( M y 0 + F 6 ) + M a 3 cos( F 6 )] F 5 − ( F 2 + M 2 ) F 1 4( F 2 a 1 − β ) M [ a 2 a 3 sin( F 2 ) M − ( M 2 a 2 3 + 2 a 2 2 )(cos( M y 0 + F 2 ) − cos( F 2 )) + M (2 a 2 y 0 + a 3 )(2 M 2 a 2 y 2 0 + 7 2 M 2 a 3 y 0 − a 2 ) sin( M y 0 + F 2 ))], e 6 = − F 1 ( M 2 + F 2 ) M [cos( F 2 ) − cos( M y 0 + F 2 )], e 7 = − y 0 6 [(2 y 2 0 F 2 − 12) a 2 τ + 3 y 0 F 2 a 3 τ + 6 F 2 a 4 τ ], and M , K, C i , ( i = 1 , 2 , ..., 7) are arbitrary constan ts. There is also a necessary relation b e- t w een F 2 ∼ F 4 : F 3 [2 M 2 ( λ 0 2 a 1 − β ) √ − C 1 + C 2 τ (2 a 2 sin( F 4 ) − 2 a 2 sin( M y 0 + F 4 ) − a 3 M cos( F 4 ) + M (2 y 0 a 2 + a 3 ) cos( M y 0 + F 4 ))] − F 2 1 ( F 2 + M 2 )[ a 2 M 3 √ − C 1 + C 2 τ (cos( F 2 ) sin( F 2 ) − cos( M y 0 + F 2 ) sin( M y 0 + F 2 ) + M y 0 ) − C 3 ( F 2 a 1 − β )( F 2 + M 2 )(cos( F 2 ) − cos( M y 0 + F 2 )) 2 ] = 0. I I I. ANAL YTIC AL DIAGNOSIS O N DB E V OLUTI ON In this section w e analyze a DB ev olution case under certain parameters. Set e 4 = 0 and a 4 = − y 0 2 a 3 + ( 2 F 2 − y 2 0 3 ) a 2 (and then e 6 = 0). Hence, one p ossible appro ximate solution to (1) is in the form of ψ ≈ ǫa 2 y 2 + ( a 1 + ǫa 3 ) y + a 0 − y 0 2 ǫa 3 +  2 F 2 − y 2 0 3  ǫa 2 − sin( M y + F 2 ) [cos( F 2 ) − cos( M y 0 + F 2 )] × ǫM ( M 2 + F 2 ) √ − C 1 + ǫ 3 C 2 t  ǫC 2 2 C 3 ( x − c 0 t ) + C 4 M 1 C 2 3 C 1 / 3 7 + 12 C 3 M 1 K 2 C 2 / 3 7 × s ech 2 " K √ − C 1 + ǫ 3 C 2 t ǫC 3 C 1 / 3 7 M 1 ( x − c 0 t ) + 8 K 2 C 3 3 C 7 − 2 C 4 C 2  + K ( C 5 − 4 K 2 C 6 )  , (4) with M 1 = ( F 2 a 1 − β ) 1 / 3 ( F 2 + M 2 ) 2 / 3 . F rom the analytical solution of streamfunction ψ it is easily to derive the bac kground w est- erly flow u = − ∂ ψ 0 ∂ y = u 0 + δ y , where u 0 = − ( a 1 + ǫa 3 ( t )) and δ = − 2 ǫa 2 ( t ). As w e kno wn, the w eather o r climate p erio dically c hanges on either larg e or small scales. Thu s, it is reasonable to choose the time-dep endent unkno wn functions in the bac kgro und flow as p erio dic, so the sine function is in use. Th us w e choose a 2 ( t ) = − a 20 sin ( k 2 t ) , a 3 ( t ) = − a 30 sin ( k 3 t ). The basic-state w esterlies hav e cyclonic shear when δ < 0 or a 20 > 0, but correspo nd to anticy- clonic w esterly shear when δ > 0 or a 20 < 0. Here w e consider φ 0 = 60 o N in that the blo c king systems are mainly situated in the mid-high latitudes. F or example, when w e tak e the func- tions and para meters in our ana lytical solution (4) as a 20 = 0 . 1 , k 2 = 2 . 1 , a 30 = 0 . 5 , k 3 = k 2 , F 2 = arccos(1 0 − 8 cosh(0 . 75 t − 4 . 5)) , ǫ = 0 . 1 , a 0 = 2 , a 1 = − 0 . 2 , C 1 = − 30 , C 2 = 8 FIG. 1: A d ip ole b lo c king life cycle from the theoretical solution (4) with a 2 = − 0 . 1 sin(2 . 1 t ) , a 3 = − 0 . 5 s in (2 . 1 t ) , ǫ = 0 . 1 , a 1 = − 0 . 2. The con tour inte r v al CI=0.2. 0 . 1 , C 3 = − 10 , C 4 = − 16 , C 5 = 4 , C 6 = − 7 , C 7 = 0 . 1 , K = 0 . 2 , F = 64 , M = π 6 , a dip ole-ty p e blo c king life cycle app ears and is sho wn in F ig. 1. Figure 1 clearly rev eals the onset, dev elopmen t, main tenance, and deca y pro cesses–a whole life cycle of a dip ole-type blo c king ev ent. The streamlines ar e gradually deformed and split in to t wo branc hes a t the second da y , with the an ticyclonic high in the nor t h dev eloping first (Fig. 1a). Then the cut-off lo w dev elops south of the high, f orming a pa ir of high-low dip ole patt ern where the cen ter o f the hig h is lo cated at the higher latitude than that of the lo w (Fig. 1b) . They are strengthened daily . A t around the sixth day (Fig. 1c), they are at their strongest stage and then b ecome w eak er and ev en tually disapp ear after the elev enth da y (Fig. 1d-f ). Ob viously , Fig. 1 p ossesses the phenomenon’s salien t features including their spatial-scale and structure, amplitude, life cycle and duration. The refo r e, F ig . 1 is a v ery t ypical dip ole-type blo c king episo de. More imp ortantly , it corresp o nds quite well to a real observ a t io nal blo c king case ( F ig. 2) that happ ened o v er the Pacific during 2-12 Jan ua ry 1996, whic h is o btained from the Natio nal Cen ters for En vironmen tal Prediction-National Cen ter for A tmospheric R esearch (NCEP-NCAR) reanalysis data. Because blo ck ing is a large-scale atmo spheric phenomenon, the geop oten tial heigh t fields in Fig. 2 w ere filtered b y preserving w av e n umbers 0-4 in a harmonic ana lysis around latitudes in order to filter high-frequency synoptic-scale p erturbations. It is easily found in F ig. 2 that the life cycle of this blo c king lasts almost 10 da ys, exp eriencing three stages: onset (2-5 Jan ua ry , 19 9 6), maturity (7-9 Jan uary , 1996 ) and deca y (10-12 Jan uary , 1996). This blo c king ev en t dev elop ed within a n atmospheric long R o ssb y 9 FIG. 2: Filtered geop oten tial heigh t at 500-hP a pressure lev el of a blo cking case dur ing 2-11 Jan 1996. Con tour inte r v al is 4 gp dm. T he x -axis is longitude, and the y -axis is latitude. w av e ridge extending north we stw ard and an upstream cutoff low mo ving southeast ward (Fig. 2a, b). With the strengthening of the blo c king high and the cutoff lo w, tw o closed cen ters of high and low app eared for ming a north- south dip ole- lik e pattern with the closed high in the north and the closed lo w in the south during the mature stage (Fig. 2c, d). Then the high cen ter deca y ed and v anished first, as w ell as the cutoff low deca y ed subseq uently (Fig. 2e,f ). At last the blo ck b ecame a trav elling Rossb y w a ve ridge and mov ed eastw ar d (figure omitted). A. Effect of the mean bac kground westerlies Previous researc h sho w ed that the back g round w esterlies are a necess a r y precondition f or the onset of an ticyclonic blo c king and they a lso play ed a n imp ortant role in the blo c king life cycle (Sh utts 1983, 1986; Luo and Ji 1991 ; Luo 1 9 94; Luo et al. 2001). F or DB ev olution during its life cycle, the r o le of the mean background w esterlies in the DB ev o lut io n is not clear in a V CKdV soliton system, while the t ypical KdV R o ssb y solito n can only represen t a steady-state DB pa t t ern instead of an ev olution of DB life cycles. T o simplify the question, here only v ariation of the mean flow is considered without w esterly shear 10 FIG. 3: The ev olutions of DB life cycles u nder different bac kgrou n d weste r lies (a) a 1 = − 0 . 2; (b) a 1 = − 0 . 5; (c) a 1 = − 0 . 8; (d) a 1 = − 1 . 0. CI=0.4. ( a 2 ( t ) = 0) and time-dep endent bac kground w esterly term ( a 3 ( t ) = 0). Only the parameter a 1 v aries in differen t constan ts indicating the v ariation of the mean bac kground w esterlies. The ev olutions of DB life cycles under differen t mean back gr o und we sterlies are sho wn in Fig. 3. It is sho wn in F ig. 3 that the in tensit y , zonal and meridional scales, p osition, and p erio d of D B during its life episo de alter with the increasing mean bac kground w esterlies. Under 11 the w eak mean we sterly a 1 = − 0 . 2 condition (Fig. 3a), the DB p erio d is ab out 10 days from t = 1 to t = 10, consisten t with t ypical p erio d of blo c king observ ed [ ? ? ? ? ? ? ? ]. At time t = 2 an incipien t dip o le-lik e en v elop Rossb y soliton pattern a pp ears and b egins to enhance, for ming closed high and lo w cen ters at t = 4. Then the high and low cen ters strengthen dramatically fro m t = 4 to t = 6 and reac h their p eaks at t = 6 . Subseq uently , the high and low cen ters in D B decrease gradually fr om t = 7 to t = 1 0. The meridional scale of DB ( distance betw een the high and lo w cen ters) increases during the dev eloping p erio d ( t = 1 to t = 6) o f D B life cycle and decreases during the deca y pro cess ( t = 7 to t = 10). It is noticeable that the high and lo w cen ters in the DB streamlines pattern seem an tisymmetric along the north-south direction as w ell as the evolution pro cess is symmetric with resp ect to the mature phase of DB at t = 6 under the high idealized theoretical solution. Ho w eve r , the main c haracters of the DB streamlines pattern and its ev olution, including the onset, dev elopmen t, mature, and deca y pro cesses, are w ell captured. Comparing the D B ev olution pro cesses under differen t mean w esterlies (Fig. 3b-d), it is easily found that the incipien t dip ole-lik e en v elop at time t = 3 w eak ens with the mean w esterly increasing. The closed streamlines of high and low cen ters in DB under we a k w est- erly conditions (Fig. 3 a,b) at t = 4 disapp ear and are replaced b y the gra dually decreasing en v elop streamline pattern (Fig. 3c,d) when t he para meter a 1 v aries from a 1 = − 0 . 8 to a 1 = − 1 . 0. That means the p erio d of the DB life cycle shortens when the mean w esterlies increas. A t time t = 4 or t = 8, it is very clear tha t the DB in tensit y represen ted by v alues of the DB hig h a nd low cente r s also w eak ens with t he mean flow increasing. This feature is also clearly seen in Fig. 4, show ing the inte nsity of D B high (Fig. 4a) and low (F ig. 4b) cen ter v a rying with resp ect to time t . It is ob vious that the high center o f DB w eake ns while the lo w cen ter strengthens when the mean w esterlies increas throughout the p erio d of DB life cycles. Nev ertheless, the b ehav ior s of the high and lo w cen ters app ear inconsisten t during the differen t dev eloping stages of DB life episo de. That is, the high cen ter (Fig. 4a) decreases greatly during t he onset ( t = 1 to t = 4) and deca y ( t = 8 to t = 11 ) stages of DB life episo de, and decreases slo wly in the DB mature phase ( t = 5 to t = 7). On the con trary , the lo w cen ter (F ig . 4b) deepens g reatly during the mature stage of DB lif e cycle, but strengthens r elat ively sligh tly at the o nset and deca y stages. T o see more clearly the mo ve ment of DB high cen ter with resp ect to time t , Fig . 5 sho ws space-time ( x − t or y − t ) cro ss sections crossing the high center. F rom the x − t 12 (a) (b) FIG. 4: The in tensity of the DB high (a) and lo w (b ) cen ter v arying w ith resp ect to time t . cross section (Fig. 5a) it is found that the blo c king high mo ve s we stw ard and the zonal scale of DB enlarges with the strengthening of the mean we sterlies. This f eature can also b e iden tified in Fig. 3. F o r the y − t cross section crossing the high cen ter (Fig. 5b), the meridional scale of the DB shortens and the ev olut io n crossing t he dip ole high/low cen ter with resp ect to t displa ys a shrinking dip ole-lik e pattern, suggesting shortening of DB p erio d of life cycles under the condition of the mean w esterlies increasing. This r esult implies that the DB could not main ta in stationary long time in strong w esterlies (Sh utts 1 983). B. Effect of the bac kground westerly shear In this section effects of the w esterly shears including cyclonic shear and an ticyclonic shear on ev olution o f DB a r e inv estigated. As men tio ned ab o ve, the w esterly shear pa r am- eter δ is expressed as δ = − 2 ǫa 2 ( t ); δ < 0 or a 2 ( t ) > 0 denotes the cyclonic bac kground w esterly shear, and δ > 0 or a 2 ( t ) < 0 represen ts the anticyc lo nic shear. As w e kno w, a 2 ( t ) is a function with resp ect to time t . T o simplify the question, here a 2 is assumed to b e a small constan t, suggesting a time-indep enden t w eak linear shear sup erp osed on the ba c k- 13 FIG. 5: The sp ace-time x − t (a) and y − t (b) cross sections crossing th e high center of DB under differen t basic mean w esterlies. CI=0.2. ground mean flo w. The mean wes terlies without the time-dep enden t t erm ( a 3 ( t ) = 0) is also assumed. Conseq uently , the background w esterly with linear shear is u = u 0 + δ y , where the mean flow u 0 = − a 1 = 0 . 6 . 1. The cyclonic westerly she ar The bac kground w esterly shear is a lw ay s introduced in to a theoretical mo del (T ung and Linzen 1979; McWilliams 1980; Malguzzi and Malanotte-Rizzoli 1984 ; Haines and Marshall 1987; Butc hart et al. 1989; Luo 1994 ; Haines and Holland 199 8 ; Luo et al. 2001; Luo 2005b) 14 in the study of blo c king, but the role of the cyclonic w esterly shear (CWS) in ev o lution o f DB during its life cycle is not clear, although Luo et al. (2001 ) and Luo (2005b) emphasized the imp ortant role of the cyclonic shear of ba c kground w esterly wind in blo c k onset. Recen tly , Dong and Colucci (2 005) also demonstrated the imp orta n t effect of cyclonically sheared flow on forcing w eak ening w esterlies asso ciated with the Southern Hemisphere blo c king onset. T o compare the influence of differen t cyclonic shear on a D B episo de, the DB ev olutions at parameter a 2 = 0 . 03 , 0 . 15 , 0 . 3 , 0 . 45 and 0 . 6, corresp o nding to 1%, 5%, 10%, 15% and 2 0% of the mean w esterlies u 0 , a re inv estigated resp ectiv ely in Fig. 6. Results show that w eak CWS is f av orable for t he onset o f DB, similar to the conclusion b y Luo et al. (2001), Luo (2005b), and D ong and Colucci (2 005). Ho w ev er, the D B life p erio d shortens a nd the DB high/lo w cen ters w eak en sim ultaneously when the weak CWS enhances during the ev olution of DB episo de (Figs. 7a, c). The in tensit y v ariation of the low cen ter app ears differen t b eha viors, considering the CWS comparisons with and without shear (F ig. 4). That is, the low we a k ens alo ng with increasing CWS (Fig. 7c), and strengthens with increasing bac kground mean w esterlies without shear (Fig. 4b). A noticeable phenomenon is that the CWS destro ys the an tisymmetric structure o f the D B high/low p oles. The low dev elops more slo wly than the high at the D B onset stage ( t = 4), while it trails off faster than the high during the deca y p erio d ( t = 8) with the CWS increasing (Fig. 6 ). This feature is a lso indicated in Figs. 7a,c. 2. The anticyclonic westerly she ar Luo (1994) rev ealed that strong a n ticyclonic w esterly shear (A WS) w as disadv an tageous for establishmen t of dip ole-ty p e blo c king, but his solution could not interpret the o nset and deca y pro cess o f DB. Recen tly , Luo et al. (2001) a nd Luo (2005b) also p oin t ed out that the an ticyclonic bac kground w esterly shear w eak ened the precursor blo c king ridge considerably; therefore, the formation of a blo c king an ticyclone w as difficult. Ho wev er, the role of the A WS on DB life cycles is not clear. In our mo del, the influence o f differen t w eak A WS on DB episo de with parameters a 2 = − 0 . 03 , − 0 . 15 , − 0 . 3 , − 0 . 45, and − 0 . 6, respective ly , are compared with each other in Fig. 8. Results sho w that only v ery w eak a nticyclonic shear is more pr eferable for the onset and maintenance of DB, resulting longer life of DB (Fig s. 8a-c). There exists a threshold v alue of a 2 limiting the anticy clonic shear that con tr ols the 15 FIG. 6: The stream fu nction p atterns of DB ev olution under differen t cyclonic w esterly sh ears. (a) a 2 = 0 . 03; (b) a 2 = 0 . 15; (c) a 2 = 0 . 3; (d) a 2 = 0 . 45; (e) a 2 = 0 . 6. CI=0.4. app earance o f DB streamline patterns. Under the mean wes terlies u 0 = 0 . 6 , the threshold of a 2 is ab out − 0 . 45, or the threshold shear δ c = 0 . 09, denoting the slop e of lines on the ( u 0 − y ) plane. This means the DB is easily established under w eak A WS condition when δ < δ c . F or δ < δ c , the DB life p erio d is prolonged ob viously and DB establishes (decays ) earlier (later) with the increasing of A WS ( F igs. 8a-c). F or δ > δ c , the dip ole-like circulation can still dev elop, but nev er b e a blo ck ing pattern splitting the wes terlies in to tw o northw ard and 16 (a) (d) (c) (b) FIG. 7: The in tensit y of the DB high (a, b) and lo w (c, d) cen ter v aryin g w ith resp ect to time t under different conditions of westerly shears. (a) and (c) are for cyclonic shear, (b) and (d) are for an ticyclonic shear. south w ard branche s (Figs. 8d,e). Instead, the separate no rth ward w esterlies a r ound the high disapp ear and the dev elop ed high in the dip ole comes from the most northern b oundary , whic h indicates a cutoff high from the North P ole region mo ving southw ar d, forming the dip ole circulation together with the low in the south. Actually , in a da ily synoptic c hart, the cutoff high from the North Pole region is often seen moving southw a r d. The threshold v alue of A WS decreases along with decreasing of the mean bac kgro und w esterlies u 0 . It is also ob vious that the A WS induces the asymmetry deve lo pment of the high/low cen ters in DB during different stages of its life cycle (Fig . 8). Esp ecially at DB onset ( t = 3 , 4) and decay ( t = 8 , 9) phases, the high cen ter is appar ently stronger than the 17 FIG. 8: T he stream function patterns of DB ev olution und er different antic yclonic we s terly shears. (a) a 2 = − 0 . 03; (b) a 2 = − 0 . 15; (c) a 2 = − 0 . 3; (d) a 2 = − 0 . 45; (e) a 2 = − 0 . 6. CI=0.4. lo w cen ter, a nd the high/lo w cen ter tends t o strengthen alo ng with the strengthening of the A WS. The D B high/lo w cen ter strengthens sim ultaneously when the A WS strengthens during the ev olution of DB episo de (Figs. 7 b, d) , whic h app ears to ha ve opp o site b ehav ior than that with CWS (F igs. 7 a, c). In addition, the inte nsity of the DB dipo le cen ters with A WS is stronger tha n that of CWS. 18 C. Effect of the time-dep endent back ground w esterlie s In the real atmospheric circulation, the bac kground w esterly is not alwa ys a constant during the lif e p erio d of blo c king. Actually there is interaction b et we en blo c king and the bac kground w esterlies; that is, the w eak back g r ound w esterlies modulated b y transien t eddies are a precondition fo r the onset of blo c king; mean while, after the blo c king is established, it prev en ts the w esterlies passing through as a result of a w eakene d we sterly (Berggren et al. 1 9 49; Elliott and Smith 1949 ; Egger et al. 198 6; Mullen 1987; Long 196 4; Sh utts 198 3 ; Colucci 19 85; Holopainen and F ortelius 1987; Dole 1989; Luo et al. 2001). Therefore, the background w esterlies ma y b e a time-dep enden t function during the episo de in teracting with blo c king. F or simplification, the wes terly shear ( a 2 = 0) is not considered here, so the bac kground flow , u = u 0 − ǫa 3 ( t ), here is u 0 = − a 1 = 0 . 5. Sev eral ty p es of a 3 ( t ) profiles sho wn in Fig. 9 are attempted t o inv estigate the influence of time-dep enden t back g round w esterlies (TDW) on DB evolution. Case 1 indicates t he condition that the bac kground w esterlies b ecome weak est at DB onset stage and increases g radually througho ut the blo c k life cycle. It is f ound that the v aria tions of the curv e slop e do not impact the ev olutio n of DB in Case 1. Case 2 and 3 show the w esterly profiles r eflecting the in teraction b et wee n the bac kground w esterlies a nd blo cking with the wes terlies decreasing during the dev eloping p erio d b efore the DB p eak at t = 6 and increasing throughout t he deca y stage after the DB p eak. The la tter is conceptually consisten t with the actual v ariation of the we sterlies during a blo ck ing ev olution (Elliott and Smith 1949). Fig. 10 presen ts the inten sity of the DB high (Fig. 10a) and low (Fig. 10b) cen t er v arying with resp ect to time t under differen t t yp es of TDW a s sho wn in Fig. 9 during the DB life cycle. F or case 1, the TD W v arying from the w eak est to the strongest stage during the DB life cycle, shortens the life p erio d of the D B and w eakens the in tensit y of the DB. T ests c hanging slop es of the ( u − t ) pro file indicate that v aria tion rate of the bac kground w esterlies during the DB episo de do es not act as an influence on the DB ev olution (F igures omitted). This result implies that the time-dep enden t w esterlies v arying accordantly b efore and after the DB reaching its p eak phase play a similar role in t he DB ev olution. F or the TD W reflecting interaction b et we en blo cking and bac kground flo w in case 2, the ev olution of DB has b een considerably impacted. The most no t iceable feature is that the time of the DB onset and deca y lags tha t without TD W term, either for the high (Fig. 10a) o r for t he 19 FIG. 9: Diffe r en t t yp es of TBW profiles durin g the DB life cycle. The x-axis is time t in DB life episo de and y-axis denotes u . Case1 rep resen ts u = 0 . 5 − 0 . 3 cos (0 . 26 t ), Case2 denotes u = 0 . 5 − 0 . 3 cos (0 . 52( t − 6)), Case3 denotes u = 0 . 5 − 0 . 3 cos (0 . 37( t − 6)). lo w cen ter (Fig. 10b) in the dip ole structure. The deca y pro cess from the p eak phase of DB to its disapp earance is longer than the DB dev eloping pro cess from the onset of D B to its p eak phase, r esulting in t he asymmetric evolution of the DB during its life episo de. In addition, the inte nsity of the DB high/low cen ters at its p eak phase strengthens sligh tly . Ho w eve r , the intens ity of D B dep ends o n the slop e of the profile. F or example, when the slop e is not so sharp (case 3 in Fig. 9) as in case 2, the high (lo w) p eak drops down (go es up) indicating the w eake ning o f the DB (dot- dashed line in Fig. 10), but the lag and asymmetric c haracteristics are not changed. The detailed ev olutions o f D B impacted b y TDW are presen ted in Fig. 1 1. F or case 1 (Fig. 11a), the DB ev olution pro cess is similar to that without the TD W term in Fig. 3b, exhibiting a symmetric feature ab out the p eak phase in the ev olution o f the DB episo de. The most differences b et w een F ig . 3 b and Fig. 11a are in the differen t in tensity o f DB a t an y phase during the DB lif e cycle and zonal and meridional scales. The high/lo w in tensit y w eak ens in case 1 and the zonal and meridional scales of DB all less ens a little, suggesting the DB p erio d shortens, whic h is coheren t with the results discussed ab o ve from Fig. 10. F or case 2 the dev eloping pro cess from the onset ( t = 3) of DB to its p eak phase ( t = 6) resem bles that in case 1, but the decay pro cess from the DB p eak phase t o blo cking v anishing ( t = 10) app ears ve ry different. Compared to the DB ev olution pro cess without TDW (Fig. 3b), 20 (a) (b) FIG. 10: T he intensit y of the DB h igh (a) and lo w (b) cente r v arying with resp ect to time t under differen t types of TBW as shown in Fig. 9 du ring the DB life cycle. Lab el ”Ctrl” means the case of b ac kground w esterly u without time-dep endent term a 3 ( t ) = 0. the dip ole cen ter of the blo ck ing in case 2 is stronger than that in Fig. 3b at the b eginning deca y stage ( t = 7 , 8). A t t = 9 the dip ole-like pattern with closed high/low cen ters still main tains although the streamlines b ecome sparser in case 2, while it disapp ears and only sho ws a dip ole-lik e env elop in Fig. 3b. At t = 10 the almo st straig h t streamlines in F ig. 3b are instead of dip ole-like env elop in Fig. 11b with sparser streamlines. Ab o ve all, the effect of the TD W on evolution of DB is o wing to altering the DB life p erio d and leading to the asymmetry of the DB life cycle ev olution with resp ect to its p eak phase. IV. OBSER V A TIO NAL FEA T URES O F DB IN THE NO R THERN HE MISPHERE A. Data and the DB definition The data source in this study is the NCEP-NCAR reanalysis data from Jan uary 1958 to Decem b er 1997 , whic h uses a state-of-the-a rt global data assimilation system (Kalnay et al. 1996). The v ariables used in t his study are daily mean geop oten tial heigh t at 1000 and 500 21 FIG. 11: T he stream function patterns of DB evol u tion und er differen t time-dep en d en t westerly profiles app eared in Fig. 9. (a) Case 1; (b) Case 2. CI=0.4. hP a and the w est-east wind comp onent at 500 hPa, on a 2 . 5 ◦ latitude × 2 . 5 ◦ longitude grid, from 20 ◦ N to 90 ◦ N around the Northern Hemisphe re. Although t he blo c king phenomenon is well known in the meteorological communit y , there is no generally accepted definition of a blo c king eve nt (Lejen¨ and ∅ kland 1983 ) . Commonly used definitions can b e divided in to four categories of metho ds to iden tif y blo c king. The first is a sub jective tec hnique first put forw ard b y Rex (19 5 0a,b) in the 1950s. This was then inherited and mo dified by Sumner (1954 ) and White and Clark (1975). They iden tified the blo c king ev en ts sub jectiv ely b y visual insp ection a nd then used semiob jective criteria to determine the exact dates of initiation and duration of blo c king based on examination of the daily weather c ha rts. The second category uses ob jectiv e criteria first used by Elliott and Smith (1949) based on mag nitude and p ersistence of pressure a nomalies. Later, Hartman and Ghan (198 0), Dole and Gor do n (1 9 83), D ole (1986 , 1989), Sh ukla and Mo (1983), Huang (2002), and Huang et al. (2002b) extended the analysis from sea leve l pressure departure to the o ccurrence of p ersisten t p ositiv e geop otential height ano malies at the upp er lev el. The 22 useful ob jectiv e criteria fo r iden tif ying blo ck ing ev en ts w ere designed b y L ejen¨ and ∅ kland 1983, using t he north- south geop ot ential heigh t gradient based on the coherence b etw een the o ccurrence of p ersisten t anomalous mid-latitude easterly flo w and blo c king. The adv a n tage of this ob jectiv e metho d is its simplicit y f o r automat ic calculation, a nd it has therefore b een used widely (Tibaldi a nd Molteni 1990 ; DAndrea et al. 1998; Huang et al. 200 2a). Recen tly , P elly and Hoskins (2003a,b) constructed a new dynamical blo c king index using a meridional θ difference on a p oten tia l v orticit y ( PV) surface. They reve a l that their PV- θ index is b etter able to detect Ω blo cking tha n con ven tional height field indices. Since w e are fo cusing on dip ole-ty p e blo c king, the new est PV- θ index is not suitable for us in this study . All o f the definitions of blo c king mentioned ab o ve do not wholly address DB, so new criteria for iden tifying DB hav e to b e established. In this study the DB is distinguished b y the following criteria. 1) A t least on pair of closed high/lo w con tours app ears sim ultaneously at 4 (2.5 ) geop o- ten tial decameters (dam) con tour inte rv al at 500 hPa ( 1 000 hPa), with the distance b et w een the high a nd lo w cen ters is no larger than 30 longitudes. 2) The we sterlies m ust split into tw o branc hes a t 500 hP a , and the distance b etw een the div aricatio n p o in t and the meeting p oin t is no less than 45 longitudes. 3) The closed high or lo w cen ter la sts at least 5 da ys. 4) The mo ving sp eed of the blo cking do es not exceed ten degrees of longit ude p er da y . 5) The high center is lo cated at least north of 40 ◦ N . 6) The blo cking index (BI) is no less than 20 ms − 1 . The BI is calculated a s B I = U W M − 3 × U E M ms − 1 , where U E M is the maxim um of the 20- p oint running mean of geostrophic easterlies b ounded by 20 degrees of longitude in the east and we st of the high cen ter, resp ective ly , and 30 degrees of lat it ude south o f the high cen t er; U W M is the maxim um of the w esterlies south o f the U E M p osition b ounded by 30 degrees of latitude. Because the easterlies south of the blo c king an ticyclone are usually ab out 3 times less than the w esterlies, 3 times U E M is calcu la ted in BI. The geostrophic wind U is calculated as U = − ( g /f ) ∂ z / ∂ y ( f is the Coriolis parameter, g is the gra vitat io nal acceleration, and z is the geop otential height at 500 hPa). The latit ude a nd longitude of the high cen ter denote the p osition of the high, whic h is pr escrib ed similarly to that of Do le (1 986) as p ersisten t p ositiv e anomalies (with resp ect to latitude mean) greater tha n 70 gpm at 50 0 hP a. The n um b er of times that dip ole-t yp e blo c king app ear s is coun ted and the latitude where the 23 high cen ter of a dip ole blo ck is lo cated is regarded as the meridional p osition of the DB. B. Climatological features of DB and p ossible reasons Man y studies hav e deriv ed a comprehensiv e set o f climatological statistical c har a cter- istics of blo ck ing anticy clones using sub j ectiv e or ob jectiv e techn iques, including lo cat io n, frequency , duration, in t ensity , size, and distribution (Elliott and Smith 1949 ; Rex 1950a,b; White and Clark 1975; Lup o and Smith 19 95a; L ejen¨ and ∅ kland 1983; Huang et al. 2002a; Huang 2 0 02). Preferred Northern Hemisphere lo cations for blo c king are the northeastern b oundaries of the P acific and Atlan tic Oceans and the Ural area (Elliott and Smith 1949; Rex 1950b; Lejen¨ and ∅ kland 1983; D ole and Gor don 1983; Shuk la and Mo 1 9 83; Huang 2002). F or the statistical characteris tics of DB, only Luo and Ji (1991) p erfor med an ob- serv ational study of dip ole-type blo c king in the atmosphere during 1 969-84 . How eve r, the time series a re to o short to include the in terdecadal v ariability of blo c king (Huang et al. 2002b). The av ailabilit y of t he NCEP-NCAR reanalysis data (Kalnay et al. 1996) includes a considerably long time series, and so pro vides a solid basis f o r understanding b ehavior of the atmospheric midlatitude blo cking. In this pap er, statistical features of DB in the Northern Hemisphere are inv estigated based on a 40-yr p erio d da t a spanning f r om 1958C97. Results show that there are t hr ee preferable regions: t he north wes tern P a cific, the A tlantic, and the Ural area, where DB frequen tly o ccurred, agreemen t with the results of Luo and Ji (19 91). The P acific is the most preferable region, where a to tal of 1982 da ys of DB o ccurred, far more than the A tlantic ( 848 da ys) and Ural areas (533 da ys). It is interes ting to note that the pap ers b y Elliott and Smith ( 1 949) and Rex (19 5 0b), whic h w ere published within a ye a r of eac h other, gav e contradictory results o n the relativ e frequency of blo ck ing in the Atlan tic and P acific. White and Clark (1975), ho wev er, obtained a result that w a s not in agreemen t with Rex (1950b). In the r esearch of Elliott and Smith ( 1949), they found that the total n um b er of blo c king da ys in cen t r a l P acific is 4 times of tha t in the northeastern A tlantic. Notice that the previous results did not distinguish the D B or monop o le blo c king, Elliott and Smiths (19 49) results might include more DBs, while the statistics of Rex (19 5 0b) ma y include more mo no p ole blo c ks since they used differen t dat a in differen t p erio ds of time o r their definitions of blo c king ma y not b e exactly the same. 24 FIG. 12: T he latitudinal distribution of DB o v er three fa v orable regions. The la titudinal distribution of DB ov er the three f av orable regio ns is display ed in Fig . 12. It is shown tha t the latitude of D B ov er the P acific is mo stly concen trated at 65 ◦ N ∼ 80 ◦ N , while the D B o v er A tlantic fo cuses at 55 ◦ N ∼ 65 ◦ N , ab out 10 ◦ south w ard from that in the Pacific. The Ural area has the least D B, mainly at 60 ◦ N ∼ 70 ◦ N , and it is also more south w ard than that o v er the Pacific . F rom the statistics ab o ve it is f o und that the Pacific is the most fav o rable place for DB o ccurrence, but few er blo ck ing ev ents (include DB and Ω blo ck ing) o ccurred ov er the P acific compared to the Atlan tic and Ural a reas (see Hua ng 2002). Why DB seems to prefer the P acific? This is really a puzzling question. According to t he ana lysis in section I I I, we know the in tensit y of the bac kground mean w esterlies is associated with the D B dev elopmen t. Therefore, the observ at io nal we sterlies are examined first. Figure 13 displa ys the climatological seasonal cycle of the w esterlies av eraged in the high latitudes (55 ◦ − 8 0 ◦ N ) for the Nor thern Hemispheric DB preferable latitudes at 500 hP a and annual mean w esterly profiles with resp ect to latitudes av eraged ov er three fa vorable regions fo r DB o ccurrence: the Pacific (120 ◦ E − 15 0 ◦ W ), the A tla ntic (60 ◦ W − 30 ◦ E ) and the Ural ( 3 0 ◦ E − 90 ◦ E ) areas, resp ectiv ely . It is noticeable t ha t the w esterlies where the high cen ter of DB lo cates preferably o v er the high latitudes of the Pacific region are almost all less than that ov er the Atlan tic a nd Ural areas (Fig. 13a). The climatological ann ual mean we sterly o v er the P acific is ab out 3 . 25 m/s , not reac hing half of that o ve r the A tla ntic (7 . 35 m/s ) or Ural areas (7 . 77 m/s ). Hence it is not strange that many more DBs o ccurred o ver the P a cific than o ver the Atlan tic a nd Ural regions. F rom a climato lo gical p oin t of view, the weak w esterlies are 25 FIG. 13: Climatological s easonal cycle of w esterlies (a) and weste r ly profiles with resp ect to lati- tudes (b) a v eraged o v er P acific (P AC or WP A C, 120 ◦ E ∼ 150 ◦ W ), A tlant ic (A TL, 60 ◦ W ∼ 30 ◦ E ) and Ural (URL, 30 ◦ E ∼ 90 ◦ E ) areas. Label ”EP A C” denotes the eastern P acific a v eraged fr om 180 ◦ to 90 ◦ W , and ”GLB” is the mean of P A C, A TL and URL. one of the most imp ortant factors that impact the DB o ccurrence. Ho w eve r , the w eak we sterly is a precondition for onset of blo c king, including DB and monop ole blo c ks (Shutts 1983, 1986; Luo and Ji 1991; Luo 1994; Luo et al. 2001). The inconsistency b etw een DB (the most lo cated at north we stern P acific) and tot al blo ck ings (the least o ccurred at northeastern Pac ific) implies that the we a k we sterly is a necessary condition for blo c ks, but not a sufficie nt condition for DB onset. Since the w esterly shears pla y a considerably impo rtan t r o le in DB ev olution, the annual mean w esterly profiles with resp ect to latitudes av eraged o ver the three DB preferable regions are also studied (Fig. 13b). The figure rev eals that w esterly profile ov er the global DB regions app ears app ears to b e a w eak cyclonic wes terly shear from 40 ◦ N to 80 ◦ N , which is adv antageous for D B establishmen t at these latitudes. The Pacific has the strongest CWS from 4 0 ◦ N to 60 ◦ N and a considerably we a k A WS from 60 ◦ N to 80 ◦ N , whic h fav ors more D B pro duced at higher latitudes and less DB reduced at low er lat it udes, corresp o nding to the latitudinal distribution o f the DB shown in Fig. 12 . Observ ational studies also giv e t he seasonal v aria bilit y of globa l D B a s sho wn in Fig . 14a. The n umbers of DB decrease from winter (Decem b er t o F ebruary) to spring (March to Ma y), 26 summer (June to August), and autumn (September to Nov em b er) successiv ely . Win ter is the most fav orable season for D B pro ducing, consisten t with t he results for total blo c king b y White and Clark (1975), Lejen¨ and v ar nothing kland (198 3), Sh ukla and Mo (1983) , and Huang ( 2 002), but con tradictory to results by Rex (195 0a); while autumn is t he quiet season for D B o nset, whic h disagrees with the common statistical results for to t a l blo c king (White and Clark 1 9 75; Lejen¨ and v ar nothing kland 1983; Sh ukla and Mo 1983; Huang 2002). The mean w esterlies av eraged o ver the three preferable DB regions in eac h season (Fig. 14b) indicate that the autumn w esterlies are the strongest, and are asso ciat ed with the least DB o ccurring in autumn according to the theory discussed ab o ve . The summer w esterly is w eak est, but do es not correspond to the largest D B frequency in summer. Nev ertheless, the seasonal v ariability of the background w esterlies ov er the P a cific region (120 ◦ E ∼ 150 ◦ W ) in Fig. 14c app ears to hav e a completely negativ e correlation to the seasonal cycle of the global DB o ccurrence in F ig. 14a. This suggests that the seasonal v ariation of the globa l DB is mostly due to the seasonal cycle of the Pacific DB, whic h is mostly determined by the mean back g r o und we sterlies o v er the Pacific . F or the A tlantic and Ural DB, the seasonal v aria bilit y o f the mean w esterlies ma y pla y imp or tan t but indecisiv e role in the DB seasonal cycle. The seasonal wes terly pro files with resp ect to latitudes o ve r the P acific region of DB (Fig. 1 4 d) sho w that the w esterly shears are v ery fav orable for DB establishmen t in win ter, spring and autumn, with a CWS a t 45 ◦ − 60 ◦ N and ve ry w eak A WS at higher latitudes, while in summer the A WS at 60 ◦ − 75 ◦ N seems to o strong to b e unfav orable for D B pattern dev eloping. It is r emark able that among the v ery w eak A WSs in win ter, spring and autumn, the A WS in win ter is the relatively strongest one, whic h is fav orable for strong DB dev elopmen t, and results in the most DB o ccurring in win ter. Therefore, the in tensit y of the bac kground mean w esterlies and their shear structures closely asso ciate with the establishmen t of DB, leading to the north wes tern P acific b eing the most preferable region for DB and its seasonal v ariability . V. CONC LUDING REMARKS In this pap er, a v ariable co efficien t Kortewe g de V ries (V CKdV) system is deriv ed b y considering the time dep enden t bac kground flow and b oundary conditio ns fro m the non- linear, in viscid, no ndissipativ e, and equiv alent barotropic v orticity equation in a b eta-plane 27 FIG. 14: Seasonal v ariations of (a) th e num b er of DB d a ys, (b) the mean we sterlies a v eraged o ver the thr ee DB preferable regions, (c) weste r lies a v eraged o ve r the Pa cific (120 ◦ E ∼ 150 ◦ W ), and (d) seasonal mean w esterly pr ofiles with resp ect to latitude a verage d o ve r the Pac ifi c. Lab el ”Win” denotes winter season (Decem b er to F ebruary), ”Sum” denotes summer (June to August), ”Spr” denotes sp ring (Marc h to Ma y), and ”Aut” denotes autu mn (September to No vem b er). c hannel. The analytical solution obtained from the V CKdV equation can b e successfully used to explain the ev olution of atmospheric dip ole- t yp e blo c king life cycle. Analytical di- agnoses are analyzed and three factors that may influence the evolution of atmospheric DB are inv estigated. Theoretical results sho w that the bac kground mean w esterlies hav e great influence on ev olution o f DB during its lif e episode. W eak w esterlies are necessary for blo ck ing deve l- opmen t a nd the high/lo w cen ters o f blo ck ing decrease and mo v e south ward and w estw ar d, the horizon t a l scale enlarges and meridional span reduces, as w ell as the blo c king life p erio d shortens with resp ect to the enhanced w esterly . Shear of the back g round w esterlies also plays an imp ortant role in ev olution of DB. The CWS is preferable for the dev elopmen t o f DB, whic h agrees with the recen t result from Luo (2005b). When the cyclonic shear increases, the intens ity o f D B decreases and its life p erio d b ecomes shorter. W eak A WS is also fa v or a ble for DB fo rmation, but a critical threshold shear 28 exists, b ey ond whic h a cutoff anticyc lo ne from the North Pole region dev elops dramatically instead of a n en v elop e R ossb y soliton forming the hig h a n ticyclone cen ter of DB. Inside the critical threshold of t he an ticyclonic shear, the intens ity of D B increases a nd the life p erio d of D B pro longs when the anticyc lo nic shear increases. F or the role of CWS in D B ev olution, our result is inconsisten t with that of Luo (2005b), who rev ealed tha t an isolated v ortex pair blo c k excited resonan tly b y synoptic-scale eddies is more lik ely suppressed in an an t icyclonic shear en vironmen t. The difference ma y lie on differen t nonlinear systems considered b y Luo (2005b) a nd us. That is, Luo (2005b) considered an eddy-forced nonlinear Sc hr¨ odinger Rossb y soliton system in the atmosphere, while we established a nonlinear system based on a nonlinear KdV equation without for cing, and our results sho w that the no nlinear effect of the free atmosphere could also induce blo c king circulatio n by its in trinsic nonlinear in teraction without external eddy forcing. Time-dependent v ariation of bac kground flo w in the life cycle o f DB has some mo dulatio ns on blo cking life p erio d and intens ity due to the b eha vior of the mean wes terlies. The effect of TD W, esp ecially the st yle of TDW, reflecting interaction b et w een the bac kground we sterlies and blo c king, with t he we sterlies decreasing during the dev eloping p erio d b efore DB p eak and increasing throughout the deca y stage after D B p eak, is caused b y the alteration of the DB life p erio d and leads to the a symmetry of the D B life cycle ev olution. Statistical analysis of climatological features of observ ed DB is inv estigated based on 4 0-yr geop otential heigh t fields fro m the NCEP-NCAR reanalysis data during 1958-97. Observ a- tional results sho w that there are three preferable regions of DB in the Northern Hemisphere, lo cated in the northw estern P acific, the northeastern Atlan tic and the Ural Mountain areas, in a greemen t with the results of Luo and Ji (1991). The n um b er o f DB days o ccuring ov er the P acific is far larger than to tal of that o ver the other tw o regions, corresp onding to the w eak er w esterlies ov er the north wes tern P acific and the particular wes t erly shear structure o ve r P acific. The Pacific DB prefers to b e established at higher latitudes tha n the A tla n tic and Ural regions b y ab out 10 ◦ , whic h ma y b e caused by its havin g the strongest CWS at mid-latitudes and w eak er A WS at higher latitudes ov er t he P acific region. Seasonal v ariabil- it y of the global DB is also asso ciated with the seasonal cycle of the mean w esterlies ov er the P acific and the w esterly shear structure. Therefore, the intens ity of the mean w esterlies and the shear structure of the w esterly profiles are t w o imp ortant conditions a sso ciated with the climatological features of the DB in the Northern Hemisphere, whic h may pla y crucial role 29 in the DB life cycles. The type o f TD W could also impact the D B life episo de concluded from analytical resolution, but it is y et to b e demonstrated from the o bserv ational studies in the futur e. Ac kno wledgments The w ork w as supp orted by the Natio nal Natural Science F oundation of China (Grants 40305009 , 1047 5055, 1054124 , and 90203001), Program for New Cen tury Excellen t T alen ts in Unive rsity (NCET-05- 0591), Program 973 (No. 2005CB422301) , and Shanghai Pos t - Do ctora l F o undat ion. REFEREN CES Berggren, R., B. Bolin, and C. G. 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