Physics of the Earth and Planetary Interiors

Transcription

1 Physcs of the Eath and Planetay Inteos ( Contents lsts avalable at ScVese ScenceDect Physcs of the Eath and Planetay Inteos jounal homepage: The vscosty stuctue of the D laye of the Eath s mantle nfeed fom the analyss of Chandle wobble and tdal defomaton Masao Nakada a,, Chho Iguch b, Shun-cho Kaato c a Depatment of Eath and Planetay Scences, Faculty of Scence, Kyushu Unvesty, Fukuoka , Japan b Depatment of Eath and Planetay Scences, Gaduate School of Scences, Kyushu Unvesty, Fukuoka , Japan c Depatment of Geology and Geophyscs, Yale Unvesty, New Haven, CT 652, USA atcle nfo abstact Atcle hstoy: Receved 31 Mach 212 Receved n evsed fom 29 June 212 Accepted 2 July 212 Avalable onlne 14 July 212 Edted by Ke Hose Keywods: Chandle wobble Tdal defomaton D laye Coe mantle bounday Vscosty Maxwell body The vscosty stuctue of the D laye of the Eath s mantle s nfeed fom the decay tme of the Chandle wobble and sem-dunal to 18.6 yeas tdal defomatons combned wth model vscosty depth pofles coespondng to a ange of tempeatue depth models. We use two typcal tempeatue pofles of the D laye by consdeng ts dynamc state: ( bottom themal bounday laye of the mantle convecton (TBL model and ( vgoously small-scale convectng laye (CON model. Thee possble models ae deved fom the compason between the numecal and obsevatonally nfeed decay tmes of Chandle wobble and tdal defomaton. The fst and second models ae those wth a vscosty of 1 16 Pa s at the coe mantle bounday. The tempeatue gadent fo the fst one, TBL model wth a thckness of the D laye (Lof2 km, s nealy constant wthn the D laye. The second one, TBL and CON models wth L 3 km, eques that the tempeatue gadent of the lowe pat (1 km thckness s lage than that of the uppe pat. The tempeatue nceases wthn the D laye fo these two models ae lage than 15 K. The thd model has a constant low vscosty laye (1 km thckness and vscosty smalle than Pa s at the bottom of the D laye n TBL (L 2 and 3 km and CON (L 3 km models. The tempeatue nceases would be 1 16 K dependng on the vscosty at the top of the D laye ( Pa s. The heat flows fom the coe to the mantle fo these thee models ae estmated to be lage than 5 TW. The thd model may be pefeable afte compehensvely takng account of the ftness of the decay tme of the Chandle wobble and the tdal defomatons fo each model. Ó 212 Elseve B.V. All ghts eseved. 1. Intoducton D laye of the Eath s mantle, the lowemost laye n the Eath s mantle, plays an mpotant ole n the dynamcs and evoluton of the Eath. In patcula, ts heologcal popetes ae mpotant n dscussng a numbe of geodynamc pocesses, but t s dffcult to estmate ts vscosty stuctue based on commonly used methods, glacal sostatc adjustment (GIA due to the last deglacaton (Pelte and Andews, 1976 and flow models nfeed fom global long geod anomales (Hage, The GIA obsevatons fo the elatve sea level (RSL dung the postglacal phase have lttle senstvty to the vscosty of the mantle deepe than 12 km (Mtovca and Pelte, In the latte appoach, Eath s suface gavty sgnals ae used to estmate the vscous esponse to the lateal densty vaatons nsde the mantle nfeed fom sesmc tomogaphy (Hage, 1984; Hage et al., The appoach s, n Coespondng autho. Tel.: ; fax: E-mal addesses: mnakada@geo.kyushu-u.ac.jp (M. Nakada, 2SC11121Y@ s.kyushu-u.ac.jp (C. Iguch, shun-cho.kaato@yale.edu (S.-. Kaato. pncple, effectve n estmatng the vscosty stuctue fo the whole mantle, but the velocty-to-densty conveson facto s uncetan n the deep mantle such as the D laye whee the tempeatue senstvty of sesmc wave veloctes deceases due to hgh pessue, and also chemcal heteogenety may cause velocty heteogenety (Kaato and Kak, 21; Kaato, 28. Moe ecently, Nakada and Kaato (212 showed that the decay tme of the Chandle wobble and tdal defomaton wth typcal peods longe than.1 yea, whch ae elated to the defomaton n the deep mantle (Smth and Dahlen, 1981, can be ntepeted as vscoelastc esponses fo the Maxwell body (e.g., Pelte, 1974 and also povde new constants on the heologcal popetes of the D laye. Regadng the exctaton of the Chandle wobble, thee s gowng consensus that the souce s a combnaton of atmosphec, oceanc and hydologc pocesses (e.g., Goss, 27. Once the Chandle wobble s excted, the ampltude of Chandle wobble decays wth a decay tme of s CW ¼ 2Q CW T CW =2p n the absence of exctaton, n whch T CW and Q CW ae the peod and qualty facto of Chandle wobble, espectvely (e.g., Munk and Macdonald, 196; Smth and Dahlen, The values of /$ - see font matte Ó 212 Elseve B.V. All ghts eseved.

2 12 M. Nakada et al. / Physcs of the Eath and Planetay Inteos ( T CW and Q CW, ecently evewed and ecommended by Goss (27, ae: T CW of 433 ± 1.1 (1 sdeeal days and Q CW of 179 wth a 1 ange of estmated by Wlson and Vcente (199, coespondng to the decay tmes of 3 3 yeas. We befly summaze the esults by Nakada and Kaato (212. The decay tme of Chandle wobble povdes nfomaton on the effectve vscosty of the D laye wth the thckness of 3 km. Moeove, the vscosty of the bottom pat of 1 km thckness s constaned moe tghtly by usng the tdal defomatons acoss the sem-dunal to 18.6 yeas tdes as well. These defomatons combned wth the GIA constants by elatve sea level obsevatons suggest that the effectve vscosty of the D laye (3 km thckness s Pa s, and that fo the bottom pat of the D laye (1 km thckness s less than Pa s. The estmates by Nakada and Kaato (212 ae, howeve, based on smple one- o two-laye vscosty model. If we consde that the tempeatue gadent s lkely hgh n the D laye as nfeed fom the double-cossng of sesmc ays of the phase bounday between peovskte and post-peovskte (Henlund et al., 25 and the vscosty s hghly senstve to tempeatue (e.g., Kaato, 28, then t s mpotant to examne these defomaton pocesses based on the models wth tempeatue (T dependent vscosty stuctue. In examnng these defomaton pocesses based on a T-dependent vscosty model, we adopt two typcal tempeatue pofles by consdeng that the D laye s a bottom themal bounday laye of the mantle convecton (e.g., Tucotte and Schubet, 1982, o vgoous small-scale convecton occus n the D laye (Solomatov and Moes, 22. These numecal esults ae dscussed n Sectons 3 and 4, espectvely. The pesent esults, consequently, povde mpotant constants on both the depthdependent vscosty stuctue of the D laye, tempeatue of the coe mantle bounday ( (Boehle, 2; Alfè et al., 22 and heat flow fom the coe to mantle (e.g., Lay et al., 28. We dscuss these ponts n Secton Numecal method Hee we adopt the Maxwell vscoelastc Eath s model, and befly explan the method to estmate the decay tme of Chandle wobble and the esponse functon to tdal focng (Nakada and Kaato, 212. The densty and elastc constants ae based on the PREM (Dzewonsk and Andeson, 1981 model. The thckness of elastc lthosphee s 1 km and uppe mantle vscosty s 1 21 Pa s. The lowe mantle vscosty above the D laye s assumed to be 1 22 Pa s. Ths model s smla to a heologcal model explanng elatve sea level obsevatons fo the postglacal ebound aound the Austalan egon (Nakada and Lambeck, Howeve, the choce of the backgound model does not affect the conclusons on the vscosty of the lowemost laye of the mantle so much. The Chandle wobble s smulated by a lneazed Louvlle equaton descbng the pola moton m ¼ m 1 þ m 2 ( m <<1 usng the Maxwell model, whee the quanttes m 1 and m 2 descbe the dsplacement of the otaton axs n the dectons and 9 E, espectvely (e.g., Sabadn and Pelte, 1981; Wu and Pelte, 1984: _ mðtþ þ mðtþ¼ 1 C A ðdðtþþkl ðtþþ! DIðtÞ D_ IðtÞ X þ kt ðtþ mðtþ k f whee astesk ( denotes convoluton, d(t s the delta functon, X s the mean angula velocty of the Eath, ¼ðC AÞX=A, DI ¼ DI 13 ðtþþdi 23 ðtþ, A and C ae the equatoal and pola moments of neta, espectvely. DI 13 (t and DI 23 (t ae focng neta elements fo the pola moton. Love numbes k L (t and k T (t n Eq. ð1þ (1 depend on the densty and vscoelastc stuctue of the Eath, and chaacteze the tme-dependent Eath defomaton to suface loadng and that to the potental petubaton, espectvely (Pelte, k f s the flud Love numbe chaactezng the hydostatc state of the Eath and s defned by 3GðC AÞ=ða 5 X 2 Þ, whee G s the gavtatonal constant (Munk and MacDonald, 196. We numecally smulate the Chandle wobble excted by the pulse-lke focng functon of DI 13 (t and DI 23 (t and estmate ts decay tme (Nakada, 29; Nakada and Kaato, 212. The defomaton by the lun-sola tdal foce s senstve to the anelastc popetes of deep mantle (Smth and Dahlen, 1981, and the esponses to the focngs wth peods longe than.1 yea can be examned based on the Maxwell vscoelastc model Nakada and Kaato, 212. Hee we do not consde the effects of the coe mantle couplng such as electomagnetc couplng (Buffett et al., 22. Then, the Eath s esponse R(x,t to a peodc focng of Fðx; tþ /e xt wth fequency x s gven by Rðx; tþ ¼k T ðtþfðx; tþ usng the tdal Love numbe k T (t, ðxþfðx; tþ (Lambeck and Nakboglu, 1983; Sabadn et al., The Love numbe, k T;P ðxþ, takes a complex fom of k T;P ðxþ ¼k T;P ðxþþk T;P ðxþ, and depends on the vscoelastc stuctue, patculaly on the vscosty stuctue of the deep mantle. The ampltude of the esponse s chaactezed by ts modulus, jk T;P j, and the phase dffeence between the esponse and focng, D/, s gven by D/ ¼ tan 1 Þ. That s, we can estmate the Love numbe as a functon of fequency by analyzng the tme sees of geodetc obsevatons, R(x,t. In ode to dscuss the decay tme of the Chandle wobble and tdal defomatons, we adopt the T-dependent vscosty stuctue, g(z, fo the D laye gven by: ð k T;P =k T;P gðzþ ¼g expðh =RTÞ whee z s the depth and H s the actvaton enthalpy (e.g., Kaato, 28. The depth of the top of the D laye s z top and that fo the bottom s gven by z, and theefoe the thckness of the D laye, L, s L = z z top. In the PREM model wth z = 2891 km, L = 3 km fo z top = 2591 km. Hee we put g(z top =g top and T(z top =T top. Then we get g = g top exp( H /RT top, and Eq. (2 s consequently expessed as: gðzþ ¼g top exp H 1 1 ð3þ R T top TðzÞ Hee we put T(z=T top + DT(z, then Eq. (3 takes a fom of: gðzþ ¼g top exp H DTðzÞ=T top RT top 1 þ DTðzÞ=T top Eq. (4 s used to estmate the depth-dependent vscosty stuctue, whch s a functon of g top, H /RT top and DT/T top. Also we denote the vscosty and tempeatue ncease at the base of the D laye as gðz Þ¼g and DTðz Þ¼DT, then we get: g ¼ g top exp H RT top Eq.(5 s also wtten as: In g top ¼ g H RT top DT =T top 1 þ DT =T top ð2þ ð4þ ð5þ DT =T top 1 þ DT =T top ð6þ The elatonshp between H =RT top and DT =T top gven by Eq. (6 s used to dscuss the tempeatue at the n Secton 5.

3 M. Nakada et al. / Physcs of the Eath and Planetay Inteos ( Results fo a bottom themal bounday laye model 3.1. Settng of paamete values The vscosty stuctue of the D laye s deved fom Eq. (4 by gvng values of g top, T top, H and depth-dependent DT(z. Fstly, the tempeatue gadent, dt/dz, s assumed to be constant fo the whole D laye,.e., dt/dz = a (constant. Then we get the vscosty stuctue fo DT(z=a(z z top at an abtay depth. In ths study, howeve, we examne the decay tme of the Chandle wobble and tdal defomatons (two data sets as a functon of vscosty at the base of D laye g, usng a elatonshp of Eq. (6 that the tempeatue gadent s detemned fo gven values of g top, T top, H and g. Fo numecal calculatons, we dvde the D laye nto a numbe of sub-layes wth constant thckness and vscosty, and the vscosty of each sub-laye s fxed to the value at the mddepth. Accodng to numecal expements wth the thckness of 25 and 5 km, dffeences of the pedctons fo two data sets ae less 1% fo a model wth 3 km thckness of D laye (L egadless of the values of g. In cases of L = 2 and 25 km, we also get suffcently accuate pedctons fo a model wth 25 km thckness, and theefoe the sub-laye thckness s fxed to 25 km. The vscosty pofles fo TBL1 model (see Table 1 ae shown n Fg. 1 as a functon of vscosty at the base of the D laye g fo a model wth H = 5 kj mol 1, T top = 26 K, g top =1 22 Pa s and L = 3 km, and Fg. 2 shows the pedcted decay tmes of the Chandle wobble fo each model. The tempeatue above the D laye s assumed to be adabatc. The value of H coesponds to that by Yamazak and Kaato (21 and the uncetanty s 1 kj mol 1. The depth-dependent tempeatue dstbuton fo a specfc valueg, fo example, g ¼ 2 Pa s, s affected by the uncetantes of H and T top, whch may sgnfcantly affect the pedctons fo two data sets. Hee we have examned two data sets based on the vscosty models wth H = 4, 5 and 6 kj mol 1 and T top = 26, 29 and 32 K, n whch g s fxed to a specfc value. Although we do not show the esults hee, the dffeences ae neglgbly small and those fo the decay tme of Chandle wobble ae seveal yeas at most. We theefoe show the pedctons fo H = 5 kj mol 1 and T top = 26 K. In ths study, we adopt g top =1 21 and 1 22 Pa s. The esults fo g top >1 22 Pa s ae nfeed fom those fo g top =1 22 Pa s. On the othe hand, the conclusons fo the vscosty stuctue of the D laye wth g top 1 2 Pa s ae essentally the same as those fo a unfom one-laye model adopted by Nakada and Kaato (212, Table 1 Tempeatue and vscosty stuctues of the D laye fo a bottom themal bounday laye model (TBL model. The bottom of the D laye s 2891 km depth and the thckness of the lowe laye s 1 km. The tempeatue gadents fo the uppe and lowe layes ae denoted by (dt/dz u (constant and (dt/dz l (constant, espectvely. In these models, the lthosphec (elastc thckness s 1 km, and the uppe and lowe mantle vscostes except fo the D laye ae 1 21 and 1 22 Pa s, espectvely. The vscosty of km depth s 1 21 Pa s fo TBL3 model. Model name Thckness of the D laye (km Vscosty at the top of the D laye (g top (Pa s Vscosty of the lowe laye (1 km thckness of the D laye Tempeatue gadent stuctue TBL vaables (dt/dz u =(dt/dz l TBL vaables (dt/dz u =(dt/dz l TBL vaables (dt/dz u =(dt/dz l TBL vaables (dt/dz u =(dt/dz l TBL vaables (dt/dz u =(dt/dz l TBL1a Pa s at the bottom (dt/dz u <(dt/dz l TBL1b Pa s at the bottom (dt/dz u <(dt/dz l TBL2a Pa s at the bottom (dt/dz u <(dt/dz l TBL2b Pa s at the bottom (dt/dz u <(dt/dz l TBL1c Pa s fo km depth (dt/dz u,(dt/dz l = TBL1d Pa s fo km depth (dt/dz u,(dt/dz l = TBL1e Pa s fo km depth (dt/dz u,(dt/dz l = TBL1f Pa s fo km depth (dt/dz u,(dt/dz l = TBL2c Pa s fo km depth (dt/dz u,(dt/dz l = TBL2d Pa s fo km depth (dt/dz u,(dt/dz l = TBL4c Pa s fo km depth (dt/dz u,(dt/dz l = TBL4d Pa s fo km depth (dt/dz u,(dt/dz l = TBL4f Pa s fo km depth (dt/dz u,(dt/dz l = TBL5c Pa s fo km depth (dt/dz u,(dt/dz l = TBL5d Pa s fo km depth (dt/dz u,(dt/dz l = (a TBL (b TBL2 Depth (km Depth (km Vscosty (Pa s Vscosty (Pa s Fg. 1. Vscosty pofles fo the D laye of TBL1 and TBL2 models wth H = 5 kj mol 1 and T top = 26 K.

4 14 M. Nakada et al. / Physcs of the Eath and Planetay Inteos ( Decay tme of the Chandle wobble (yea Q =179 CW TBL1 TBL2 TBL3 mtbl3 TBL4 CON1 CON2 Q =789 CW Q =74 CW Bottom vscosty of D" laye, (Pa s Fg. 2. Decay tmes of the Chandle wobble fo seveal vscosty models as a functon of the bottom vscosty of the D laye g and the estmates fo 74 6 Q CW by Wlson and Vcente (199. The paamete values fo each model ae shown n Tables 1 and 2. In mtbl3 model, the vscosty fo km depth ange s 1 23 Pa s and the vscosty stuctue except fo ths depth ange s the same as that fo TBL3. whch appoxmately coesponds to a convectng D laye model (see Secton 4 wth vey thn uppe and lowe themal bounday layes. In a model of TBL2 wth g top =1 21 Pa s (Table 1 and Fg. 1a fo the vscosty stuctue of the D laye, the vscosty at z = z top s dscontnuous. To examne the effect of ths jump on the decay tme, we compute the decay tmes of the Chandle wobble fo a model wth the lowe mantle vscosty of 1 22 Pa s fo 67 6 z < 2291 km depth and 1 21 Pa s fo z < 2591 km depth (TBL3 n Table 1. The dffeence n the decay tmes between models TBL2 and TBL3 s detected fo g > Pa s (Fg. 2. Howeve, the pemssble vscosty ange fo obsevatonally nfeed decay tmes of 3 3 yeas (Wlson and Vcente, 199 s smla fo the vscosty stuctues TBL2 and TBL3. We theefoe adopt TBL2 n the case of g top =1 21 Pa s. Hee we shotly comment about the effect of a non-unfom lowe mantle vscosty pofle wth vscosty 1 23 Pa s aound 18 km depth (e.g., Mtovca and Fote, 24 on the decay tme of the Chandle wobble. We adopt a vscosty model of mtbl3, n whch the vscosty fo z < 291 km depth s 1 23 Pa s and the vscosty stuctue except fo ths depth ange s the same as that fo TBL3. The dffeence n the decay tmes between mtbl3 and TBL3 models shown n Fg. 2 s neglgbly small, suggestng that such vscosty statfcaton of the lowe mantle does not alte the esults pesented hee (see also Nakada and Kaato ( Infeence of the D vscosty stuctue based on a constant tempeatue gadent Fg. 2 shows the decay tmes of the Chandle wobble (s CW fo models TBL1 (g top =1 22 Pa s and TBL2 (g top =1 21 Pa s wth the D laye of 3 km thckness, n whch the egon wth vscosty lage than Pa s nsgnfcantly affects the decay of the Chandle wobble (Nakada and Kaato, 212. Although we do not show hee, the decay tmes fo g < 1 16 Pa s ae shote than 3 yeas. The decay tmes fo obsevatonally nfeed Q CW -value by Wlson and Vcente (199 ae 3 3 yeas wth the optmum values of Q CW = 179 and 68 yeas. Fo models TBL1 and TBL2, pemssble values of g satsfyng the decay tmes of 3 3 yeas ae 3 6 g Pa s and 4 6 g < Pa s, and g -values fo s CW = 68 yeas ae 8 and 1 19 Pa s, espectvely. These esults ndcate that the decay tmes ae less senstve to the g top -value adopted hee. To examne the effect of thckness of the D laye (L on the decay tme, we have evaluated the decay tmes fo L = 3, 25 and fo the Chandle wobble s fo Dckman and Nam (1998. The mddle and ght estmates n k T;P (left and ght ones n k T;P ae based on the ognal data fo Vcente and Wlson (1997 and Fuuya and Chao (1996, and the Q CW -values used fo the estmates of k T;P ae 179 and 49, espectvely. The Eath s esponses shown n Fg. 3 ndcate jk T;P jk T;P and the eal pat s domnated by the elastc esponse, mplyng that the eal pat, k T;P, descbes the ampltude esponse and the phase lag, D/, s mostly detemned by the magnay pat, k T;P. Although we plot geodetcally nfeed magnay pats fo the decay tmes by Vcente and Wlson (1997 (left estmate n Fg. 3 and Fuuya and Chao (1996 (ght estmate n Fg. 3, we use the decay tme 2 km. The thcknesses of 3 km and 2 km may coespond to cold and hot mantle by Henlund et al. (25, espectvely. The decay tmes fo models TBL4 wth L = 2 km ae shown n Fg. 2, and pemssble values of g satsfyng s CW of 3 3 yeas natually decease wth deceasng the thckness (Nakada and Kaato, 212. Consequently, TBL4 model wth g 1 16 Pa s pedcts the decay tme of 3 yeas (also TBL5 model shown n Fg. 6b, coespondng to the obsevatonally nfeed mnmum estmate. Fo these pedctons, we should note that the functonal type of decay tme as a functon of g takes a smla fom of paabola as ndcated by Nakada and Kaato (212. Ths eflects that the decay of the Chandle wobble s manly detemned by the uppe pat: when the lowe pat wth a vscosty smalle than Pa s behaves as an nvscd laye to the defomaton fo the Chandle wobble (see also Fg. 8 fo a smple two-laye vscosty model by Nakada and Kaato (212. We next dscuss the tdal esponses descbed by the eal pat, k T;P, and the magnay pat, k T;P, of the Love numbes. The Love numbes examned hee ae geodetcally nfeed Love numbes fo sem-dunal (M 2 (Ray et al., 21, nne-day (M 9 (Dckman and Nam, 1998, fotnghtly (M f (Dckman and Nam, 1998; Benjamn et al., 26, monthly (M m (Dckman and Nam, 1998; Benjamn et al., 26, 18.6 yeas tde (Benjamn et al., 26 and Chandle wobble coected fo the ocean effects (Dckman and Nam, 1998; Benjamn et al., 26 (Fg. 3. The estmates fo Chandle wobble coespond to the esponse to the accompanyng vaatons n centfugal foce (e.g., Benjamn et al., 26. The estmates fo 18.6 yeas tde (Benjamn et al., 26 ae deved fom the degee-two and ode-zeo gavty component fo satellte lase angng fom 1979 to 24. In these fgues, geodetcally nfeed Love numbes fo 18.6 yeas tde ae shown afte the coecton fo seveal factos (left: afte atmosphec effect coecton, mddle: afte atmosphec and oceanc cculaton effect coecton, ght: afte the coecton used fo the mddle estmate and contnental wate + snow + ce effect coecton (Benjamn et al., 26. These estmates fo 18.6 yeas tde may ndcate that the magnay pat s hghly senstve to the coecton factos, but not fo the eal. The left estmate n k T;P fo the ampltude esponses (Fg. 3a and c n dscussng the vscosty stuctue fom the Chandle wobble. The Love numbes fo TBL1 and TBL2 ae shown n Fg. 3. The by Wlson and Vcente (199 (Fg. 2 pedctons of k T;P fo both models (Fg. 3a and b ae nealy dentcal fo peods less than 1 yea, and the magntude of 18.6 yeas tde fo TBL2 (g top =1 21 Pa s s only slghtly lage than those fo TBL1,.e., at most.4 fo k T;P. Howeve, the numecal expements ndcate that the esponse ampltude at peods of 18.6 yeas fo TBL2 wth a cetan value of g s nealy the same as that fo TBL1 model wth g =2. Fo example, we get k T;P (g ¼ Pa s fo TBL2 k T;P (g =5 Pa s fo TBL1 (g = Pa s fo TBL2 k T;P (g =5 Pa s fo TBL1. We theefoe dscuss the vscosty stuctue of the D laye based on the Love numbes fo vscosty models wth g top =1 22 Pa s.

5 M. Nakada et al. / Physcs of the Eath and Planetay Inteos ( Real pat of tdal esponse (k Real pat of tdal esponse (k M2 data MfMm M9 { { CW (a { 18.6 {.3 TBL1 (dashed lnes TBL2 (sold lnes.28 Peod (yea (c TBL1 (dashed lnes TBL4 (sold lnes.28 Peod (yea Imagnay pat of tdal esponse (k Imagnay pat of tdal esponse (k.1 (b TBL1 & TBL Peod (yea.1 (d TBL1 & TBL Peod (yea Fg. 3. Real (a, c and magnay (b, d pats of tdal esponses fo TBL1, TBL2 and TBL4 models and geodetcally nfeed estmates fo sem-dunal tde (M 2 (Ray et al., 21, nne-day tde (M 9 (Dckman and Nam, 1998, fotnghtly tde (M f (Dckman and Nam, 1998; Benjamn et al., 26, monthly tde (M m (Dckman and Nam, 1998; Benjamn et al., 26, Chandle wobble (Dckman and Nam, 1998; Benjamn et al., 26 and 18.6 yeas tde (Benjamn et al., 26 as a functon of the peod. To clealy show each estmate, we plot the data wth appopate shft of poston of the peod. The estmates fo 18.6 yeas tde wee deved fom the degee two (n = 2 and ode zeo (m = gavty component fo satellte lase angng (SLR fom 1979 to 24 (26 yeas. These data nclude the effects assocated wth atmosphec, oceanc and hydologc pocesses. The left, mddle and ght estmates ae coected fo atmosphec effects, atmosphec and ocean cculaton effects, and atmosphec, ocean cculaton and contnental wate + snow + ce effects, espectvely (Benjamn et al., 26. In the estmates fo the Chandle wobble, the left estmate n k T;P s fo Dckman and Nam (1998. The mddle and ght estmates n k T;P used fo the estmates of k T;P (left and ght ones n k T;P ae 179 and 49, espectvely. ae based on the ognal data fo Vcente and Wlson (1997 and Fuuya and Chao (1996, and the Q CW -values Fg. 3c and d show the esponses fo TBL1 and TBL4, n whch the thcknesses of the D laye (L ae 3 and 2 km, espectvely. We do not show the esults fo 25 km thckness because the esults ae ntemedate between these pedctons. The esults fo k T;P ae sgnfcantly senstve to ts thckness, and the magntude at peods 1.2 and 18.6 yeas fo TBL4 s.1 and.15 smalle than that fo TBL1, espectvely. The magntude of k T;P fo TBL4 s also smalle than that fo TBL1. That s, the vscoelastc Eath s esponses fo peods examned hee ae appoxmately popotonal to the thckness of vscoelastc D laye fo vscosty models wth an dentcal vscosty (g, and the magntude fo TBL4 s 2/3 fo that of TBL1. Ths elatonshp s also tue fo the pedctons of k T;P between TBL2 and TBL5 wth g top =1 21 Pa s (see Table 1. As dscussed by Nakada and Kaato (212, the defomatons fo peods longe than.1 yea ae ntepeted as the vscoelastc esponses fo the Maxwell model. That s, the esponses fo the Chandle wobble and 18.6 yeas tde would be explaned by the e-, fo TBL1 and TBL2 models, the pemssble ange of g fo 18.6 yeas tde s g 6 Pa s, and that fo the Chandle wobble s g Pa s (Fg. 3a. The g -value of Pa s satsfyng these defomatons s, howeve, sgnfcantly smalle than the vscosty nfeed fom the decay tmes as shown n Fg. 2. Consequently, Nakada and Kaato (212 has poposed a low sponses examned hee. In the eal pat of Love numbe, k T;P vscosty zone wth Pa s and 1 km thckness at the base of the D laye to esolve the dscepancy fo two data sets. On the othe hand, the esponses fo k T;P at peods longe than.1 yea can be explaned by TBL4 model wth g 1 16 Pa s, whch also pedcts the decay tme of 3 yeas fo the mnmum estmate by Wlson and Vcente (199. Howeve, ths model cannot explan the magnay pat. Ths s tue fo a vscosty model of TBL5 wth g top =1 21 Pa s. We wll dscuss geophyscal mplcatons fo these models n Secton 5. In the next secton, we examne two data sets based on the vscosty models wth depth-dependent tempeatue gadents fo the D laye, and also examne whethe a low vscosty zone at the vey bottom of the D laye poposed by Nakada and Kaato (212 s equed n explanng both data sets smultaneously Infeence of the D vscosty fo models wth depth-dependent tempeatue gadents and constant low vscosty n the lowe pat of the D laye In ths secton, we examne two cases, ( dffeent tempeatue gadents fo the uppe and lowe pats n the D laye, and ( ncluson of a constant low vscosty laye at the bottom of the D laye. We fst examne case ( fo the D laye wth 3 km thckness only because the depth-dependent tempeatue gadent s not equed fo models wth 2 km thckness as nfeed fom

6 16 M. Nakada et al. / Physcs of the Eath and Planetay Inteos ( Depth (km (a TBL1a Vscosty (Pa s Decay tme of the Chandle wobble (yea 1 (b TBL1 TBL1a TBL1b TBL2a TBL2b Vscosty * (Pa s Real pat of tdal esponse (k (c data 1 19 * TBL1 (dashed lnes TBL1a (sold lnes.28 Peod (yea Imagnay pat of tdal esponse (k (d TBL1 & TBL1a -.4 Peod (yea Real pat of tdal esponse (k (e.3 TBL1 (dashed lnes TBL1b (sold lnes.28 Peod (yea Imagnay pat of tdal esponse (k.1 (f TBL1 & TBL1b Peod (yea Fg. 4. Results fo vscosty models wth depth-dependent tempeatue gadent wthn the D laye. The thckness of the D laye s 3 km. (a Vscosty pofles fo TBL1a model, (b decay tmes of the Chandle wobble as a functon of g (see text, and eal (c, (e and magnay (d, (f pats of tdal esponses as a functon of peod. The paamete values fo each model ae shown n Table 1. the esults fo TBL4 model. The bounday depths (z b of a change n tempeatue gadent (dt/dz ae assumed to be 2791 and 2691 km, and we have examned two cases of these models. The vscosty dstbuton fo the uppe laye (z 6 z b s detemned by gvng the vscosty as fo models of TBL1 and TBL2, and that fo the lowe laye (z > z b s deved fom the vscosty at z = z b, g(z b, and a specfc vscosty of the (g, 1 16 o Pa s. The tempeatue gadents ae assumed to be constant n each laye. The tempeatue pofle fo ths model s smla to a model by Henlund et al. (25 studyng a doublng of the post-peovskte phase bounday n the D laye fo the cold mantle. Fg. 4a shows the vscosty pofles of TBL1 (dashed lnes and TBL1a (sold lnes wth g top =1 22 Pa s and g =1 16 Pa s. The TBL1a model has a moe dstnct low vscosty zone elatve to that fo TBL1. The vscosty pofle of TBL1a s chaactezed by the values of g top and g, and also an extapolated vscosty, g, coespondng to the vscosty fo the TBL1 wth a constant dt/dz value fo the whole laye. Fg. 4b shows the decay tmes of the Chandle wobble fo models TBL1a, TBL1b (g = Pa s, TBL2a and TBL2b (Table 1. The decay tme becomes longe wth deceasng g -value, mplyng that the decay s pedomnantly contolled by the vscous esponse fo the uppe laye as the lowe laye becomes nvscd n tems of the decay of the Chandle wobble (Nakada and Kaato, 212. The g -values fo models satsfyng obsevatonally nfeed decay tmes ae g P Pa s fo TBL1a and g P Pa s fo TBL1b. Although we do not show the esults fo models wth a bounday depth of 2691 km, the decay tmes fo those models ae shote than 3 yeas and cannot explan the obsevatonally nfeed decay tmes. Fg. 4c f show the pedctons fo k T;P fo TBL1, TBL1a and TBL1b. Dffeences of the pedctons between TBL1, TBL1a and TBL1b ae clealy seen n the pedcted k T;P. The magntude of k T;P at peods less than 1 1 yeas fo TBL1a and TBL1b s lage than that fo TBL1 and ts effect eaches to much shote peod ange deceasng g -value. Consequently, the geodetcally nfeed tdal defomatons of k T;P fo.1.1 yea can be explaned by the pedctons fo TBL1a wth g ¼ 1 16 Pa s. Although t may be dffcult to clealy descbe the dffeences fo pedcted k T;P, the

7 M. Nakada et al. / Physcs of the Eath and Planetay Inteos ( Depth (km (a TBL1d Vscosty (Pa s Decay tme of the Chandle wobble (yea 1 1 TBL1 TBL1c TBL1d TBL1e TBL1f TBL2c TBL2d (b Vscosty * (Pa s Real pat of tdal esponse (k Real pat of tdal esponse (k (c data 1 19 * TBL1 (dashed lnes TBL1c (sold lnes.28 Peod (yea (e.3 TBL1 (dashed lnes TBL1d (sold lnes.28 Peod (yea Imagnay pat of tdal esponse (k Imagnay pat of tdal esponse (k.1 (d TBL1 & TBL1c Peod (yea.1 (f TBL1 & TBL1d Peod (yea Fg. 5. Results fo vscosty models wth a constant low vscosty laye at the bottom of the D laye. The thcknesses of the D laye and the constant vscosty laye ae 3 and 1 km, espectvely. (a Vscosty pofles fo TBL1d model, (b decay tmes of the Chandle wobble as a functon of g (see text, and eal (c, (e and magnay (d, (f pats of tdal esponses as a functon of peod. The paamete values fo each model ae shown n Table 1. magntude of k T;P at a peod of 18.6 yeas becomes smalle than that fo TBL1. These chaactestcs fo k T;P ae caused by a moe dstnct low vscosty zone elatve to that fo TBL1. Consequently, the pedctons fo TBL1a model (g =1 16 Pa s wth g 118 Pa s can explan the geodetcally nfeed defomatons fo peods longe than.1 yea and also the decay tme of the Chandle wobble. The pedcted decay tme fo such a model s, howeve, 3 yeas, coespondng to the mnmum estmate by Wlson and Vcente (199. Next we dscuss two models wth a constant vscosty laye at the bottom of the D laye (see TBL1d model n Fg. 5a. Ths model coesponds to a case whee the heologcal popetes ae moe senstve to factos othe than tempeatue. We fst dscuss the esults fo D laye model wth 3 km thckness. Fg. 5b depcts the decay tmes of the Chandle wobble fo seveal models wth a constant vscosty laye of 1 km thckness (see Table 1 fo the paamete values. Although we have examned based on vscosty models wth ts thckness of 5 and 1 km, the models wth 1 km thckness ae equed to explan the tdal defomatons as stated below. The g -values fo models satsfyng obsevatonally nfeed decay tmes ae g P 2 Pa s fo TBL1c, g P Pa s fo TBL1d and g P Pa s fo TBL1e, and the pedcted decay tmes fo model TBL1f wth a constant vscosty (g of5 Pa s ae shote than the obseved estmates. It s noted that TBL1c and TBL1d models, wth a developed channel-lke low vscosty laye at the bottom of the D laye, can pedct the decay tmes of 7 yeas coespondng to the optmum value by Wlson and Vcente (199. fo such vscosty models s constant fo a specfc peod ange and ts peod ange nceases wth deceasng g -value, whch clealy dffes fom the tendency detected fo TBL1a and Fg. 5c f show the k T;P fo TBL1, TBL1c and TBL1d, n whch the thckness of a constant vscosty laye s 1 km. The value of k T;P TBL1b. Fo example, the k T;P -value fo TBL1c model wth g ¼ 119 Pa s s.315 fo.1 3 yeas. Fo models wth 5 between the elastc one fo peods smalle than.1 yea fo 1 yea,, s about half fo that of 1 km thckness as ndcated by km thckness, howeve, the change of k T;P Dk T;P

8 18 M. Nakada et al. / Physcs of the Eath and Planetay Inteos ( Depth (km Real pat of tdal esponse (k (a TBL4d Vscosty (Pa s (c data 1 16 * TBL4 (dashed lnes TBL4d (sold lnes.28 Peod (yea Decay tme of the Chandle wobble (yea Imagnay pat of tdal esponse (k 1 1 TBL4 TBL5 TBL4c TBL4d TBL4f TBL5c TBL5d (b Vscosty * (Pa s (d -.4 Peod (yea TBL4 & TBL4d Fg. 6. Results fo vscosty models wth a constant low vscosty laye at the bottom of the D laye. The thcknesses of the D laye and the constant vscosty laye ae 2 and 1 km, espectvely. (a Vscosty pofles fo TBL4d model, (b decay tmes of the Chandle wobble as a functon of g (see text, and eal (c and magnay (d pats of tdal esponses as a functon of peod. The paamete values fo each model ae shown n Table 1. Nakada and Kaato (212. In the pedctons fo k T;P, ts magntude at a peod of 18.6 yeas becomes smalle than that fo TBL1. The pedcted k T;P fo models wth g Pa s satsfyng obsevatonally nfeed decay tmes ae consstent wth the geodetcally nfeed defomatons fo peods longe than.1 yea. Moeove, the decay tme fo models wth g 118 Pa s s nealy dentcal to the optmum value by Wlson and Vcente (199. We befly dscuss the esults fo D laye model wth 2 km thckness. In the models wth a constant low vscosty laye of 5 km thckness and vscostyg, the pemssble vscosty stuctue s smla to that fo TBL4 model as nfeed fom the vscosty stuctue of TBL4d shown n Fg. 6a,.e., g 116 Pa s and g 6 Pa s, and the pedcted decay tme ae also 3 yeas. Fg. 6 depcts the esults fo models wth 1 km thckness (see Table 1 fo model paametes. Pedctons fo TBL4c and TBL4d (g top =1 22 Pa s wth g Pa s can explan obsevatonally nfeed decay tme of the Chandle wobble and k T;P, but not k T;P. The decay tmes fo g Pa s ae 4 5 yeas. Fo models wth g top =1 21 Pa s (TBL5c and TBL5d, the pemssble ange s g Pa s and the decay tmes fo g 117 Pa s ae 6 7 yeas. 4. Results fo a convectng laye model In case of convectng D laye, tempeatue gadents of the uppe and lowe themal bounday layes ae sgnfcantly hghe than that fo the ntelaye. Hee we show the esults fo models wth 1 km thckness fo thee layes,.e., uppe themal bounday laye, sothemal laye and bottom themal bounday laye. Although we have examned seveal vscosty models wth dffeent thckness fo each laye, those esults ae essentally the same as the esults shown hee. To obtan the tempeatue dstbuton fo such a convectng D laye, we fst detemne DT fo a specfc value of g usng Eq. (6 as fo TBL1 and TBL2 models. Then we detemne the tempeatue dstbutons fo both themal bounday layes by assumng that the gadents fo both layes ae constant and dentcal, n whch the tempeatue fo the ntelaye s fxed to that fo the bottom of the uppe laye. Fg. 7 shows the vscosty pofles fo CON1 model wth g top =1 22 Pa s and CON2 wth g top =1 21 Pa s, and Fg. 2 shows the pedcted decay tmes (s CW as a functon of g (see Table 2 fo convectng D laye models. The pemssble g values fo CON1 and CON2 satsfyng s CW (3 3 yeas ae 2 6 g Pa s and 2 < g Pa s, and g -values fo s CW 7 yeas ae 5 Pa s and 8 Pa s, espectvely. These values ae slghtly smalle than those fo the TBL models. Fg. 8a and b show the esults fo k T;P based on TBL1 and CON1 models wth seveal g values. As easly seen, fo example, fom the dffeences of k T;P fo both models wth g ¼ Pa s, the esponse fo CON1 s moe effcent fo peods longe than 3 yeas and less fo smalle than 3 yeas. Ths tendency may be effectve to solve the dscepances fo TBL models. Howeve, the pedctons fo CON1 model satsfyng obsevatonally nfeed decay tmes cannot explan the obsevatons wth peods longe than 1 yea, whch s also tue fo CON2 model as shown n Fg. 8c and d. We dscuss two data sets based on vscosty models wth tempeatue gadent of the lowe laye detemned by specfc vscostes of 1 16 and Pa s (g. That s, the tempeatue gadent of the lowe laye s lage than that fo the uppe laye.

9 M. Nakada et al. / Physcs of the Eath and Planetay Inteos ( (a CON (b CON2 Depth (km Depth (km Vscosty (Pa s Vscosty (Pa s Fg. 7. Vscosty pofles fo the D laye of CON1 and CON2 models wth H = 5 kj mol 1 and T top = 26 K. Table 2 Tempeatue and vscosty stuctues of the D laye fo a convectng laye model (CON model. The thckness of the D laye s 3 km and the bottom of the D laye s 2891 km depth. The thckness of the uppe and lowe themal bounday layes and the sothemal ntelaye s 1 km. The tempeatue gadents fo the uppe and lowe bounday layes ae denoted by (dt/dz u (constant and (dt/dz l (constant, espectvely. In these models, the lthosphec (elastc thckness s 1 km, and the uppe and lowe mantle vscostes except fo the D laye ae 1 21 and 1 22 Pa s, espectvely. Model name Vscosty at the top of the D laye (g top (Pa s Vscosty of the lowe laye of the D laye Tempeatue gadent stuctue CON vaables (dt/dz u =(dt/dz l CON vaables (dt/dz u =(dt/dz l CON1a Pa s at the bottom (dt/dz u <(dt/dz l CON1b 1 22 Pa s at the bottom (dt/dz u <(dt/dz l CON2a Pa s at the bottom (dt/dz u <(dt/dz l CON2b 1 21 Pa s at the bottom (dt/dz u <(dt/dz l CON1c Pa s fo km depth (dt/dz u,(dt/dz l = CON1d 1 22 Pa s fo km depth (dt/dz u,(dt/dz l = CON1e Pa s fo km depth (dt/dz u,(dt/dz l = CON1f Pa s fo km depth (dt/dz u,(dt/dz l = CON2c Pa s fo km depth (dt/dz u,(dt/dz l = CON2d 1 21 Pa s fo km depth (dt/dz u,(dt/dz l = Fg. 9 shows the esults fo such models. The vscosty pofles fo CON1a wth g ¼ 1 16 Pa s ae shown n Fg. 9a and b shows the decay tmes fo seveal such models. The pedcted decay tmes fo these models ae shote than 6 yeas as also pedcted fo the same sots of TBL model. The g -values fo models satsfyng obsevatonally nfeed decay tme ae g P Pa s fo CON1a and g P Pa s fo CON1b (g ¼ Pa s. Among these pemssble vscosty models, the model of CON1a wth g 118 Pa s can also explan geodetcally nfeed defomatons fo peods longe than.1 yea and poduces the decay tme 4 yeas (Fg. 9c and d. We obtan a smla concluson about the vscosty stuctue fo vscosty models wth g top = Pa s. Such a concluson has also been deved fom TBL models of TBL1a and TBL2a (see Fg. 3. Consequently, these esults fo tempeatue-dependent vscosty models, fo both TBL and CON models, ndcate that the vscosty at the bottom of the D laye s 1 16 Pa s fo g top =1 21 and 1 22 Pa s. Fnally, we show the esults fo vscosty models wth the lowe laye of constant vscosty (g, n whch the g -values fo models wth g top =1 22 Pa s ae ,, 2 and 5 Pa s fo CON1c, CON1d, CON1e and CON1f models (Table 2, espectvely. The decay tmes fo CON1f ae smalle than the obsevatonally nfeed values (Fg. 1b. The esults shown n Fg. 1b d ndcate that the pedctons fo CON1d model wth 2 < g Pa s can explan both obsevatonally nfeed estmates, whch s also tue fo CON1c model. In patcula, the pedcted decay tmes fo CON1c model wth g (5 1 Pa s and fo CON1d model wth g 118 Pa s ae 7 yeas fo the optmum estmate by Wlson and Vcente (199. These conclusons ae also applcable to models wth g top =1 21 Pa s. We befly state the esults fo the D laye model wth 2 km thckness, n whch the uppe and lowe layes have 5 km thckness and the ntelaye has 1 km. In these models, the pemssble vscosty anges satsfyng obsevatonally nfeed decay tmes ae g > Pa s and g < 117 Pa s, espectvely. That s, we could not fnd pemssble vscosty model satsfyng both data sets even f we consde a 5 km constant low vscosty laye at the base of the D laye. 5. Implcatons fo the D laye and coe mantle bounday egon We summaze the numecal esults fo TBL and CON models n Table 3. Vscosty models of the D laye satsfyng obsevatonally nfeed decay tmes of the Chandle wobble (ampltude esponse fo peods longe than.1 yea eque ethe followng condton: ( tempeatue gadent s nealy constant fo TBL model wth the D laye of 2 km thckness, ( tempeatue gadent of the lowe pat (1 km thckness s lage than that of the uppe pat fo TBL model (D laye wth 2 o 3 km thckness and fo CON model (D laye wth 3 km thckness and the vscosty (g s1 16 Pa s fo both models, and ( vscosty of the lowe pat (1 km thckness s constant and smalle than Pa s fo TBL model (D laye wth 2 o 3 km thckness and fo

10 2 M. Nakada et al. / Physcs of the Eath and Planetay Inteos ( Real pat of tdal esponse (k (a data CON1 (dashed lnes TBL1 (sold lnes.28 Peod (yea Imagnay pat of tdal esponse (k.1 (b CON1 & TBL Peod (yea Real pat of tdal esponse (k (c CON1 (dashed lnes CON2 (sold lnes.28 Peod (yea Imagnay pat of tdal esponse (k.1 (d CON1 & CON Peod (yea Fg. 8. Real (a, c and magnay (b, d pats of tdal esponses fo CON1 and CON2 models as a functon of peod. Depth (km (a CON1a Vscosty (Pa s Decay tme of the Chandle wobble (yea 1 1 (b CON1 CON2 CON1a CON1b CON2a CON2b Vscosty * (Pa s Real pat of tdal esponse (k (c data 1 19 * CON1 (dashed lnes CON1a (sold lnes.28 Peod (yea Imagnay pat of tdal esponse (k.1 (d CON1 & CON1a Peod (yea Fg. 9. Results of vscosty models fo convectng D laye model. The thckness of the D laye s 3 km, and the thcknesses of uppe themal bounday laye, sothemal laye and bottom themal bounday laye ae 1 km. (a Vscosty pofles fo CON1a model, (b decay tmes of the Chandle wobble as a functon of g (see text, and eal (c and magnay (d pats of tdal esponses as a functon of peod. The paamete values fo each model ae shown n Table 2.

11 M. Nakada et al. / Physcs of the Eath and Planetay Inteos ( Depth (km (a CON1d Vscosty (Pa s Decay tme of the Chandle wobble (yea 1 1 (b CON1 CON1c CON1d CON1e CON1f CON2 CON2c CON2d Vscosty * (Pa s Real pat of tdal esponse (k (c data 1 19 * CON1 (dashed lnes CON1d (sold lnes.28 Peod (yea Imagnay pat of tdal esponse (k.1 (d CON1 & CON1d Peod (yea Fg. 1. Results fo vscosty models wth a constant low vscosty laye at the bottom of the D laye. The thcknesses of the D laye and the constant vscosty laye ae 3 and 1 km, espectvely, and those fo uppe themal bounday laye and sothemal laye ae 1 km. (a Vscosty pofles fo CON1d model, (b decay tmes of the Chandle wobble as a functon of g (see text, and eal (c and magnay (d pats of tdal esponses as a functon of peod. The paamete values fo each model ae shown n Table 2. CON model (D laye wth 3 km thckness. Fg. 11 also shows the pefeed vscosty stuctues of the D laye deved fom ou numecal expements, n whch the egon wth vscosty lage than Pa s nsgnfcantly affects the decay of the Chandle wobble (Nakada and Kaato, 212. The pedcted k T;P fo models satsfyng the condton ( o ( also explans the obsevatonally nfeed value fo 18.6 yeas tde, but TBL4 and TBL5 models wth the D laye of 2 km thckness cannot explan the obsevatonally nfeed k T;P value fo 18.6 yeas tde. Howeve, f we consde that the phase esponse at 18.6 yea tde may be explaned by the electomagnetc couplng at the coe mantle bounday (Buffett et al., 22 and s also hghly senstve to coecton factos (Benjamn et al., 26, then such models (TBL4 and TBL5 would be possble vscosty stuctues of the D laye. We fst dscuss the tempeatue ncease wthn the D laye (DT fo TBL4 and TBL5 models (see Fg. 11a wth the pedcted decay tmes of 3 yeas, whch may coespond to the hot mantle by Henlund et al. (25. These models eque the vscosty (g of 1 16 Pa s. The elatonshp between H =RT top and DT =T top usng Eq. (6 s shown n Fg. 12 as a functon of g. The pemssble values of H =RT top ae 2 3 (Kaato, 28 and ts values fo H = 5 kj mol 1 and T top = 26 K ae The estmates of DT deved fom the elatonshp fo g ¼ 1 16 Pa s ae as follows: ( T top fo g top ¼ 1 22 Pa s and ( T top fo g top ¼ 1 21 Pa s. If we assume T top = 26 K, then DT ae lage than 22 K fo g top =1 22 Pa s and lage than 15 K fo g top =1 21 Pa s, n whch the tempeatues of 22 K and 15 K coespond to H =RT top 3. Recent estmates of the tempeatue at the top of the coe (T, whch ae nfeed fom the on meltng tempeatue detemnatons fo the nne-coe bounday (Boehle, 2; Alfè et al., 22, ae K (see also Henlund et al. (25 and Lay et al. (28. Then, DT 15 K fo g top =1 21 Pa s and T top of K may be a possble soluton (TBL5 n Fg. 11a, whch may coespond to the wam mantle by Lay et al. (28. If we also assume a commonly used estmate of themal conductvty of 1 W m 1 K 1 (Stacey, 1992, then the aveage heat flow fom the coe to the mantle s estmated to be 11 TW, 3 4 tmes lage than the estmate of 3 TW by Stacey (1992. The tempeatue ncease of 15 K s also obtaned fo models of TBL2a and CON2a fo the D laye wth 3 km thckness, whch, howeve, coespond to the cold mantle by Lay et al. (28. The aveage heat flow fo these models s 7 8 TW, 2 3 tmes lage than the estmate by Stacey (1992. That s, vscosty models wth no constant low vscosty laye at the base of the D laye eque hgh tempeatue ncease of 15 K wthn the D laye and also suggest sgnfcantly hgh coe heat flow lage than 3 TW estmated by Stacey (1992. We next dscuss geophyscal mplcatons deved fom the vscosty stuctues wth a constant (channel-lke low vscosty laye at the base of the D laye such as TBL4c, TBL5c, TBL1d, TBL2d, CON1d and CON2d (see Fg. 11. These models wth the D laye of 3 km thckness explan both data sets and also pedct the decay tme of the Chandle wobble smla to the optmum value by Wlson and Vcente (199 (see Table 3. It s dffcult to estmate the tempeatue ncease wthn the D laye fo these models. Howeve, the mnmum value may be nfeed fom the pefeed extapolated vscosty, g (see, fo example, Fg. 5a. The g -values fo TBL4c and TBL5c ae 116 Pa s, and

12 22 M. Nakada et al. / Physcs of the Eath and Planetay Inteos ( Table 3 Summaes of the esults based on TBL and CON models. The pemssble vscosty anges satsfyng obsevatonally nfeed estmates ae gven by g -value fo TBL1, TBL2, TBL4, TBL5, CON1 and CON2 models and g -value fo othe models. Model name Pemssble vscosty ange fo s CW (Pa s Vscosty fo optmum s CW 7 yeas (Pa s Pemssble vscosty ange fo k at peods longe than.1 yea (Pa s Pemssble vscosty ange satsfyng s CW and k fo peods longe than.1 yea (Pa s, and optmum (o nea optmum s CW (yea and ts vscosty (Pa s (wthn the paenthess TBL1 2.5 None No TBL2 3 None No TBL (3 y fo 1 16 Pa s No TBL (3 y fo 1 16 Pa s No TBL1a P7 None (3 y fo Pa s Yes TBL1b P6 None None No TBL2a P1.5 None 2 2 (3 y fo 2 Pa s Yes TBL2b P None 2 None No TBL1c P2 2 (7 y fo Pa s Yes TBL1d P (7 y 2 Pa s Yes TBL2c P (7 y fo 2 Pa s Yes TBL2d P (55 y fo 2 Pa s Yes TBL4c P (6 y fo Pa s No TBL4d P (5 y fo Pa s No TBL5c P (6 y fo Pa s No TBL5d > (5 y fo Pa s No CON None No CON None No CON1a P5 None 5 (4 y fo Pa s Yes CON1b P2.5 None None No CON2a > None 2 2 (35 y fo 2 Pa s Yes CON2b >8 None 2 None No CON1c P2 7 2 (7 y fo 7 Pa s Yes CON1d > (6 y fo Pa s Yes CON2c P (7 y fo 2 Pa s Yes CON2d P (55 y fo 2 Pa s Yes Does the pemssble vscosty ange satsfyng s CW and k also satsfy k fo 18.6 yeas tde? the esults ae smla to those fo TBL4 and TBL5. Those fo models wth 3 km thckness ae g ( Pa s fo g top =1 22 and 1 21 Pa s. Then, the estmates fo DT (Fg. 12 ae as follows: (.4.8 T top fo g top =1 22 Pa s and (.3.5 T top fo g top = Pa s. If we assume T top = 26 K, then DT ae 1 21 K fo g top =1 22 Pa s and 8 13 K fo g top =1 21 Pa s. Fo H = 5 kj mol 1 (H =RT top ¼ 23:1, DT 16 K and T 42 K fo g top =1 22 Pa s and DT 1 K and T 37 K fo g top =1 21 Pa s, and the aveage heat flows ae 8 and 5TW, espectvely, whch ae also lage than 3 TW. These models have a constant (channel-lke low vscosty laye at the base of the D laye wth 1 km thckness and ts vscosty smalle than Pa s. Ths laye may be elated to the ultalowvelocty zone (ULVZ detected just above the n ples o layes a few tens of klometes thck (Ganeo et al., Henlund and Jellnek (21 agued that melt wth a dffeent densty than the suoundng mateals could be dynamcally suppoted by flow (see also Lay et al., 28. Such a mechansm s, howeve, hghly senstve to the vscosty, and theefoe t would be necessay to examne the valdty n the case of Pa s fo the bottom pat (1 km thckness vscosty of the D laye. Smlaly, Kanda and Stevenson (26 suggested that on-ch melt may penetate nto the mantle by the pessue gadent caused by the dynamc topogaphy. The depth of penetaton of on-ch melt s agan contolled by the vscosty of the bottom of the D laye, and the penetaton depth wll be neglgble (<1 m f the vscosty thee s less than Pa s as suggested by ths study. 6. Conclusons We have examned the decay tme of the Chandle wobble and sem-dunal to 18.6 yeas tdal defomatons to estmate the tempeatue-dependent vscosty stuctue of the D laye. The tempeatue dstbuton depends on ts dynamc state, and we theefoe adopt two typcal models,.e., bottom themal bounday laye of the mantle convecton (TBL model and vgoously smallscale convectng laye (CON model. In these models, we assume the vscosty at the top of the D laye g top to be 1 21 and 1 22 Pa s. Howeve, the choce of the vscosty at the top of the D laye does not affect the concluson on the vscosty stuctue so much. Thee possble models ae deved fom the compason between the numecal and obsevatonally nfeed decay tmes of Chandle wobble and tdal defomatons. The fst model coesponds to nealy constant tempeatue gadent wthn the D laye wth ts thckness (L of 2 km n TBL model, and the vscosty at the (g s1 16 Pa s. The tempeatue ncease wthn the D laye DT s lage than 15 K, and the tempeatue at the top of the coe T coesponds to the ecent estmate of K. The second model eques that the tempeatue gadent of the lowe pat (1 km thckness s lage than that of the uppe pat and the g 1 16 Pa s n TBL and CON models wth L = 3 km. The tempeatue dstbuton causes a moe dstnct low vscosty zone at the base of the D laye, and DT s also lage than 15 K. The thd model has a channel-lke (constant

13 M. Nakada et al. / Physcs of the Eath and Planetay Inteos ( Depth (km (a TBL4 TBL4c TBL5 TBL5c T /T top (a x1 16 2x 1 19 Depth (km Depth (km (b Vscosty (Pa s TBL1a TBL1d TBL2a TBL2d Vscosty (Pa s (c CON1a CON1d CON2a CON2d Vscosty (km T /T top Acknowledgments top =122 Pa s H*/RT top (b x1 16 2x 1 19 top =121 Pa s H*/RT top Fg. 12. The elatonshp between H =RT top and DT =T top usng Eq. (6 as a functon of g : (a fo g top =1 22 Pa s and (b fo g top =1 21 Pa s. We thank H. Czkova and an anonymous evewe fo the helpful comments. Ths wok was patly suppoted by the Japanese Mnsty of Educaton, Scence and Cultue (Gand-n-Ad fo Scentfc Reseach No , and patly by the Natonal Scence Foundaton of USA (to SK. Fg. 11. Pefeed vscosty stuctues of the D laye obtaned n ths study: (a fo TBL model wth the thckness of 2 km, (b fo TBL model wth 3 km thckness and (c fo CON model wth 3 km thckness. low vscosty laye (1 km thckness at the bottom of the D laye wth ts vscosty smalle than Pa s n TBL (L = 2 and 3 km and CON (L = 3 km models. The plausble estmates fo the tempeatue ncease ae 16 K fo g top =1 22 Pa s and 1 K fo g top =1 21 Pa s. The heat flows fom the coe to the mantle fo these thee models appea to be sgnfcantly lage than 3 TW estmated by Stacey (1992. Among these thee models, the pedcted decay tmes fo the fst and second models ae close to the mnmum estmate (3 yeas by Wlson and Vcente (199 and those fo the thd model ae 7 yeas coespondng to the optmum estmate. Also, the thd model explans the geodetcally nfeed eal (ampltude and magnay (phase lag pats fo the tdal defomatons fo peods longe than.1 yea. Consequently, the thd model s a pefeed model wthn the lmted ange of ou numecal expements. It would be mpotant to examne the elatonshp between the channel-lke low vscosty laye and the ultalowvelocty zone. Refeences Alfè, D., Gllan, M.J., Pce, G.D., 22. Composton and tempeatue of the Eath s coe constaned by combnng ab nto calculatons and sesmc data. Eath Planet. Sc. Lett. 195, Benjamn, D., Wah, J., Ray, R.D., Egbet, G.D., Sesa, S.D., 26. Constants on mantle anelastcty fom geodetc obsevatons, and mplcatons fo the J 2 anomaly. Geophys. J. Int. 165, Boehle, R., 2. Hgh-pessue expements and the phase dagam of lowe mantle and coe consttuents. Rev. Geophys. 38, Buffett, B.A., Mathews, P.M., Heng, T.A., 22. Modelng of nutaton and pecesson: effects of electomagnetc couplng. J. Geophys. Res dx.do.og/1.129/2jb56. Dckman, S.R., Nam, Y.S., Constants on Q at long peods fom Eath s otaton. Geophys. Res. Lett. 25, Dzewonsk, A.M., Andeson, D.L., Pelmnay efeence Eath model (PREM. Phys. Eath Planet. Inte. 25, Fuuya, M., Chao, B.F., Estmaton of peod and Q of the Chandle wobble. Geophys. J. Int. 127, Ganeo, E.J., Revenaugh, J., Wllams, Q., Lay, T., Ultalow velocty zone at the coe mantle bounday. In: Guns, M., Wysesson, M.E., Knttle, E., Buffett, B.A. (Eds., The Coe-Mantle Bounday Regon, Geodynamcs Sees, vol. 28, Amecan Geophyscal Unon, pp Goss, R.S., 27. Eath otaton vaatons long peods. In: Heng, T. (Ed., Teatse on Geophyscs, vol. 3. Geodesy, Elseve, pp Hage, B.H., Subducted slabs and the geod: constants on mantle heology and flow. J. Geophys. Res. 89, Hage, B.H., Clayton, R.W., Rchads, M.A., Come, R.P., Dzewonsk, A.M., Lowe mantle heteogenety, dynamc topogaphy and the geod. Natue 313,