Content-Type: multipart/mixed; boundary="-------------0104251443200" This is a multi-part message in MIME format. ---------------0104251443200 Content-Type: text/plain; name="01-157.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-157.comments" To appear in: Stochastics and Dynamics http://ejournals.worldscientific.com.sg/sd/sd.html ---------------0104251443200 Content-Type: text/plain; name="01-157.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-157.keywords" Stochastic geophysical flow models, convergence in probability, random dynamical systems, random wind forcing ---------------0104251443200 Content-Type: application/postscript; name="layer.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="layer.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: 2layer14.dvi %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments 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y(ORN)g(SCHMALFUSS)754 1315 y Fv(Abstra)n(ct.)43 b Fw(The)25 b(t)n(w)n(o-la)n(y)n(er)g(quasigeostrophic)i(\015o)n(w)e (mo)r(del)f(is)i(an)f(in)n(termidi-)754 1406 y(ate)31 b(system)e(b)r(et)n(w)n(een)g(the)h(single-la)n(y)n(er)g(2D)g (barotropic)h(\015o)n(w)f(mo)r(del)f(and)h(the)754 1498 y(con)n(tin)n(uously)d(strati\014ed,)g(3D)f(baro)r(clinic)i(\015o)n(w)f (mo)r(del.)37 b(This)27 b(mo)r(del)f(is)h(widely)754 1589 y(used)c(to)g(in)n(v)n(estigate)h(basic)f(mec)n(hanisms)f(in)g (geoph)n(ysical)i(\015o)n(ws,)h(suc)n(h)d(as)h(baro-)754 1680 y(clinic)28 b(e\013ects,)g(the)f(Gulf)h(Stream)e(and)g (subtropical)i(gyres.)39 b(The)27 b(wind)g(forcing)754 1772 y(acts)i(only)e(on)h(the)g(top)f(la)n(y)n(er.)41 b(W)-6 b(e)27 b(consider)i(the)e(t)n(w)n(o-la)n(y)n(er)h (quasigeostrophic)754 1863 y(mo)r(del)g(under)e(sto)r(c)n(hastic)j (wind)e(forcing.)41 b(W)-6 b(e)27 b(\014rst)h(transformed)f(this)g (system)754 1954 y(in)n(to)e(a)g(coupled)g(system)f(of)i(random)e (partial)h(di\013eren)n(tial)h(equations)f(and)f(then)754 2046 y(sho)n(w)f(that)e(the)g(asymptotic)h(probabilistic)h(dynamics)d (of)j(this)e(system)g(dep)r(ends)754 2137 y(only)h(on)g(the)f(top)h (\015uid)f(la)n(y)n(er.)34 b(Namely)-6 b(,)21 b(in)h(the)f(probabilit)n (y)h(sense)h(and)e(asymp-)754 2228 y(totically)-6 b(,)27 b(the)d(dynamics)g(of)h(the)g(t)n(w)n(o-la)n(y)n(er)g(quasigeostrophic) h(\015uid)e(system)g(is)754 2320 y(determinied)29 b(b)n(y)g(the)g(top)h (\015uid)f(la)n(y)n(er,)i(or,)h(the)d(b)r(ottom)g(\015uid)g(la)n(y)n (er)h(is)g(sla)n(v)n(ed)754 2411 y(b)n(y)24 b(the)f(top)h(\015uid)g(la) n(y)n(er.)34 b(This)25 b(conclusion)g(is)f(true)g(pro)n(vided)g(that)f (the)h(Wiener)754 2502 y(pro)r(cess)29 b(and)f(the)f(\015uid)g (parameters)g(satisfy)i(a)f(certain)g(condition.)41 b(In)27 b(partic-)754 2594 y(ular,)33 b(this)d(latter)h(condition)g(is)g (satis\014ed)g(when)f(the)g(trace)h(of)g(the)f(co)n(v)l(ariance)754 2685 y(op)r(erator)g(of)g(the)f(Wiener)g(pro)r(cess)h(is)f(con)n (troled)g(b)n(y)f(a)h(certain)h(upp)r(er)e(b)r(ound,)754 2776 y(and)20 b(the)f(Ekman)g(constan)n(t)h Fu(r)h Fw(is)f(su\016cien)n (tly)g(large.)34 b(Note)19 b(that)h(the)f(generalized)754 2867 y(time)k(deriv)l(ativ)n(e)f(of)i(the)e(Wiener)i(pro)r(cess)f(mo)r (dels)g(the)g(\015uctuating)f(part)h(of)h(the)754 2959 y(wind)34 b(stress)f(forcing)i(on)e(the)f(top)h(\015uid)f(la)n(y)n(er,) k(and)c(the)h(Ekman)f(constan)n(t)h Fu(r)754 3050 y Fw(measures)28 b(the)g(rate)g(for)h(v)n(orticit)n(y)f(deca)n(y)f(due)h(to)g(the)g (friction)h(in)f(the)f(b)r(ottom)754 3141 y(Ekman)e(la)n(y)n(er.)p 456 4759 499 4 v 555 4850 a Ft(Date)6 b Fw(:)27 b(Marc)n(h)f(31,)g (2001.)555 4942 y(1991)37 b Ft(Mathematics)g(Subje)l(ct)g(Classi\014c)l (ation.)i Fw(Primary)34 b(60H15,)k(76U05;)i(Secondary)34 b(86A05,)456 5033 y(34D35.)555 5124 y Ft(Key)23 b(wor)l(ds)g(and)f(phr) l(ases.)40 b Fw(Sto)r(c)n(hastic)21 b(geoph)n(ysical)g(\015o)n(w)f(mo)r (dels,)h(con)n(v)n(ergence)f(in)f(probabilit)n(y)-6 b(,)456 5216 y(random)24 b(dynamical)h(systems,)h(random)f(wind)g(forcing.)1932 5315 y Fs(1)p eop %%Page: 2 2 2 1 bop 456 251 a Fs(2)362 b(IGOR)24 b(CHUESHO)n(V,)f(JINQIA)n(O)h (DUAN,)e(AND)h(BJ)2368 234 y(\177)2358 251 y(ORN)h(SCHMALFUSS)1583 450 y Fr(1.)46 b Fq(Intr)n(oduction)456 612 y Fr(The)19 b(con)m(tin)m(uously)h(strati\014ed,)h(three)g(dimensional)c(\(3D\))22 b(baro)s(clinic)c(quasigeostrophic)456 720 y(\015o)m(w)25 b(mo)s(del)g(describ)s(es)e(large)j(scale)g(geoph)m(ysical)g(\015uid)d (motions)i(in)f(the)i(atmosphere)456 828 y(and)33 b(o)s(ceans.)54 b(This)32 b(mo)s(del)i(is)f(m)m(uc)m(h)i(simpler)d(than)i(the)g (primitiv)m(e)f(\015o)m(w)h(mo)s(del)f(or)456 936 y(rotating)27 b(Na)m(vier-Stok)m(es)h(\015o)m(w)f(mo)s(del.)38 b(When)26 b(the)h(\015uid)d(densit)m(y)i(is)g(appro)m(ximately)456 1044 y(constan)m(t,)h(this)d(mo)s(del)f(reduces)h(to)i(the)f (barotropic,)g(single-la)m(y)m(er,)h(t)m(w)m(o)g(dimensional)456 1152 y(\(2D\))f(quasigeostrophic)f(mo)s(del.)38 b(The)23 b(t)m(w)m(o-la)m(y)m(er)k(quasigeostrophic)d(\015o)m(w)g(mo)s(del,)h (in)456 1260 y(whic)m(h)i(the)i(\015uid)e(consists)h(of)i(t)m(w)m(o)g (homogeneous)f(\015uid)e(la)m(y)m(ers)i(of)h(uniform)c(but)i(dis-)456 1367 y(tinct)g(densities)g Fp(\032)1087 1381 y Fs(1)1155 1367 y Fr(and)g Fp(\032)1377 1381 y Fs(2)1417 1367 y Fr(,)h(is)f(an)h(in)m(termidiate)e(system)i(b)s(et)m(w)m(een)h(the)f (single-la)m(y)m(er)456 1475 y(2D)d(barotropic)f(\015o)m(w)g(mo)s(del)f (and)h(the)h(con)m(tin)m(uously)e(strati\014ed,)i(3D)g(baro)s(clinic)d (\015o)m(w)456 1583 y(mo)s(del.)456 1691 y(The)d(t)m(w)m(o-la)m(y)m(er) j(quasigeostrophic)e(\015o)m(w)g(mo)s(del)f(has)g(b)s(een)h(used)f(as)h (a)h(theoretical)f(and)456 1799 y(n)m(umerical)26 b(mo)s(del)h(to)i (understand)d(basic)h(mec)m(hanisms,)h(suc)m(h)g(as)g(baro)s(clinic)e (e\013ects)456 1907 y([29)q(],)40 b(wind-drv)m(en)35 b(circulation)h([4)q(,)h(5)q(],)j(the)e(Gulf)e(Stream)i([24)q(],)i (\015uid)35 b(stabilit)m(y)h([3)q(])456 2015 y(and)29 b(subtropical)g(gyres)h([30)r(,)g(28)q(],)h(in)e(large)h(scale)h(geoph) m(ysical)f(\015o)m(ws.)456 2123 y(W)-8 b(e)31 b(consider)e(the)i(t)m(w) m(o-la)m(y)m(er)i(quasigeostrophic)c(\015o)m(w)i(mo)s(del)e(\([29)q(],) i(p.)40 b(423\):)1154 2369 y Fp(q)1195 2383 y Fs(1)p Fo(t)1280 2369 y Fr(+)20 b Fp(J)9 b Fr(\()p Fp( )1524 2383 y Fs(1)1564 2369 y Fp(;)15 b(q)1645 2383 y Fs(1)1704 2369 y Fr(+)20 b Fp(\014)5 b(y)s Fr(\))26 b(=)f Fp(\027)6 b Fr(\001)2183 2331 y Fs(2)2222 2369 y Fp( )2281 2383 y Fs(1)2341 2369 y Fr(+)20 b Fp(f)29 b Fr(+)2634 2346 y(_)2597 2369 y Fp(W)38 b(;)1154 2521 y(q)1195 2535 y Fs(2)p Fo(t)1280 2521 y Fr(+)20 b Fp(J)9 b Fr(\()p Fp( )1524 2535 y Fs(2)1564 2521 y Fp(;)15 b(q)1645 2535 y Fs(2)1704 2521 y Fr(+)20 b Fp(\014)5 b(y)s Fr(\))26 b(=)f Fp(\027)6 b Fr(\001)2183 2483 y Fs(2)2222 2521 y Fp( )2281 2535 y Fs(2)2341 2521 y Fn(\000)20 b Fp(r)s Fr(\001)p Fp( )2611 2535 y Fs(2)2675 2521 y Fp(;)456 2445 y Fr(\(1\))456 2680 y(where)42 b(p)s(oten)m(tial)g(v)m(orticities)h Fp(q)1601 2694 y Fs(1)1640 2680 y Fr(\()p Fp(x;)15 b(y)s(;)g(t)p Fr(\),)47 b Fp(q)2036 2694 y Fs(2)2075 2680 y Fr(\()p Fp(x;)15 b(y)s(;)g(t)p Fr(\))44 b(for)e(the)h(top)h(la)m(y)m(er)f(and)f (the)456 2788 y(b)s(ottom)29 b(la)m(y)m(er)h(are)g(de\014ned)e(via)h (stream)h(functions)d Fp( )2352 2802 y Fs(1)2392 2788 y Fr(\()p Fp(x;)15 b(y)s(;)g(t)p Fr(\),)31 b Fp( )2790 2802 y Fs(2)2830 2788 y Fr(\()p Fp(x;)15 b(y)s(;)g(t)p Fr(\),)30 b(resp)s(ec-)456 2896 y(tiv)m(ely)-8 b(,)1423 3036 y Fp(q)1464 3050 y Fs(1)1528 3036 y Fr(=)25 b(\001)p Fp( )1759 3050 y Fs(1)1819 3036 y Fn(\000)19 b Fp(F)1967 3050 y Fs(1)2028 3036 y Fn(\001)h Fr(\()p Fp( )2167 3050 y Fs(1)2227 3036 y Fn(\000)g Fp( )2377 3050 y Fs(2)2417 3036 y Fr(\))p Fp(;)1423 3176 y(q)1464 3190 y Fs(2)1528 3176 y Fr(=)25 b(\001)p Fp( )1759 3190 y Fs(2)1819 3176 y Fn(\000)19 b Fp(F)1967 3190 y Fs(2)2028 3176 y Fn(\001)h Fr(\()p Fp( )2167 3190 y Fs(2)2227 3176 y Fn(\000)g Fp( )2377 3190 y Fs(1)2417 3176 y Fr(\))p Fp(:)456 3113 y Fr(\(2\))456 3335 y(Here)35 b Fp(x;)30 b(y)37 b Fr(are)e(Cartesian)f(co)s(ordinates) h(in)e(zonal)h(\(east\),)k(meridional)32 b(\(north\))i(di-)456 3443 y(rections,)40 b(resp)s(ectiv)m(ely;)j(\()p Fp(x;)15 b(y)s Fr(\))39 b Fn(2)g Fp(O)j Fr(:=)d(\(0)p Fp(;)15 b(L)p Fr(\))27 b Fn(\002)e Fr(\(0)p Fp(;)15 b(L)p Fr(\),)42 b(where)c Fp(L)h Fr(is)e(a)i(p)s(ositiv)m(e)456 3551 y(n)m(um)m(b)s(er;)29 b Fp(F)870 3565 y Fs(1)910 3551 y Fp(;)15 b(F)1008 3565 y Fs(2)1078 3551 y Fr(are)31 b(p)s(ositiv)m(e)f(constan)m(ts)h(de\014ned)e(b)m(y)h(\(see)i(also)e ([31)q(],)h(p.87\))1597 3765 y Fp(F)1655 3779 y Fs(1)1720 3765 y Fr(=)1848 3703 y Fp(f)1903 3670 y Fs(2)1893 3728 y(0)p 1826 3744 139 4 v 1826 3827 a Fp(g)s(h)1924 3841 y Fs(1)2083 3703 y Fp(\032)2130 3717 y Fs(0)p 1984 3744 284 4 v 1984 3827 a Fp(\032)2031 3841 y Fs(2)2091 3827 y Fn(\000)20 b Fp(\032)2229 3841 y Fs(1)2278 3765 y Fp(;)1597 4018 y(F)1655 4032 y Fs(2)1720 4018 y Fr(=)1848 3956 y Fp(f)1903 3923 y Fs(2)1893 3980 y(0)p 1826 3997 139 4 v 1826 4080 a Fp(g)s(h)1924 4094 y Fs(2)2083 3956 y Fp(\032)2130 3970 y Fs(0)p 1984 3997 284 4 v 1984 4080 a Fp(\032)2031 4094 y Fs(2)2091 4080 y Fn(\000)g Fp(\032)2229 4094 y Fs(1)2278 4018 y Fp(;)456 4215 y Fr(with)36 b Fp(g)42 b Fr(the)c(gra)m(vitational)g(acceleration;)43 b Fp(h)2051 4229 y Fs(1)2090 4215 y Fp(;)15 b(h)2182 4229 y Fs(2)2260 4215 y Fr(the)38 b(depth)g(of)g(top)g(and)f(b)s(ottom) 456 4323 y(la)m(y)m(ers,)31 b Fp(\032)784 4337 y Fs(1)824 4323 y Fp(;)15 b(\032)911 4337 y Fs(2)982 4323 y Fr(the)31 b(densities)e(\()p Fp(\032)1591 4337 y Fs(2)1657 4323 y Fp(>)d(\032)1801 4337 y Fs(1)1841 4323 y Fr(\))31 b(of)g(top)g(and)g (b)s(ottom)g(la)m(y)m(ers,)h(resp)s(ectiv)m(ely;)456 4431 y(and)j Fp(L;)15 b(\032)787 4445 y Fs(0)862 4431 y Fr(the)36 b(c)m(haracteristic)h(scales)f(for)f(horizon)m(tal)h (length)f(and)g(densit)m(y)g(of)h(the)456 4539 y(\015o)m(ws,)31 b(resp)s(ectiv)m(ely;)h Fp(f)1275 4553 y Fs(0)1334 4539 y Fr(+)21 b Fp(\014)5 b(y)34 b Fr(\(with)c Fp(f)1849 4553 y Fs(0)1888 4539 y Fp(;)15 b(\014)37 b Fr(constan)m(ts\))c(is)d (the)i(Coriolis)d(parameter)456 4647 y(and)d Fp(\014)33 b Fr(is)26 b(the)i(meridional)d(gradien)m(t)i(of)h(the)f(Coriolis)e (parameter,)k(and)e Fp(\027)k(>)25 b Fr(0)j(is)e(vis-)456 4758 y(cosit)m(y)-8 b(..)44 b(Note)33 b(that)f Fp(r)d Fr(=)e Fp(f)1401 4772 y Fs(0)1556 4719 y Fo(\016)1587 4730 y Fm(E)p 1450 4737 296 4 v 1450 4791 a Fs(2\()p Fo(h)1553 4800 y Fl(1)1588 4791 y Fs(+)p Fo(h)1684 4800 y Fl(2)1718 4791 y Fs(\))1786 4758 y Fr(is)k(the)g(Ekman)g(constan)m(t) i(\()e([30)q(],)h(p.29\))h(whic)m(h)456 4884 y(measures)42 b(the)g(in)m(tensit)m(y)f(of)i(friction)d(at)j(the)g(b)s(ottom)f(b)s (oundary)e(la)m(y)m(er)j(\(the)g(so-)456 4992 y(called)27 b(Ekman)g(la)m(y)m(er\))i(or)f(the)g(rate)h(for)e(v)m(orticit)m(y)i (deca)m(y)g(due)e(to)i(the)f(friction)f(in)f(the)456 5108 y(Ekman)g(la)m(y)m(er.)40 b(Here)26 b Fp(\016)1268 5122 y Fo(E)1354 5108 y Fr(=)1450 5030 y Fk(p)p 1541 5030 226 4 v 78 x Fr(2)p Fp(\027)6 b(=f)1727 5122 y Fs(0)1793 5108 y Fr(is)25 b(the)i(Ekman)f(la)m(y)m(er)h(thic)m(kness)e(\([29)r (],)j(p.188\).)456 5216 y(Moreo)m(v)m(er,)h Fp(J)9 b Fr(\()p Fp(h;)15 b(g)s Fr(\))28 b(=)d Fp(h)1320 5230 y Fo(x)1364 5216 y Fp(g)1407 5230 y Fo(y)1461 5216 y Fn(\000)12 b Fp(h)1596 5230 y Fo(y)1637 5216 y Fp(g)1680 5230 y Fo(x)1751 5216 y Fr(is)25 b(the)i(Jacobi)f(op)s(erator)h(and)e (\001)g(=)g Fp(@)3051 5230 y Fo(xx)3147 5216 y Fr(+)12 b Fp(@)3278 5230 y Fo(y)r(y)3383 5216 y Fr(is)p eop %%Page: 3 3 3 2 bop 1043 251 a Fs(STOCHASTIC)31 b(TW)n(O-LA)-6 b(YER)30 b(GEOPHYSICAL)h(FLO)n(WS)584 b(3)456 450 y Fr(the)32 b(Laplacian)f(op)s(erator.)46 b(Finally)-8 b(,)32 b Fp(f)10 b Fr(\()p Fp(x;)15 b(y)s(;)g(t)p Fr(\))32 b(is)f(the)i(mean)f (\(deterministic\))e(wind)456 558 y(forcing)f(with)g(a)m(v)m(erage)k (zero:)1517 485 y Fk(R)1560 590 y Fo(O)1635 558 y Fp(f)10 b(dO)27 b Fr(=)e(0)456 682 y(An)38 b(imp)s(ortan)m(t)f(part)i(of)g(the) f(ab)s(o)m(v)m(e)i(equation)e(is)g(the)h(white)e(noise)h(term)3196 659 y(_)3159 682 y Fp(W)12 b Fr(,)41 b(de-)456 790 y(scribing)34 b(the)j(\015uctuating)f(part)h(of)g(the)g(external)g(forcing)f(in)g (the)h(top)g(\015uid)d(la)m(y)m(er,)456 898 y(see)g(Hasselmann)g([22)q (])g(and)g(Arnold)e([2)q(].)53 b(An)33 b(example)h(of)h(these)f(short)g (time)g(scale)456 1006 y(in\015uences)29 b(is)h(the)i(w)m(eather)g(v)-5 b(ariabilit)m(y)28 b(or)k(wind)d(forcing.)42 b(A)31 b(white)g(noise)f (is)g(giv)m(en)456 1114 y(as)36 b(the)h(generalized)f(time)h(deriv)-5 b(ativ)m(e)36 b(of)h(a)g(Wiener)f(pro)s(cess)g Fp(W)13 b Fr(\()p Fp(t)p Fr(\))36 b(in)g(a)h(function)456 1222 y(space.)456 1330 y(W)-8 b(e)27 b(assume)g(p)s(erio)s(dic)d(b)s (oundary)g(conditions)h(for)h Fp( )k Fr(in)c Fp(x)g Fr(and)g Fp(y)j Fr(with)d(p)s(erio)s(d)e Fp(L)p Fr(.)39 b(In)456 1437 y(addition,)28 b(w)m(e)j(imp)s(ose)e(that)1701 1499 y Fk(Z)1751 1705 y Fo(O)1826 1623 y Fp( )s(dO)g Fr(=)c(0)p Fp(:)456 1831 y Fr(W)-8 b(e)31 b(also)f(assume)h(an)f(appropriate)f (initial)f(condition)1561 2012 y Fp(q)s Fr(\()p Fp(x;)15 b(y)s(;)g Fr(0\))27 b(=)e Fp(q)2064 2026 y Fs(0)2103 2012 y Fr(\()p Fp(x;)15 b(y)s Fr(\))p Fp(:)456 2194 y Fr(Sto)s(c)m(hastically)29 b(forced)h(QGE)g(has)g(b)s(een)f(used)g(to)i (in)m(v)m(estigate)g(v)-5 b(arious)29 b(phenomena)456 2301 y(in)f(geoph)m(ysical)h(\015o)m(ws)g([23)q(,)h(27)q(,)f(33)q(,)h (21)q(,)f(15)q(,)h(9].)41 b(Recen)m(tly)30 b(Salmon)e([32)q(])i(in)m (tro)s(duced)456 2409 y(some)g(generalized)g(t)m(w)m(o-la)m(y)m(er)j(o) s(cean)e(\015o)m(w)f(mo)s(dels.)456 2517 y(It)i(is)g(our)h(aim)f(to)h (study)f(the)h(long)g(time)f(dynamics)f(of)i(the)g(sto)s(c)m(hastic)h (di\013eren)m(tial)456 2625 y(equation)29 b(with)f(random)g(co)s (e\016cien)m(ts)j(but)d(without)h(white)f(noise)h(whic)m(h)f(w)m(e)i (obtain)456 2733 y(after)j(a)g(random)f(co)s(ordinate)h (transformation.)47 b(The)32 b(solution)f(of)i(this)f(di\013eren)m (tial)456 2841 y(equation)24 b(generates)h(a)g(random)e(dynamical)g (system.)39 b(The)23 b(structure)h(of)g(this)f(di\013er-)456 2949 y(en)m(tial)29 b(equation)g(enables)f(us)h(to)h(pro)m(v)m(e)g(the) f(dissipativit)m(y)e(of)i(the)h(system.)40 b(In)29 b(order)456 3057 y(to)c(in)m(v)m(estigate)h(the)f(long)f(time)g(dynamics)g(w)m(e)h (will)d(apply)h(the)i(metho)s(d)f(of)h Fj(determin-)456 3165 y(ing)i(functionals)f Fr(in)d(a)j(v)m(ersion)e(for)h(random)f (dynamical)f(systems)i(whic)m(h)f(is)g(based)g(on)456 3273 y(the)31 b(con)m(v)m(ergence)i(in)d(probabilit)m(y)-8 b(,)30 b(see)h(Ch)m(uesho)m(v)g(et.)44 b(al)31 b([13)q(].)43 b(It)31 b(follo)m(ws)f(that)i(the)456 3381 y(asymptotic)d(long)f(time)h (b)s(eha)m(vior)e(is)h(determined)g(b)m(y)g(the)h(asymptotic)g(b)s(eha) m(vior)f(of)456 3489 y(\014nitely)g(man)m(y)j(functionals.)40 b(F)-8 b(or)31 b(instance,)f(for)h(these)g(functionals)d(w)m(e)j(can)g (c)m(ho)s(ose)456 3597 y(the)g(F)-8 b(ourier)31 b(mo)s(des)f(giv)m(en)i (b)m(y)f(the)g(eigenfunctions)f(of)h(the)g(Laplacian.)43 b(According)456 3705 y(to)27 b(this)f(prop)s(ert)m(y)g(w)m(e)h(can)g (further)f(sho)m(w)g(that)i(the)f(functionals)e(de\014ned)g(on)i(the)g (top)456 3812 y(\015uid)33 b(la)m(y)m(er)j(alone)g(can)g Fj(determine)h Fr(the)e(asymptotical)h(b)s(eha)m(vior)f(of)h(the)g (complete)456 3920 y(t)m(w)m(o-la)m(y)m(er)30 b(system,)f(when)e(the)i (\015uid)d(parameters)i(and)g(the)g(Wiener)g(pro)s(cess)g(satisfy)456 4028 y(a)34 b(certain)g(condition,)g(suc)m(h)f(as)i(the)f(Ekman)f (constan)m(t)j Fp(r)g Fr(is)d(su\016cien)m(tly)g(large)h(and)456 4136 y(the)h(trace)i(of)f(the)f(co)m(v)-5 b(ariance)37 b(op)s(erator)f(of)f(the)h(Wiener)f(pro)s(cess)g(is)g(con)m(troled)g(b) m(y)456 4244 y(some)28 b(upp)s(er)d(b)s(ound.)38 b(Note)29 b(that)f(the)g(generalized)f(time)h(deriv)-5 b(ativ)m(e)27 b(of)h(the)g(Wiener)456 4352 y(pro)s(cess)38 b(mo)s(dels)g(the)i (\015uctuating)e(part)h(of)g(the)h(wind)d(stress)i(forcing)g(on)g(the)g (top)456 4460 y(\015uid)27 b(la)m(y)m(er,)k(and)e(the)h(Ekman)f (constan)m(t)i Fp(r)h Fr(measures)d(the)h(rate)h(for)e(v)m(orticit)m(y) h(deca)m(y)456 4568 y(due)f(to)i(the)g(friction)e(in)g(the)i(b)s(ottom) f(Ekman)g(la)m(y)m(er.)456 4676 y(W)-8 b(e)42 b(\014rst)f(recall)g (some)h(basic)f(facts)h(in)f(random)g(dynamical)f(systems)h(in)g (Section)456 4784 y(2.)49 b(In)32 b(Section)h(3,)h(w)m(e)g(establish)d (the)i(w)m(ell-p)s(osedness)f(of)h(the)g(sto)s(c)m(hastic)h(t)m(w)m (o-la)m(y)m(er)456 4892 y(quasigeostrophic)27 b(mo)s(del)f(b)m(y)i (transforming)f(it)g(in)m(to)h(a)g(coupled)f(system)h(of)g(random)456 5000 y(partial)37 b(di\013eren)m(tial)g(equations.)65 b(The)38 b(main)f(results)g(on)i(asymptotic)f(probabilis-)456 5108 y(tic)f(determining)f(functionals)g(are)h(presen)m(ted)h(in)e (Sections)i(4)g(and)f(5.)62 b(Finally)36 b(w)m(e)456 5216 y(summarize)29 b(our)h(conclusions)f(in)g(Section)h(6.)p eop %%Page: 4 4 4 3 bop 456 251 a Fs(4)362 b(IGOR)24 b(CHUESHO)n(V,)f(JINQIA)n(O)h (DUAN,)e(AND)h(BJ)2368 234 y(\177)2358 251 y(ORN)h(SCHMALFUSS)590 450 y Fr(2.)46 b Fq(Random)33 b(d)n(ynamical)g(system)g(and)g (determining)h(functionals)456 612 y Fr(It)c(is)f(our)g(goal)i(to)f (study)g(the)g(long)f(time)h(dynamics)f(of)h(\(1\))h(whic)m(h)e(is)g (in\015uenced)e(b)m(y)456 720 y(random)37 b(forces.)66 b(Appropriate)38 b(to)s(ols)g(to)h(treat)h(this)e(equation)g(are)h(giv) m(en)g(b)m(y)f(the)456 828 y(theory)30 b(of)h Fj(r)-5 b(andom)35 b(dynamic)-5 b(al)35 b(systems)p Fr(.)456 936 y(A)c(random)f(dynamical)f(system)j(consists)e(of)h(t)m(w)m(o)i (comp)s(onen)m(ts.)42 b(The)31 b(\014rst)f(comp)s(o-)456 1044 y(nen)m(t)f(is)g(a)h Fj(metric)i(dynamic)-5 b(al)34 b(system)c Fr(\(\012)p Fp(;)15 b Fn(F)9 b Fp(;)15 b Fi(P)p Fp(;)g(\022)s Fr(\))29 b(as)h(a)g(mo)s(del)e(for)i(a)f(noise,)h(where) 456 1152 y(\(\012)p Fp(;)15 b Fn(F)9 b Fp(;)15 b Fi(P)p Fr(\))30 b(is)f(a)h(probabilit)m(y)e(space)i(and)g Fp(\022)i Fr(is)d(a)h Fn(F)f(\012)19 b(B)s Fr(\()p Fi(R)p Fr(\))p Fp(;)d Fn(F)45 b Fr(measurable)29 b(\015o)m(w:)41 b(w)m(e)456 1260 y(ha)m(v)m(e)1281 1371 y Fp(\022)1324 1385 y Fs(0)1389 1371 y Fr(=)25 b(id)n Fp(;)198 b(\022)1826 1385 y Fo(t)p Fs(+)p Fo(\034)1974 1371 y Fr(=)25 b Fp(\022)2113 1385 y Fo(t)2163 1371 y Fn(\016)20 b Fp(\022)2271 1385 y Fo(\034)2339 1371 y Fr(=:)26 b Fp(\022)2504 1385 y Fo(t)2533 1371 y Fp(\022)2576 1385 y Fo(\034)456 1502 y Fr(for)36 b Fp(t;)31 b(\034)46 b Fn(2)35 b Fi(R)p Fr(.)66 b(The)36 b(measure)h Fi(P)e Fr(is)h(supp)s(osed)f(to)j(b)s(e)e(ergo)s(dic)g (with)g(resp)s(ect)g(to)i Fp(\022)s Fr(.)456 1610 y(The)g(second)g (comp)s(onen)m(t)h(of)f(a)h(random)f(dynamical)f(system)h(is)g(a)h Fn(B)s Fr(\()p Fi(R)3052 1577 y Fs(+)3117 1610 y Fr(\))25 b Fn(\012)h(F)35 b(\012)456 1718 y(B)s Fr(\()p Fp(H)7 b Fr(\))p Fp(;)15 b Fn(B)s Fr(\()p Fp(H)7 b Fr(\)-measurable)30 b(mapping)f Fp(')i Fr(satisfying)e(the)h Fj(c)-5 b(o)g(cycle)31 b Fr(prop)s(ert)m(y)860 1867 y Fp(')p Fr(\()p Fp(t)21 b Fr(+)f Fp(\034)5 b(;)15 b(!)s(;)g(x)p Fr(\))26 b(=)f Fp(')p Fr(\()p Fp(t;)15 b(\022)1703 1881 y Fo(\034)1747 1867 y Fp(!)s(;)g(')p Fr(\()p Fp(\034)5 b(;)15 b(!)s(;)g(x)p Fr(\)\))p Fp(;)200 b(')p Fr(\(0)p Fp(;)15 b(!)s(;)g(x)p Fr(\))27 b(=)e Fp(x;)456 2017 y Fr(where)32 b(the)i(phase)e(space)i Fp(H)40 b Fr(is)32 b(a)i(separable)e(metric)h(space)h(and)e Fp(x)h Fr(is)g(c)m(hosen)g(arbi-)456 2125 y(trarily)28 b(in)h Fp(H)7 b Fr(.)41 b(W)-8 b(e)32 b(will)27 b(denote)k(this)f (random)f(dynamical)g(system)i(b)m(y)f(sym)m(b)s(ol)f Fp(')p Fr(.)456 2233 y(A)f(standard)g(mo)s(del)f(for)h(suc)m(h)h(a)f (noise)g Fp(\022)j Fr(is)c(the)i(t)m(w)m(o-sided)g Fj(Br)-5 b(ownian)33 b(motion)p Fr(:)41 b(Let)456 2340 y Fp(U)f Fr(b)s(e)30 b(a)g(separable)g(Hilb)s(ert)f(space.)41 b(W)-8 b(e)31 b(consider)f(the)g(probabilit)m(y)e(space)1435 2490 y(\()p Fp(C)1535 2504 y Fs(0)1575 2490 y Fr(\()p Fi(R)p Fp(;)15 b(U)10 b Fr(\))q Fp(;)15 b Fn(B)s Fr(\()p Fp(C)2015 2504 y Fs(0)2060 2490 y Fr(\()p Fi(R)q Fp(;)g(U)10 b Fr(\)\))q Fp(;)15 b Fi(P)p Fr(\))456 2639 y(where)26 b Fp(C)780 2653 y Fs(0)819 2639 y Fr(\()p Fi(R)p Fp(;)16 b(U)10 b Fr(\))33 b(is)25 b(the)i(F)-8 b(r)m(\023)-43 b(ec)m(het)29 b(space)e(of)g(con)m(tin)m(uous)f(functions)f(on)h Fi(R)33 b Fr(of)27 b(uniform)456 2747 y(con)m(v)m(ergence)39 b(on)e(compact)h(in)m(terv)-5 b(als)36 b(whic)m(h)g(are)i(zero)g(at)f (zero)h(and)f Fn(B)s Fr(\()p Fp(C)3127 2761 y Fs(0)3166 2747 y Fr(\()p Fi(R)p Fp(;)16 b(U)10 b Fr(\)\))456 2855 y(is)37 b(the)h(corresp)s(onding)e(Borel)j Fp(\033)s Fr(-algebra.)64 b(Supp)s(ose)37 b(that)h(w)m(e)h(ha)m(v)m(e)g(a)g(co)m (v)-5 b(ariance)456 2963 y(op)s(erator)34 b Fp(Q)f Fr(on)h Fp(U)10 b Fr(.)51 b(Then)33 b Fi(P)f Fr(denotes)j(the)f Fj(Wiener)h(me)-5 b(asur)g(e)35 b Fr(with)e(resp)s(ect)h(to)g Fp(Q)p Fr(.)456 3071 y(Note)d(that)g Fi(P)e Fr(is)h(ergo)s(dic)g(with)f (resp)s(ect)h(to)h(the)g(\015o)m(w)456 3221 y(\(3\))525 b Fp(\022)1139 3235 y Fo(t)1169 3221 y Fp(!)28 b Fr(=)d Fp(!)s Fr(\()p Fn(\001)20 b Fr(+)g Fp(t)p Fr(\))h Fn(\000)f Fp(!)s Fr(\()p Fp(t)p Fr(\))p Fp(;)197 b Fr(for)30 b Fp(!)e Fn(2)d Fp(C)2521 3235 y Fs(0)2561 3221 y Fr(\()p Fi(R)p Fp(;)15 b(U)10 b Fr(\))456 3370 y(whic)m(h)29 b(is)g(called)h(the)g Fj(Wiener)j(shift)p Fr(.)456 3478 y(A)27 b(main)f(source)i(of)f(a)h(random)e(dynamical)g(system)i(is)e(a) i(random)e(di\013eren)m(tial)g(equa-)456 3586 y(tion.)42 b(F)-8 b(or)32 b(example,)g(let)f(us)f(consider)g(the)i(follo)m(wing)d (ev)m(olution)i(equation)g(in)f(some)456 3694 y(Hilb)s(ert)e(space)456 3886 y(\(4\))1408 3824 y Fp(du)p 1408 3865 100 4 v 1410 3948 a(d)15 b(t)1542 3886 y Fr(=)25 b Fp(F)13 b Fr(\()p Fp(u;)i(\022)1879 3900 y Fo(t)1909 3886 y Fp(!)s Fr(\))p Fp(;)107 b(u)p Fr(\(0\))26 b(=)f Fp(x;)456 4066 y Fr(o)m(v)m(er)46 b(some)f(metric)g(dynamical)e(system)i(\(\012)p Fp(;)15 b Fn(F)9 b Fp(;)15 b Fi(P)p Fp(;)g(\022)s Fr(\).)84 b(If)44 b(\(4\))i(is)e(w)m(ell-p)s(osed)g(for)456 4174 y(ev)m(ery)35 b Fp(!)h Fn(2)c Fr(\012)j(and)f(solutions)g Fp(u)p Fr(\()p Fp(t;)15 b(!)s Fr(;)g Fp(x)p Fr(\))36 b(dep)s(ends)d(measurably)g(on)i (\()p 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b(is)h Fn(1)h Fr(whic)m(h)f(describ)s(es)g(the)h(degenerated)i(case)f (of)f(sta-)456 2275 y(tionarit)m(y)-8 b(,)30 b(see)h(Arnold)e([1)q(],)i (page)g(164)g(f.)456 2491 y(Our)24 b(main)h(purp)s(ose)g(will)e(b)s(e)i (to)i(estimate)g(the)f(degree)h(of)f(freedom)g(of)g(the)g(long)g(time) 456 2599 y(dynamics)38 b(of)h(a)h(random)e(dynamical)g(systems)h(whic)m (h)f(stem)i(from)f(the)g(t)m(w)m(o-la)m(y)m(er)456 2707 y(\015o)m(w)d(problem)g(of)g(the)h(o)s(cean)h(in)m(tro)s(duced)d(ab)s (o)m(v)m(e.)61 b(W)-8 b(e)38 b(will)c(apply)h(the)i(theory)g(of)456 2815 y(determining)29 b(functionals)h(to)i(estimate)g(the)g(degree)g (of)f(freedom.)44 b(Since)31 b(dynamics)456 2923 y(of)40 b(this)f(mo)s(del)h(is)f(in\015uenced)f(b)m(y)j(random)e(forces)i(w)m (e)g(will)d(apply)h(the)i(theory)f(of)456 3030 y(determining)28 b(functionals)h(related)h(to)h(the)f(con)m(v)m(ergence)j(in)c (probabilit)m(y)-8 b(.)456 3138 y(W)g(e)31 b(no)m(w)g(giv)m(e)f(our)g (basic)g(de\014nition:)456 3269 y Fx(De\014nition)39 b(2.1.)k Fj(L)-5 b(et)36 b Fp(V)56 b Fj(b)-5 b(e)36 b(a)f(Banach)i(sp) -5 b(ac)g(e)37 b(which)g(is)e(c)-5 b(ontinuously)37 b(emb)-5 b(e)g(dde)g(d)456 3377 y(into)27 b Fp(H)7 b Fj(.)40 b(Assume)27 b(that)i(ther)-5 b(e)28 b(exists)f Fp(\034)35 b(>)25 b Fr(0)j Fj(such)f(that)i Fp(')p Fr(\()p Fp(t;)15 b(!)s(;)g(x)p Fr(\))27 b Fn(2)e Fp(L)2923 3344 y Fs(2)2923 3405 y Fo(l)q(oc)3013 3377 y Fr(\()p Fp(\034)5 b(;)15 b Fr(+)p Fn(1)p Fr(;)g Fp(V)21 b Fr(\))456 3485 y Fj(for)27 b(almost)i(al)5 b(l)28 b Fp(!)g Fn(2)d Fr(\012)h Fj(and)i Fp(x)e Fn(2)e Fp(H)7 b Fj(.)40 b(A)27 b(set)g Fn(L)e Fr(=)g Fn(f)p Fp(l)2240 3499 y Fo(j)2277 3485 y Fp(;)41 b(j)31 b Fr(=)25 b(1)p Fp(;)15 b Fn(\001)g(\001)g(\001)32 b Fp(;)15 b(N)10 b Fn(g)28 b Fj(of)f(line)-5 b(ar)28 b(c)-5 b(on-)456 3593 y(tinuous)38 b(and)h(line)-5 b(arly)40 b(indep)-5 b(endent)39 b(functionals)h(on)e Fp(V)59 b Fj(is)38 b(c)-5 b(al)5 b(le)-5 b(d)39 b(asymptotic)-5 b(al)5 b(ly)456 3701 y(determining)33 b(in)f(pr)-5 b(ob)g(ability)35 b(if)748 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Fr(\()p Fp(!)s Fr(\))26 b Fn(2)f Fp(B)5 b Fr(\()p Fp(!)s Fr(\))33 b Fj(we)f(have)725 2127 y Fp(V)21 b Fr(\()p Fp(t;)15 b(!)s Fr(\))21 b Fn(\000)f Fp(V)g Fr(\()p Fp(s;)15 b(!)s Fr(\))26 b Fn(\024)f Fp(C)1587 2141 y Fh(L)1659 2127 y Fn(\001)1705 2003 y Fk(Z)1796 2029 y Fo(t)1755 2209 y(s)1841 2127 y Fn(N)1916 2141 y Fh(L)1968 2127 y Fr(\()p Fp(\034)5 b(;)15 b(!)s Fr(\))p Fp(d\034)1421 2383 y Fr(+)1511 2259 y Fk(Z)1602 2285 y Fo(t)1562 2465 y(s)1647 2383 y Fp(l)1674 2397 y Fh(L)1727 2383 y Fr(\()p Fp(')p Fr(\()p Fp(\034)5 b(;)15 b(!)s(;)g(x)2093 2397 y Fs(1)2134 2383 y Fr(\))p Fp(;)g(')p Fr(\()p Fp(\034)5 b(;)15 b(!)s(;)g(x)2540 2397 y Fs(2)2582 2383 y Fr(\))p Fp(;)g(\022)2700 2397 y Fo(\034)2743 2383 y Fp(!)s Fr(\))21 b Fn(\001)f Fp(V)g Fr(\()p Fp(\034)5 b(;)15 b(!)s Fr(\))p Fp(d\034)456 2250 y Fr(\(5\))456 2592 y Fj(for)33 b(al)5 b(l)33 b Fp(t)25 b Fn(\025)g Fp(s)g Fn(\025)g Fr(0)p Fj(,)33 b(wher)-5 b(e)1053 2755 y Fp(V)20 b Fr(\()p Fp(t;)15 b(!)s Fr(\))26 b(=)f Fn(k)p Fp(')p Fr(\()p 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Fs(1)2039 3339 y Fr(\))p Fp(;)g(')p Fr(\()p Fp(\034)5 b(;)15 b(!)s(;)g(x)2445 3353 y Fs(2)2487 3339 y Fr(\))p Fp(;)g(\022)2605 3353 y Fo(\034)2648 3339 y Fp(!)s Fr(\))p Fp(d\034)2840 3184 y Fk(\))2939 3339 y Fp(<)25 b Fr(0)456 3579 y Fj(for)36 b(some)g Fp(t)31 b(>)g Fr(0)p Fj(.)52 b(Then)36 b Fn(L)f Fj(is)h(a)g(set)g(of)g (asymptotic)-5 b(al)5 b(ly)39 b(determining)d(functionals)456 3687 y(in)c(pr)-5 b(ob)g(ability)35 b(for)e(r)-5 b(andom)35 b(dynamic)-5 b(al)35 b(system)e Fp(')p Fj(.)456 3868 y(Pr)-5 b(o)g(of.)43 b Fr(It)30 b(is)f(easy)i(to)h(\014nd)c(from)i (relation)g(\(5\))h(that)763 4093 y Fp(V)21 b Fr(\()p Fp(t;)15 b(!)s Fr(\))26 b Fn(\024)f Fp(V)20 b Fr(\(0)p Fp(;)15 b(!)s Fr(\))p Fp(e)1492 4002 y Ff(R)1541 4022 y Fm(t)1527 4080 y Fl(0)1580 4055 y Fo(l)1601 4066 y Fe(L)1648 4055 y Fs(\()p Fo(s;!)r Fs(\))p Fo(ds)1895 4093 y Fr(+)20 b Fp(C)2051 4117 y Fe(L)2117 3969 y Fk(Z)2208 3995 y Fo(t)2167 4175 y Fs(0)2252 4093 y Fn(N)2327 4107 y Fh(L)2380 4093 y Fr(\()p Fp(\034)5 b(;)15 b(!)s Fr(\))p Fp(e)2637 4002 y Ff(R)2685 4022 y Fm(t)2671 4080 y(\034)2725 4055 y Fo(l)2746 4066 y Fe(L)2792 4055 y Fs(\()p Fo(s;!)r Fs(\))p Fo(ds)3019 4093 y Fp(d\034)5 b(;)456 4307 y Fr(where)35 b Fp(l)r Fr(\()p Fp(t;)15 b(!)s Fr(\))35 b(=)g Fp(l)1124 4321 y Fh(L)1177 4307 y Fr(\()p Fp(')p Fr(\()p Fp(t;)15 b(!)s(;)g(x)1531 4321 y Fs(1)1572 4307 y Fr(\))p Fp(;)g(')p Fr(\()p Fp(t;)g(!)s(;)g(x)1966 4321 y Fs(2)2007 4307 y Fr(\))p Fp(;)g(\022)2125 4321 y Fo(t)2155 4307 y Fp(!)s Fr(\).)58 b(Therefore)36 b(w)m(e)g(can)g(apply)f(the)456 4415 y(argumen)m(t)30 b(giv)m(en)h(in)e(the)h(pro)s(of)g(of)h(Theorem)f (2.2)h([13)q(].)974 b Fd(\003)456 4595 y Fr(In)27 b(the)h(next)g (section,)h(w)m(e)f(return)f(to)i(the)f(t)m(w)m(o-la)m(y)m(er)i (quasigeostrophic)d(\015o)m(w)h(mo)s(del.)593 4829 y(3.)46 b Fq(Ma)-6 b(thema)g(tical)32 b(setup)h(and)g(well-posedness)e(of)j (the)g(random)1618 4937 y(tw)n(o-la)-6 b(yer)33 b(flo)n(ws)456 5099 y Fr(In)j(the)h(follo)m(wing,)h Fp(L)1219 5066 y Fs(2)1219 5121 y Fo(per)1325 5099 y Fr(,)h Fp(H)1472 5066 y Fo(s)1465 5121 y(per)1608 5099 y Fr(for)e Fp(s)f Fn(2)g Fg(R)45 b Fr(are)38 b(the)f(standard)g(Sob)s(olev)f(spaces)h(of) 456 5216 y Fp(L)p Fr(-p)s(erio)s(dic)30 b(functions)i(with)f(the)i (zero)h(mean)f(v)-5 b(alue,)33 b(i.e.)2530 5142 y Fk(R)2573 5247 y Fo(O)2648 5216 y Fp( )s(dO)g Fr(=)c(0.)49 b(Let)33 b(\()p Fn(\001)p Fp(;)15 b Fn(\001)p Fr(\))3403 5230 y Fs(0)p eop %%Page: 7 7 7 6 bop 1043 251 a Fs(STOCHASTIC)31 b(TW)n(O-LA)-6 b(YER)30 b(GEOPHYSICAL)h(FLO)n(WS)584 b(7)456 450 y Fr(and)25 b Fn(k)11 b(\001)g(k)765 464 y Fs(0)830 450 y Fr(denote)27 b(the)e(standard)g(scalar)h(pro)s(duct)e(and)h(norm)g(in)f Fp(L)2803 417 y Fs(2)2803 473 y Fo(per)2910 450 y Fr(,)i(resp)s(ectiv)m (ely)-8 b(.)456 570 y(Ev)m(ery)30 b(elemen)m(t)h Fp(u)p Fr(\()p Fp(x;)15 b(y)s Fr(\))26 b Fn(2)f Fp(L)1489 537 y Fs(2)1489 592 y Fo(per)1626 570 y Fr(can)30 b(b)s(e)g(represen)m(ted) g(in)f(the)i(form)1011 785 y Fp(u)p Fr(\()p Fp(x;)15 b(y)s Fr(\))26 b(=)1480 699 y Fk(X)1395 901 y Fo(j)t Fh(2)p Fc(Z)1525 882 y Fl(2)1554 901 y Fo(;j)t Fh(6)p Fs(=0)1711 785 y Fp(u)1763 799 y Fo(j)1800 785 y Fp(L)1862 747 y Fh(\000)p Fs(1)1971 785 y Fr(exp)2125 657 y Fk(\032)2193 785 y Fp(i)2234 724 y Fr(2)p Fp(\031)p 2234 764 101 4 v 2254 847 a(L)2345 785 y Fr(\()p Fp(j)2417 799 y Fs(1)2457 785 y Fp(x)20 b Fr(+)g Fp(j)2657 799 y Fs(2)2697 785 y Fp(y)s Fr(\))2780 657 y Fk(\033)2864 785 y Fp(;)456 1040 y Fr(where)25 b(the)h(F)-8 b(ourier)26 b(co)s(e\016cien)m(ts)g Fp(u)1684 1054 y Fo(j)1747 1040 y Fr(p)s(ossess)f(the)h(prop)s(ert)m(y) 31 b(\026)-51 b Fp(u)2626 1054 y Fo(j)2688 1040 y Fr(=)25 b Fp(u)2836 1054 y Fh(\000)p Fo(j)2953 1040 y Fr(\(bar)h(denotes)456 1148 y(the)k(complex)g(conjugation\))h(and)1166 1342 y Fn(k)p Fp(u)p Fn(k)1308 1305 y Fs(2)1308 1365 y(0)1374 1342 y Fr(:=)1495 1218 y Fk(Z)1545 1425 y Fo(O)1620 1342 y Fn(j)p Fp(u)p Fn(j)1722 1305 y Fs(2)1762 1342 y Fp(dO)e Fn(\021)2087 1256 y Fk(X)2003 1458 y Fo(j)t Fh(2)p Fc(Z)2132 1439 y Fl(2)2161 1458 y Fo(;j)t Fh(6)p Fs(=0)2318 1342 y Fn(j)p Fp(u)2395 1356 y Fo(j)2432 1342 y Fn(j)2457 1305 y Fs(2)2522 1342 y Fp(<)c Fn(1)p Fp(:)456 1607 y Fr(Note)30 b(that)g Fp(L)932 1574 y Fh(\000)p Fs(1)1042 1607 y Fr(exp)1196 1534 y Fk(\010)1249 1607 y Fp(i)1290 1571 y Fs(2)p Fo(\031)p 1290 1586 79 4 v 1305 1639 a(L)1378 1607 y Fr(\()p Fp(j)1450 1621 y Fs(1)1490 1607 y Fp(x)20 b Fr(+)g Fp(j)1690 1621 y Fs(2)1730 1607 y Fp(y)s Fr(\))1813 1534 y Fk(\011)1896 1607 y Fr(is)28 b(the)i(eigenelemen)m(t)f(of)h(the) f Fn(\000)p Fr(\001)g(with)f(the)456 1742 y(eigen)m(v)-5 b(alue)33 b Fp(\025)946 1756 y Fs(1)1007 1742 y Fn(\001)22 b Fr(\()p Fp(j)1131 1709 y Fs(2)1126 1767 y(1)1194 1742 y Fr(+)f Fp(j)1328 1709 y Fs(2)1323 1767 y(2)1368 1742 y Fr(\),)35 b(where)d Fp(\025)1781 1756 y Fs(1)1850 1742 y Fr(=)1951 1669 y Fk(\000)2002 1707 y Fs(2)p Fo(\031)p 2002 1722 V 2018 1774 a(L)2091 1669 y Fk(\001)2132 1690 y Fs(2)2205 1742 y Fr(is)g(the)h(smallest)f(eigen)m(v)-5 b(alue.)49 b(The)456 1850 y(norm)29 b(in)g Fp(H)882 1817 y Fo(s)875 1873 y(per)1011 1850 y Fr(is)h(de\014ned)f(b)m(y)h(the)h (form)m(ula)897 2058 y Fn(k)p Fp(u)p Fn(k)1039 2020 y Fs(2)1039 2080 y Fo(s)1104 2058 y Fr(:=)1225 1934 y Fk(Z)1276 2140 y Fo(O)1351 2058 y Fn(j)p Fr(\()p Fn(\000)p Fr(\001\))1604 1970 y Fm(s)p 1603 1978 31 3 v 1603 2013 a Fl(2)1652 2058 y Fp(u)p Fn(j)1729 2020 y Fs(2)1769 2058 y Fp(dO)d Fn(\021)d Fp(\025)2062 2020 y Fo(s)2062 2080 y Fs(1)2201 1971 y Fk(X)2117 2174 y Fo(j)t Fh(2)p Fc(Z)2246 2155 y Fl(2)2275 2174 y Fo(;j)t Fh(6)p Fs(=0)2417 2058 y Fr(\()p Fp(j)2494 2020 y Fs(2)2489 2080 y(1)2555 2058 y Fr(+)20 b Fp(j)2688 2020 y Fs(2)2683 2080 y(2)2728 2058 y Fr(\))2763 2020 y Fo(s)2800 2058 y Fn(j)p Fp(u)2877 2072 y Fo(j)2914 2058 y Fn(j)2939 2020 y Fs(2)2978 2058 y Fp(:)456 2307 y Fr(It)30 b(is)g(clear)g(that)456 2455 y(\(6\))464 b Fn(kr)p Fp(u)p Fn(k)1253 2418 y Fs(2)1253 2478 y Fo(s)1319 2455 y Fr(:=)25 b Fn(k)p Fp(@)1533 2469 y Fo(x)1578 2455 y Fp(u)p Fn(k)1675 2418 y Fs(2)1675 2478 y Fo(s)1735 2455 y Fr(+)20 b Fn(k)p Fp(@)1919 2469 y Fo(y)1961 2455 y Fp(u)p Fn(k)2058 2418 y Fs(2)2058 2478 y Fo(s)2123 2455 y Fr(=)25 b Fn(k)p Fp(u)p Fn(k)2361 2418 y Fs(2)2361 2478 y(1+)p Fo(s)2489 2455 y Fp(;)106 b(s)25 b Fn(2)g Fg(R)r Fp(:)456 2608 y Fr(W)-8 b(e)31 b(also)f(denote)h Fx(L)1150 2575 y Fs(2)1150 2631 y Fo(per)1282 2608 y Fr(=)25 b Fp(L)1440 2575 y Fs(2)1440 2631 y Fo(per)1566 2608 y Fn(\002)20 b Fp(L)1719 2575 y Fs(2)1719 2631 y Fo(per)1855 2608 y Fr(and)30 b Fx(H)2114 2575 y Fo(s)2114 2631 y(per)2245 2608 y Fr(=)25 b Fp(H)2424 2575 y Fo(s)2417 2631 y(per)2543 2608 y Fn(\002)20 b Fp(H)2717 2575 y Fo(s)2710 2631 y(per)2816 2608 y Fr(.)456 2728 y(W)-8 b(e)31 b(w)m(ork)g(on)f(the)g(phase)h(space)f Fx(H)1695 2695 y Fh(\000)p Fs(1)1695 2750 y Fo(per)1832 2728 y Fr(with)f(the)h(scalar)h(pro)s(duct)649 2887 y(\()p Fp(q)s(;)22 b Fr(\026)-52 b Fp(q)s Fr(\))847 2901 y Fh(\003)912 2887 y Fr(=)25 b Fp(h)1060 2901 y Fs(1)1100 2887 y Fr(\()p Fn(r)p Fp( )1270 2901 y Fs(1)1310 2887 y Fp(;)15 b Fn(r)1445 2863 y Fr(\026)1426 2887 y Fp( )1485 2901 y Fs(1)1525 2887 y Fr(\))1560 2901 y Fs(0)1620 2887 y Fr(+)20 b Fp(h)1763 2901 y Fs(2)1802 2887 y Fr(\()p Fn(r)p Fp( )1972 2901 y Fs(2)2012 2887 y Fp(;)15 b Fn(r)2147 2863 y Fr(\026)2128 2887 y Fp( )2187 2901 y Fs(2)2227 2887 y Fr(\))2262 2901 y Fs(0)2322 2887 y Fr(+)20 b Fp(p)p Fr(\()p Fp( )2553 2901 y Fs(1)2613 2887 y Fn(\000)g Fp( )2763 2901 y Fs(2)2802 2887 y Fp(;)2861 2863 y Fr(\026)2842 2887 y Fp( )2901 2901 y Fs(1)2962 2887 y Fn(\000)3071 2863 y Fr(\026)3053 2887 y Fp( )3112 2901 y Fs(2)3151 2887 y Fr(\))3186 2901 y Fs(0)3226 2887 y Fp(;)456 3035 y Fr(where)31 b Fp(q)f Fr(=)d(\()p Fp(q)965 3049 y Fs(1)1004 3035 y Fp(;)15 b(q)1085 3049 y Fs(2)1124 3035 y Fr(\),)40 b(\026)-52 b Fp(q)30 b Fr(=)d(\()7 b(\026)-52 b Fp(q)1462 3049 y Fs(1)1501 3035 y Fp(;)22 b Fr(\026)-52 b Fp(q)1582 3049 y Fs(2)1622 3035 y Fr(\))31 b(and)g Fp( )g Fr(=)c(\()p Fp( )2148 3049 y Fs(1)2188 3035 y Fp(;)15 b( )2287 3049 y Fs(2)2327 3035 y Fr(\))28 b Fn(2)f Fx(H)2560 3002 y Fs(1)2560 3057 y Fo(per)2666 3035 y Fr(,)2741 3011 y(\026)2723 3035 y Fp( )j Fr(=)d(\()2964 3011 y(\026)2945 3035 y Fp( )3004 3049 y Fs(1)3044 3035 y Fp(;)3103 3011 y Fr(\026)3084 3035 y Fp( )3143 3049 y Fs(2)3183 3035 y Fr(\).)45 b(The)456 3154 y(relation)28 b(b)s(et)m(w)m(een)i Fp(q)h Fr(\(resp.)47 b(\026)-52 b Fp(q)s Fr(\))29 b(and)g Fp( )j Fr(\(resp.)2130 3130 y(\026)2111 3154 y Fp( )t Fr(\))d(is)f(de\014ned)g(b)m(y)h(\(2\).) 41 b(Here)30 b(w)m(e)f(also)456 3262 y(use)h(the)g(notation)1645 3410 y Fp(p)25 b Fr(=)1822 3348 y Fp(f)1877 3315 y Fs(2)1867 3373 y(0)p 1822 3389 94 4 v 1846 3472 a Fp(g)2034 3348 y(\032)2081 3362 y Fs(0)p 1936 3389 284 4 v 1936 3472 a Fp(\032)1983 3486 y Fs(2)2042 3472 y Fn(\000)20 b Fp(\032)2180 3486 y Fs(1)2230 3410 y Fp(:)456 3582 y Fr(Note)31 b(that)g Fp(F)930 3596 y Fs(1)970 3582 y Fp(h)1022 3596 y Fs(1)1087 3582 y Fr(=)25 b Fp(F)1241 3596 y Fs(2)1281 3582 y Fp(h)1333 3596 y Fs(2)1398 3582 y Fr(=)g Fp(p)p Fr(.)41 b(The)29 b(norm)h(induced)e(b)m(y)j(this)e(scalar)h(pro)s(duct)873 3730 y Fn(k)p Fp(q)s Fn(k)1007 3693 y Fs(2)1007 3753 y Fh(\003)1072 3730 y Fr(=)25 b(\()p Fp(q)s(;)15 b(q)s Fr(\))1366 3744 y Fh(\003)1432 3730 y Fr(=)24 b Fp(h)1579 3744 y Fs(1)1619 3730 y Fn(kr)p Fp( )1799 3744 y Fs(1)1839 3730 y Fn(k)1884 3693 y Fs(2)1884 3753 y(0)1944 3730 y Fr(+)c Fp(h)2087 3744 y Fs(2)2127 3730 y Fn(kr)p Fp( )2307 3744 y Fs(2)2347 3730 y Fn(k)2392 3693 y Fs(2)2392 3753 y(0)2452 3730 y Fr(+)g Fp(p)p Fn(k)p Fp( )2693 3744 y Fs(1)2753 3730 y Fn(\000)g Fp( )2903 3744 y Fs(2)2942 3730 y Fn(k)2987 3693 y Fs(2)2987 3753 y(0)456 3878 y Fr(is)29 b(equiv)-5 b(alen)m(t)30 b(to)h(the)g(usual)d(norm)i(on)g Fx(H)1930 3845 y Fh(\000)p Fs(1)1930 3901 y Fo(per)2036 3878 y Fr(.)41 b(Moreo)m(v)m(er,)33 b(w)m(e)e(ha)m(v)m(e)g(the)g (estimate)456 4042 y(\(7\))153 b Fp(h)776 4056 y Fs(1)816 4042 y Fn(kr)p Fp( )996 4056 y Fs(1)1036 4042 y Fn(k)1081 4005 y Fs(2)1081 4065 y(0)1141 4042 y Fr(+)20 b Fp(h)1284 4056 y Fs(2)1324 4042 y Fn(kr)p Fp( )1504 4056 y Fs(2)1544 4042 y Fn(k)1589 4005 y Fs(2)1589 4065 y(0)1654 4042 y Fn(\024)25 b(k)p Fp(q)s Fn(k)1884 4005 y Fs(2)1884 4065 y Fh(\003)1949 4042 y Fn(\024)g Fp(a)2093 4056 y Fs(0)2148 3968 y Fk(\000)2189 4042 y Fp(h)2241 4056 y Fs(1)2281 4042 y Fn(kr)p Fp( )2461 4056 y Fs(1)2501 4042 y Fn(k)2546 4005 y Fs(2)2546 4065 y(0)2606 4042 y Fr(+)20 b Fp(h)2749 4056 y Fs(2)2789 4042 y Fn(kr)p Fp( )2969 4056 y Fs(2)3009 4042 y Fn(k)3054 4005 y Fs(2)3054 4065 y(0)3094 3968 y Fk(\001)3150 4042 y Fp(;)456 4190 y Fr(where)1007 4320 y Fp(a)1055 4334 y Fs(0)1120 4320 y Fr(=)25 b(1)20 b(+)1624 4259 y(2)p Fp(p)p 1382 4300 575 4 v 1382 4383 a(\025)1435 4397 y Fs(1)1490 4383 y Fr(min)n Fn(f)p Fp(h)1738 4397 y Fs(1)1779 4383 y Fp(;)15 b(h)1871 4397 y Fs(2)1911 4383 y Fn(g)1991 4320 y Fr(=)25 b(1)c(+)2277 4259 y(2)p 2254 4300 93 4 v 2254 4383 a Fp(\025)2307 4397 y Fs(1)2372 4320 y Fr(max)p Fn(f)p Fp(F)2644 4334 y Fs(1)2684 4320 y Fp(;)15 b(F)2782 4334 y Fs(2)2822 4320 y Fn(g)p Fp(:)456 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b(\024)f Fp(c)1556 5227 y Fs(0)1596 5213 y Fn(k)p Fr(\001)p Fp(v)s Fn(k)1809 5227 y Fs(0)1869 5213 y Fn(\001)c(kr)p Fp(u)p Fn(k)2133 5227 y Fs(0)2193 5213 y Fn(\001)f(k)p Fr(\001)p Fp(u)p Fn(k)2456 5227 y Fs(0)2497 5213 y Fp(;)108 b(u;)30 b(v)f Fn(2)c Fp(H)2979 5176 y Fs(2)2972 5236 y Fo(per)3078 5213 y Fr(;)p eop %%Page: 8 8 8 7 bop 456 251 a Fs(8)362 b(IGOR)24 b(CHUESHO)n(V,)f(JINQIA)n(O)h (DUAN,)e(AND)h(BJ)2368 234 y(\177)2358 251 y(ORN)h(SCHMALFUSS)456 450 y Fr(\(11\))121 b Fn(j)p Fr(\()p Fp(J)9 b Fr(\()p Fp(u;)15 b(v)s Fr(\))p Fp(;)g(w)r Fr(\))1207 464 y Fs(0)1250 450 y Fn(j)25 b(\024)g Fp(c)1435 464 y Fs(1)1475 450 y Fn(k)p Fr(\001)p Fp(u)p Fn(k)1693 464 y Fs(0)1754 450 y Fn(\001)20 b(k)p Fr(\001)p Fp(v)s Fn(k)2012 464 y Fs(0)2073 450 y Fn(\001)g(k)p Fp(w)r Fn(k)2275 464 y Fs(0)2316 450 y Fp(;)108 b(u;)31 b(v)d Fn(2)d Fp(H)2798 413 y Fs(2)2791 473 y Fo(per)2897 450 y Fp(;)31 b(w)c Fn(2)e Fp(L)3193 413 y Fs(2)3193 473 y Fo(per)3299 450 y Fr(;)456 654 y(\(12\))76 b Fn(j)p Fr(\()p Fp(J)9 b Fr(\()p Fp(u;)15 b(v)s Fr(\))p Fp(;)g(w)r Fr(\))1162 668 y Fs(0)1205 654 y Fn(j)26 b(\024)f Fp(c)1391 668 y Fs(1)1430 654 y Fn(kr)p Fp(u)p Fn(k)1648 668 y Fs(0)1709 654 y Fn(\001)20 b(k)p Fr(\001)p Fp(v)s Fn(k)1967 668 y Fs(0)2028 654 y Fn(\001)g(kr)p Fp(w)r Fn(k)2306 668 y Fs(0)2347 654 y Fp(;)108 b(u;)31 b(w)d Fn(2)d Fp(H)2850 616 y Fs(1)2843 676 y Fo(per)2963 654 y Fp(;)15 b(v)29 b Fn(2)c Fp(H)3245 616 y Fs(2)3238 676 y Fo(per)3344 654 y Fp(:)456 854 y Fj(Her)-5 b(e)32 b(ab)-5 b(ove)33 b Fp(c)958 868 y Fs(0)1023 854 y Fr(=)25 b(2)c(+)f(\()1311 778 y Fn(p)p 1387 778 46 4 v 76 x Fr(2)h Fn(\001)f Fp(\031)s Fr(\))1588 821 y Fh(\000)p Fs(1)1715 854 y Fj(and)34 b Fp(c)1931 868 y Fs(1)1996 854 y Fr(=)25 b Fp(c)2131 868 y Fs(0)2171 854 y Fp(\025)2224 796 y Fh(\000)2289 769 y Fl(1)p 2289 781 31 3 v 2289 822 a(2)2224 880 y Fs(1)2333 854 y Fj(.)456 1014 y(Pr)-5 b(o)g(of.)43 b Fr(W)-8 b(e)31 b(start)g(with)e(\(10\).)42 b(It)31 b(is)e(easy)i(to)g(see)g(that)974 1205 y Fp(J)9 b Fr(\()p Fp(u;)15 b(v)s Fr(\))22 b Fn(\001)e Fr(\001)p Fp(u)25 b Fr(=)1543 1143 y(1)p 1543 1184 46 4 v 1543 1267 a(2)1619 1205 y Fn(\001)20 b Fp(v)1708 1219 y Fo(y)1765 1131 y Fk(\010)1818 1205 y Fp(@)1866 1219 y Fo(x)1925 1131 y Fk(\000)1967 1205 y Fp(u)2019 1167 y Fs(2)2019 1227 y Fo(x)2083 1205 y Fn(\000)g Fp(u)2226 1167 y Fs(2)2226 1227 y Fo(y)2267 1131 y Fk(\001)2329 1205 y Fr(+)g(2)p Fp(@)2513 1219 y Fo(y)2570 1205 y Fr(\()p Fp(u)2657 1219 y Fo(x)2701 1205 y Fp(u)2753 1219 y Fo(y)2795 1205 y Fr(\))2830 1131 y Fk(\011)1462 1423 y Fr(+)1543 1361 y(1)p 1543 1402 V 1543 1485 a(2)1619 1423 y Fn(\001)g Fp(v)1708 1437 y Fo(x)1767 1349 y Fk(\010)1820 1423 y Fp(@)1868 1437 y Fo(y)1925 1349 y Fk(\000)1967 1423 y Fp(u)2019 1385 y Fs(2)2019 1445 y Fo(x)2083 1423 y Fn(\000)g Fp(u)2226 1385 y Fs(2)2226 1445 y Fo(y)2267 1349 y Fk(\001)2329 1423 y Fn(\000)g Fr(2)p Fp(@)2513 1437 y Fo(x)2573 1423 y Fr(\()p Fp(u)2660 1437 y Fo(x)2704 1423 y Fp(u)2756 1437 y Fo(y)2797 1423 y Fr(\))2833 1349 y Fk(\011)2901 1423 y Fp(:)456 1601 y Fr(Consequen)m(tly)29 b(relying)g(on)h(prop)s (ert)m(y)g(\(6\))h(w)m(e)g(ha)m(v)m(e)589 1792 y Fn(j)p Fr(\()p Fp(J)9 b Fr(\()p Fp(u;)15 b(v)s Fr(\))p Fp(;)g Fr(\001)p Fp(u)p Fr(\))1120 1806 y Fs(0)1162 1792 y Fn(j)26 b(\024)1293 1731 y Fr(1)p 1293 1771 V 1293 1854 a(2)1369 1792 y Fn(\001)20 b(k)p Fp(v)1503 1806 y Fo(y)1545 1792 y Fn(k)1590 1806 y Fs(1)1645 1718 y Fk(\000)1687 1792 y Fn(k)p Fp(@)1780 1806 y Fo(x)1840 1718 y Fk(\000)1881 1792 y Fp(u)1933 1755 y Fs(2)1933 1815 y Fo(x)1997 1792 y Fn(\000)g Fp(u)2140 1755 y Fs(2)2140 1815 y Fo(y)2182 1718 y Fk(\001)2239 1792 y Fn(k)2284 1806 y Fh(\000)p Fs(1)2399 1792 y Fr(+)f(2)p Fn(k)p Fp(@)2627 1806 y Fo(y)2685 1792 y Fr(\()q Fp(u)2773 1806 y Fo(x)2817 1792 y Fp(u)2869 1806 y Fo(y)2910 1792 y Fr(\))c Fn(k)3005 1806 y Fh(\000)p Fs(1)3100 1718 y Fk(\001)1213 2010 y Fr(+)1293 1948 y(1)p 1293 1989 V 1293 2072 a(2)1369 2010 y Fn(\001)20 b(k)p Fp(v)1503 2024 y Fo(x)1548 2010 y Fn(k)1593 2024 y Fs(1)1648 1936 y Fk(\000)1689 2010 y Fn(k)p Fp(@)1782 2024 y Fo(y)1840 1936 y Fk(\000)1881 2010 y Fp(u)1933 1972 y Fs(2)1933 2032 y Fo(x)1997 2010 y Fn(\000)g Fp(u)2140 1972 y Fs(2)2140 2032 y Fo(y)2182 1936 y Fk(\001)2239 2010 y Fn(k)2284 2024 y Fh(\000)p Fs(1)2399 2010 y Fr(+)f(2)p Fn(k)p Fp(@)2627 2024 y Fo(x)2688 2010 y Fr(\()p Fp(u)2775 2024 y Fo(x)2819 2010 y Fp(u)2871 2024 y Fo(y)2912 2010 y Fr(\))d Fn(k)3008 2024 y Fh(\000)p Fs(1)3103 1936 y Fk(\001)1213 2228 y Fn(\024)1293 2166 y Fr(1)p 1293 2207 V 1293 2290 a(2)1369 2228 y Fn(\001)k Fr(\()q Fn(k)p Fp(v)1539 2242 y Fo(x)1583 2228 y Fn(k)1628 2242 y Fs(1)1688 2228 y Fr(+)g Fn(k)p Fp(v)1868 2242 y Fo(y)1910 2228 y Fn(k)1955 2242 y Fs(1)1995 2228 y Fr(\))g Fn(\001)2096 2154 y Fk(\000)2138 2228 y Fn(k)p Fp(u)2235 2242 y Fo(x)2279 2228 y Fn(k)2324 2190 y Fs(2)2324 2256 y Fo(L)2372 2237 y Fl(4)2431 2228 y Fr(+)g Fn(k)p Fp(u)2619 2242 y Fo(y)2661 2228 y Fn(k)2706 2190 y Fs(2)2706 2256 y Fo(L)2754 2237 y Fl(4)2813 2228 y Fr(+)g(2)p Fn(k)p Fp(u)3046 2242 y Fo(x)3091 2228 y Fp(u)3143 2242 y Fo(y)3185 2228 y Fn(k)3230 2242 y Fs(0)3270 2154 y Fk(\001)1213 2406 y Fn(\024)14 b Fr(\()q Fn(k)p Fp(v)1423 2420 y Fo(x)1467 2406 y Fn(k)1512 2420 y Fs(1)1572 2406 y Fr(+)20 b Fn(k)p Fp(v)1752 2420 y Fo(y)1794 2406 y Fn(k)1839 2420 y Fs(1)1879 2406 y Fr(\))g Fn(\001)1980 2332 y Fk(\000)2022 2406 y Fn(k)p Fp(u)2119 2420 y Fo(x)2163 2406 y Fn(k)2208 2368 y Fs(2)2208 2434 y Fo(L)2256 2415 y Fl(4)2315 2406 y Fr(+)g Fn(k)p Fp(u)2503 2420 y Fo(y)2545 2406 y Fn(k)2590 2368 y Fs(2)2590 2434 y Fo(L)2638 2415 y Fl(4)2677 2332 y Fk(\001)1213 2564 y Fn(\024)1283 2485 y(p)p 1359 2485 V 79 x Fr(2)h Fn(\001)f(k)p Fr(\001)p Fp(v)s Fn(k)1683 2578 y Fs(0)1744 2564 y Fn(\001)1789 2491 y Fk(\000)1831 2564 y Fn(k)p Fp(u)1928 2578 y Fo(x)1972 2564 y Fn(k)2017 2527 y Fs(2)2017 2592 y Fo(L)2065 2573 y Fl(4)2125 2564 y Fr(+)g Fn(k)p Fp(u)2313 2578 y Fo(y)2354 2564 y Fn(k)2399 2527 y Fs(2)2399 2592 y Fo(L)2447 2573 y Fl(4)2487 2491 y Fk(\001)2543 2564 y Fp(:)456 2717 y Fr(Using)29 b(the)i(inequalit)m(y)d(\(see,)k(e.g.,)g([18)q(]\))712 2916 y Fn(k)p Fp(u)p Fn(k)854 2936 y Fo(L)902 2917 y Fl(4)967 2916 y Fn(\024)25 b Fp(a)1111 2930 y Fs(1)1171 2916 y Fn(\001)20 b(k)p Fp(u)p Fn(k)1358 2878 y Fs(1)p Fo(=)p Fs(2)1489 2916 y Fn(\001)h(kr)p Fp(u)p Fn(k)1753 2878 y Fs(1)p Fo(=)p Fs(2)1955 2916 y Fr(with)89 b Fp(a)2270 2930 y Fs(1)2335 2916 y Fr(=)2431 2815 y Fk(\020)2485 2916 y Fr(\(2)p Fp(\031)s Fr(\))2655 2878 y Fh(\000)p Fs(1)2771 2916 y Fr(+)2862 2836 y Fn(p)p 2938 2836 V 80 x Fr(2)2983 2815 y Fk(\021)3037 2838 y Fs(1)p Fo(=)p Fs(2)3163 2916 y Fp(;)456 3091 y Fr(w)m(e)30 b(obtain)g(\(10\).)456 3199 y(In)f(a)i(similar)d(w)m(a)m(y)j(w)m(e)g(can)g(establish)d(the)j (estimates)685 3363 y Fn(j)p Fr(\()p Fp(J)9 b Fr(\()p Fp(u;)15 b(v)s Fr(\))p Fp(;)g(w)r Fr(\))1155 3377 y Fs(0)1198 3363 y Fn(j)26 b(\024)f Fp(c)1384 3377 y Fs(0)1424 3363 y Fn(kr)p Fp(u)p Fn(k)1642 3315 y Fs(1)p Fo(=)p Fs(2)1642 3389 y(0)1772 3363 y Fn(\001)c(k)p Fr(\001)p Fp(u)p Fn(k)2036 3315 y Fs(1)p Fo(=)p Fs(2)2036 3389 y(0)2167 3363 y Fn(\001)f(kr)p Fp(v)s Fn(k)2425 3315 y Fs(1)p Fo(=)p Fs(2)2425 3389 y(0)2556 3363 y Fn(\001)h(k)p Fr(\001)p Fp(v)s Fn(k)2815 3315 y Fs(1)p Fo(=)p Fs(2)2815 3389 y(0)2946 3363 y Fn(\001)g(k)p Fp(w)r Fn(k)3149 3377 y Fs(0)3189 3363 y Fp(;)685 3536 y Fn(j)p Fr(\()p Fp(J)9 b Fr(\()p Fp(u;)15 b(v)s Fr(\))p Fp(;)g(w)r Fr(\))1155 3550 y Fs(0)1198 3536 y Fn(j)26 b(\024)f Fp(c)1384 3550 y Fs(0)1424 3536 y Fn(k)p Fp(u)p Fn(k)1566 3488 y Fs(1)p Fo(=)p Fs(2)1566 3562 y(0)1697 3536 y Fn(\001)20 b(kr)p Fp(u)p Fn(k)1960 3488 y Fs(1)p Fo(=)p Fs(2)1960 3562 y(0)2091 3536 y Fn(\001)g(kr)p Fp(v)s Fn(k)2349 3488 y Fs(1)p Fo(=)p Fs(2)2349 3562 y(0)2481 3536 y Fn(\001)g(k)p Fr(\001)p Fp(v)s Fn(k)2739 3488 y Fs(1)p Fo(=)p Fs(2)2739 3562 y(0)2870 3536 y Fn(\001)h(kr)p Fp(w)r Fn(k)3149 3550 y Fs(0)3189 3536 y Fp(;)456 3665 y Fr(whic)m(h)29 b(easily)g(imply)f(\(11\))k(and)e(\(12\).)1599 b Fd(\003)456 3825 y Fr(W)-8 b(e)26 b(no)m(w)e(transform)g(the)h(t)m(w) m(o-la)m(y)m(er)j(mo)s(del)23 b(\(1\))j(con)m(taining)e(white)g(noise)g (in)m(to)h(a)g(sys-)456 3933 y(tem)k(of)g(coupled)f(random)g(partial)g (di\013eren)m(tial)f(equations.)40 b(The)29 b(reason)g(for)f(taking)456 4041 y(suc)m(h)f(a)i(transformation)f(is)f(that)i(w)m(e)f(need)g(some)h (particular)d(a)j(priori)d(estimates)j(for)456 4149 y(the)j(tra)5 b(jectories)33 b(of)f(the)g(system.)46 b(Often)31 b(these)i(a)f(priori) e(estimates)i(can)h(b)s(e)e(calcu-)456 4257 y(lated)e(b)m(y)h(the)g (Gron)m(w)m(all)g(lemma.)40 b(F)-8 b(or)30 b(equations)g(con)m(taining) f(a)i(white)e(noise)g(there)456 4365 y(exists)40 b(no)g(Gron)m(w)m(all) g(lemma.)70 b(But)41 b(w)m(e)f(can)h(use)f(this)f(tec)m(hnique)i(for)f (the)g(trans-)456 4473 y(formed)29 b(random)h(partial)f(di\013eren)m (tial)g(equations.)456 4581 y(F)-8 b(or)27 b(this)f(purp)s(ose)g(w)m(e) h(in)m(tro)s(duce)f(an)h(Ornstein-Uhlen)m(b)s(ec)m(k)e(pro)s(cess)i Fp(\021)s Fr(\()p Fp(x;)15 b(y)s(;)g(t;)g(!)s Fr(\))29 b(in)456 4689 y Fp(L)518 4656 y Fs(2)518 4711 y Fo(per)624 4689 y Fr(.)38 b(This)21 b(pro)s(cess)h(is)g(de\014ned)f(b)m(y)i(the)f (solution)g(of)g(the)h(follo)m(wing)e(linear)g(sto)s(c)m(hastic)456 4797 y(partial)29 b(di\013eren)m(tial)g(equation)456 4946 y(\(13\))892 b Fp(\021)1553 4960 y Fo(t)1608 4946 y Fr(=)25 b Fp(\027)6 b Fr(\()p Fp(k)24 b Fr(+)c(1\)\001)p Fp(\021)k Fr(+)2304 4923 y(_)2268 4946 y Fp(W)12 b(;)456 5094 y Fr(with)30 b(p)s(erio)s(dic)g(b)s(oundary)g(condition)g(and)i (with)e(some)j(initial)c(condition)i Fp(x)d Fn(2)f Fp(L)3338 5061 y Fs(2)3338 5116 y Fo(per)456 5213 y Fr(where)i Fp(k)g(>)c Fr(0)30 b(is)g(a)g(free)h Fj(c)-5 b(ontr)g(ol)32 b Fr(parameter)f(and)f Fp(W)42 b Fr(is)30 b(a)g(Wiener)g(pro)s(cess)g (in)f Fp(L)3313 5180 y Fs(2)3313 5236 y Fo(per)3419 5213 y Fr(.)p eop %%Page: 9 9 9 8 bop 1043 251 a Fs(STOCHASTIC)31 b(TW)n(O-LA)-6 b(YER)30 b(GEOPHYSICAL)h(FLO)n(WS)584 b(9)456 450 y Fr(W)-8 b(e)40 b(supp)s(ose)e(that)i(the)f(co)m(v)-5 b(ariance)41 b(op)s(erator)e Fp(Q)h Fr(of)f(this)f(Wiener)h(pro)s(cess)g(has)g(a)456 558 y(\014nite)g(trace.)73 b(As)41 b(men)m(tioned)f(in)f(Section)i(2)g (suc)m(h)f(a)h(Wiener)g(pro)s(cess)f(generates)456 666 y(a)e(metric)g(dynamical)f(system)h(denoted)h(b)m(y)f(\(\012)p Fp(;)15 b Fn(F)9 b Fp(;)15 b Fg(P)p Fp(;)g(\022)s Fr(\))40 b(where)e Fp(\022)i Fr(is)d(the)i(Wiener)456 774 y(shift)28 b(in)m(tro)s(duced)g(in)h(\(3\).)42 b(It)29 b(is)g(w)m(ell)g(kno)m(wn)g (that)i(this)e(equation)g(has)h(a)g(stationary)456 882 y(solution)35 b(whic)m(h)h(is)g(generated)h(b)m(y)g(a)g Fj(Gaussian)i Fr(random)d(v)-5 b(ariable)35 b Fp(\021)41 b Fr(in)35 b Fp(H)3177 849 y Fs(1)3170 904 y Fo(per)3276 882 y Fr(.)60 b(In)456 990 y(particular,)29 b(the)h(mapping)1313 1139 y Fp(t)25 b Fn(!)g Fp(\021)s Fr(\()p Fp(\022)1613 1153 y Fo(t)1643 1139 y Fp(!)s Fr(\))g Fn(2)g Fp(L)1911 1101 y Fs(2)1911 1162 y Fo(l)q(oc)2002 1139 y Fr(\()p Fn(\0001)p Fp(;)15 b Fn(1)p Fr(;)g Fp(H)2453 1101 y Fs(1)2446 1161 y Fo(per)2552 1139 y Fr(\))456 1288 y(solv)m(es)30 b(this)f(equation.)41 b(F)-8 b(or)31 b(momen)m(ts)g(of)g Fp(\021)i Fr(w)m(e)e(obtain:)456 1489 y(\(14\))102 b Fg(E)10 b Fn(k)q Fp(\021)s Fn(k)912 1452 y Fs(2)912 1512 y(1)982 1489 y Fn(\024)1184 1428 y Fr(tr)1255 1442 y Fs(0)1294 1428 y Fp(Q)p 1088 1469 374 4 v 1088 1552 a Fr(2)p Fp(\027)c Fr(\()p Fp(k)24 b Fr(+)c(1\))1472 1489 y Fp(;)106 b Fg(E)11 b Fn(k)p Fp(\021)s Fn(k)1797 1452 y Fs(2)p Fo(n)1797 1512 y Fs(1)1910 1489 y Fn(\024)25 b Fp(C)2071 1503 y Fo(n)2133 1361 y Fk(\022)2283 1428 y Fr(tr)2354 1442 y Fs(0)2394 1428 y Fp(Q)p 2210 1469 329 4 v 2210 1552 a(\027)6 b Fr(\()p Fp(k)24 b Fr(+)c(1\))2548 1361 y Fk(\023)2615 1384 y Fo(n)2678 1489 y Fp(;)40 b(n)25 b Fn(2)g Fg(N)7 b Fp(;)46 b(C)3105 1503 y Fo(n)3177 1489 y Fp(>)25 b Fr(0)p Fp(:)456 1689 y Fr(W)-8 b(e)31 b(in)m(tro)s(duce)e (new)h(v)-5 b(ariables)456 1838 y(\(15\))423 b(~)-52 b Fp(q)1073 1852 y Fs(1)1137 1838 y Fr(:=)25 b Fp(q)1299 1852 y Fs(1)1358 1838 y Fn(\000)20 b Fp(\021)s(;)41 b(q)1604 1852 y Fs(2)1643 1838 y Fp(;)1728 1814 y Fr(~)1709 1838 y Fp( )1768 1852 y Fs(1)1833 1838 y Fr(:=)25 b Fp( )2013 1852 y Fs(1)2073 1838 y Fr(+)20 b Fp(\030)2204 1852 y Fs(1)2243 1838 y Fp(;)2327 1814 y Fr(~)2309 1838 y Fp( )2368 1852 y Fs(2)2433 1838 y Fr(:=)25 b Fp( )2613 1852 y Fs(2)2673 1838 y Fr(+)20 b Fp(\030)2804 1852 y Fs(2)2843 1838 y Fp(;)456 1987 y Fr(where)k(stationary)h(pro)s(cess)g Fp(\021)j Fr(solv)m(es)d(the)g(problem)f(\(13\))i(and)f Fp(\030)2676 2001 y Fs(1)2740 1987 y Fr(and)f Fp(\030)2951 2001 y Fs(2)3015 1987 y Fr(are)i(de\014ned)456 2095 y(suc)m(h)k(that)h (the)f(elliptic)e(equations)i(\(2\))i(remain)d(of)i(the)f(same)h(form) 1429 2252 y(~)-51 b Fp(q)1464 2266 y Fs(1)1528 2252 y Fr(=)25 b(\001)1718 2228 y(~)1700 2252 y Fp( )1759 2266 y Fs(1)1819 2252 y Fn(\000)19 b Fp(F)1967 2266 y Fs(1)2028 2252 y Fn(\001)h Fr(\()2127 2228 y(~)2108 2252 y Fp( )2167 2266 y Fs(1)2227 2252 y Fn(\000)2337 2228 y Fr(~)2318 2252 y Fp( )2377 2266 y Fs(2)2417 2252 y Fr(\))p Fp(;)1423 2408 y(q)1464 2422 y Fs(2)1528 2408 y Fr(=)25 b(\001)1718 2384 y(~)1700 2408 y Fp( )1759 2422 y Fs(2)1819 2408 y Fn(\000)19 b Fp(F)1967 2422 y Fs(2)2028 2408 y Fn(\001)h Fr(\()2127 2384 y(~)2108 2408 y Fp( )2167 2422 y Fs(2)2227 2408 y Fn(\000)2337 2384 y Fr(~)2318 2408 y Fp( )2377 2422 y Fs(1)2417 2408 y Fr(\))p Fp(;)456 2329 y Fr(\(16\))456 2558 y(The)29 b(pro)s(cesses)i Fp(\030)1076 2572 y Fs(1)1145 2558 y Fr(and)f Fp(\030)1362 2572 y Fs(2)1431 2558 y Fr(are)h(solutions)e(of)i(the)f(linear)f(elliptic)f(equations)1425 2693 y(\001)p Fp(\030)1541 2707 y Fs(1)1600 2693 y Fn(\000)20 b Fp(F)1749 2707 y Fs(1)1809 2693 y Fn(\001)g Fr(\()p Fp(\030)1929 2707 y Fs(1)1989 2693 y Fn(\000)g Fp(\030)2120 2707 y Fs(2)2159 2693 y Fr(\))26 b(=)20 b Fn(\000)g Fp(\021)s(;)1425 2833 y Fr(\001)p Fp(\030)1541 2847 y Fs(2)1600 2833 y Fn(\000)g Fp(F)1749 2847 y Fs(2)1809 2833 y Fn(\001)g Fr(\()p Fp(\030)1929 2847 y Fs(2)1989 2833 y Fn(\000)g Fp(\030)2120 2847 y Fs(1)2159 2833 y Fr(\))26 b(=0)456 2770 y(\(17\))456 2982 y(and)j(can)i(b)s(e)f(presen)m(ted)g(in)f(the)i (form)920 3169 y Fp(\030)960 3183 y Fs(1)1025 3169 y Fr(=)1261 3108 y(1)p 1130 3149 307 4 v 1130 3232 a Fp(F)1188 3246 y Fs(1)1249 3232 y Fr(+)19 b Fp(F)1397 3246 y Fs(2)1462 3096 y Fk(\010)1515 3169 y Fp(F)1573 3183 y Fs(2)1613 3169 y Fr(\()p Fn(\000)p Fr(\001\))1830 3132 y Fh(\000)p Fs(1)1945 3169 y Fr(+)h Fp(F)2094 3183 y Fs(1)2134 3169 y Fr(\()p Fn(\000)p Fr(\001)g(+)g Fp(F)2485 3183 y Fs(1)2545 3169 y Fr(+)g Fp(F)2694 3183 y Fs(2)2734 3169 y Fr(\))2769 3132 y Fh(\000)p Fs(1)2863 3096 y Fk(\011)2932 3169 y Fp(\021)920 3404 y(\030)960 3418 y Fs(2)1025 3404 y Fr(=)1235 3343 y Fp(F)1293 3357 y Fs(2)p 1130 3383 V 1130 3467 a Fp(F)1188 3481 y Fs(1)1249 3467 y Fr(+)f Fp(F)1397 3481 y Fs(2)1462 3331 y Fk(\010)1515 3404 y Fr(\()p Fn(\000)p Fr(\001\))1732 3367 y Fh(\000)p Fs(1)1847 3404 y Fn(\000)h Fr(\()p Fn(\000)p Fr(\001)g(+)g Fp(F)2289 3418 y Fs(1)2349 3404 y Fr(+)g Fp(F)2498 3418 y Fs(2)2538 3404 y Fr(\))2573 3367 y Fh(\000)p Fs(1)2668 3331 y Fk(\011)2736 3404 y Fp(\021)s(:)456 3593 y Fr(Th)m(us)34 b(the)i(pro)s(cesses)f Fp(\030)1291 3607 y Fs(1)1366 3593 y Fr(and)g Fp(\030)1588 3607 y Fs(2)1663 3593 y Fr(are)h(smo)s(other)f(in)f(spatial)h(v)-5 b(ariables)34 b(than)h Fp(\021)s Fr(.)57 b(In)456 3701 y(fact)31 b(after)g(simple)d(calculations)i(w)m(e)g(ha)m(v)m(e)i(the)f (estimates)456 3850 y(\(18\))159 b Fn(k)p Fp(\030)860 3864 y Fo(i)888 3850 y Fn(k)933 3864 y Fo(s)p Fs(+2)1086 3850 y Fn(\024)25 b(k)p Fp(\021)s Fn(k)1320 3864 y Fo(s)1358 3850 y Fp(;)106 b(i)26 b Fr(=)f(1)p Fp(;)15 b Fr(2)p Fp(;)132 b Fr(and)91 b Fn(k)p Fp(\030)2252 3864 y Fs(1)2312 3850 y Fn(\000)20 b Fp(\030)2443 3864 y Fs(2)2482 3850 y Fn(k)2527 3864 y Fo(s)p Fs(+2)2679 3850 y Fn(\024)25 b(k)p Fp(\021)s Fn(k)2913 3864 y Fo(s)3042 3850 y Fp(s)g Fn(2)g Fg(R)s Fp(:)456 3999 y Fr(No)m(w)40 b(w)m(e)g(obtain)f(the)g (coupled)g(random)f(partial)h(di\013eren)m(tial)e(equations)j(for)f (new)456 4107 y(p)s(oten)m(tial)29 b(v)m(orticities)37 b(~)-52 b Fp(q)1300 4121 y Fs(1)1339 4107 y Fr(,)31 b Fp(q)1436 4121 y Fs(2)1475 4107 y Fr(:)1134 4259 y(~)-52 b Fp(q)1168 4273 y Fs(1)p Fo(t)1233 4259 y Fr(+)o Fp(J)9 b Fr(\()1417 4235 y(~)1397 4259 y Fp( )1456 4273 y Fs(1)1517 4259 y Fn(\000)20 b Fp(\030)1648 4273 y Fs(1)1687 4259 y Fp(;)i Fr(~)-52 b Fp(q)1768 4273 y Fs(1)1828 4259 y Fr(+)20 b Fp(\021)j Fr(+)d Fp(\014)5 b(y)s Fr(\))26 b(=)f Fp(\027)6 b Fr(\001)2466 4222 y Fs(2)2523 4235 y Fr(~)2505 4259 y Fp( )2564 4273 y Fs(1)2623 4259 y Fr(+)20 b Fp(f)1324 4411 y Fn(\000)g Fp(\027)6 b Fr(\001)1542 4374 y Fs(2)1580 4411 y Fp(\030)1620 4425 y Fs(1)1680 4411 y Fn(\000)20 b Fp(\027)6 b Fr(\()p Fp(k)23 b Fr(+)d(1\)\001)p Fp(\021)29 b(;)1127 4568 y(q)1168 4582 y Fs(2)p Fo(t)1233 4568 y Fr(+)o Fp(J)9 b Fr(\()1417 4544 y(~)1397 4568 y Fp( )1456 4582 y Fs(2)1517 4568 y Fn(\000)20 b Fp(\030)1648 4582 y Fs(2)1687 4568 y Fp(;)15 b(q)1768 4582 y Fs(2)1828 4568 y Fr(+)20 b Fp(\014)5 b(y)s Fr(\))25 b(=)g Fp(\027)6 b Fr(\001)2306 4530 y Fs(2)2364 4544 y Fr(~)2345 4568 y Fp( )2404 4582 y Fs(2)2464 4568 y Fn(\000)20 b Fp(r)s Fr(\001)2693 4544 y(~)2675 4568 y Fp( )2734 4582 y Fs(2)1324 4720 y Fn(\000)g Fp(\027)6 b Fr(\001)1542 4682 y Fs(2)1580 4720 y Fp(\030)1620 4734 y Fs(2)1680 4720 y Fr(+)20 b Fp(r)s Fr(\001)p Fp(\030)1931 4734 y Fs(2)1969 4720 y Fp(:)456 4874 y Fr(W)-8 b(e)31 b(treat)h Fp(\021)s(;)15 b(\030)963 4888 y Fs(1)1003 4874 y Fp(;)g(\030)1083 4888 y Fs(2)1153 4874 y Fr(as)30 b(kno)m(wn)g(pro)s(cesses.)41 b(Using)29 b(\(17\))j(w)m(e)f(ha)m(v)m(e)906 5027 y Fn(\000)p Fp(\027)6 b Fr(\001)1104 4989 y Fs(2)1143 5027 y Fp(\030)1183 5041 y Fs(1)1242 5027 y Fn(\000)20 b Fp(\027)6 b Fr(\()p Fp(k)23 b Fr(+)d(1\)\001)p Fp(\021)29 b Fr(=)20 b Fn(\000)g Fp(\027)6 b(F)2101 5041 y Fs(1)2141 5027 y Fr(\(\001)p Fp(\030)2292 5041 y Fs(1)2351 5027 y Fn(\000)20 b Fr(\001)p Fp(\030)2558 5041 y Fs(2)2597 5027 y Fr(\))h Fn(\000)f Fp(\027)6 b(k)s Fr(\001)p Fp(\021)s(;)1469 5179 y Fn(\000)p Fp(\027)g Fr(\001)1667 5142 y Fs(2)1706 5179 y Fp(\030)1746 5193 y Fs(2)1810 5179 y Fr(=)20 b Fn(\000)g Fp(\027)6 b(F)2101 5193 y Fs(2)2141 5179 y Fr(\(\001)p Fp(\030)2292 5193 y Fs(2)2351 5179 y Fn(\000)20 b Fr(\001)p Fp(\030)2558 5193 y Fs(1)2597 5179 y Fr(\))p Fp(:)p eop %%Page: 10 10 10 9 bop 456 251 a Fs(10)327 b(IGOR)24 b(CHUESHO)n(V,)f(JINQIA)n(O)h (DUAN,)e(AND)h(BJ)2368 234 y(\177)2358 251 y(ORN)h(SCHMALFUSS)456 450 y Fr(F)-8 b(or)32 b(con)m(v)m(enience,)i(w)m(e)f(drop)d(the)j (tilde)d(and)i(rewrite)f(the)h(ab)s(o)m(v)m(e)h(system.)46 b(Th)m(us)31 b(w)m(e)456 558 y(\014nally)d(get)j(the)g(coupled)e (system)i(of)f(random)g(partial)f(di\013eren)m(tial)g(equations)793 723 y Fp(q)834 737 y Fs(1)p Fo(t)919 723 y Fr(+)20 b Fp(J)9 b Fr(\()p Fp( )1163 737 y Fs(1)1223 723 y Fn(\000)20 b Fp(\030)1354 737 y Fs(1)1393 723 y Fp(;)15 b(q)1474 737 y Fs(1)1534 723 y Fr(+)20 b Fp(\021)j Fr(+)d Fp(\014)5 b(y)s Fr(\))26 b(=)o Fp(\027)6 b Fr(\001)2146 685 y Fs(2)2185 723 y Fp( )2244 737 y Fs(1)2304 723 y Fr(+)20 b Fp(f)2040 863 y Fn(\000)g Fp(\027)6 b(F)2240 877 y Fs(1)2279 863 y Fr(\(\001)p Fp(\030)2430 877 y Fs(1)2490 863 y Fn(\000)20 b Fr(\001)p Fp(\030)2697 877 y Fs(2)2736 863 y Fr(\))g Fn(\000)g Fp(\027)6 b(k)s Fr(\001)p Fp(\021)952 1015 y(q)993 1029 y Fs(2)p Fo(t)1078 1015 y Fr(+)20 b Fp(J)9 b Fr(\()p Fp( )1322 1029 y Fs(2)1382 1015 y Fn(\000)20 b Fp(\030)1513 1029 y Fs(2)1553 1015 y Fp(;)15 b(q)1634 1029 y Fs(2)1693 1015 y Fr(+)20 b Fp(\014)5 b(y)s Fr(\))26 b(=)o Fp(\027)6 b Fr(\001)2146 977 y Fs(2)2185 1015 y Fp( )2244 1029 y Fs(2)2304 1015 y Fn(\000)20 b Fp(r)s Fr(\001)p Fp( )2574 1029 y Fs(2)2040 1155 y Fn(\000)g Fp(\027)6 b(F)2240 1169 y Fs(2)2279 1155 y Fr(\(\001)p Fp(\030)2430 1169 y Fs(2)2490 1155 y Fn(\000)20 b Fr(\001)p Fp(\030)2697 1169 y Fs(1)2736 1155 y Fr(\))g(+)g Fp(r)s Fr(\001)p Fp(\030)3042 1169 y Fs(2)456 940 y Fr(\(19\))456 1321 y(with)1435 1456 y Fp(q)1476 1470 y Fs(1)1541 1456 y Fr(=)o(\001)p Fp( )1746 1470 y Fs(1)1806 1456 y Fn(\000)g Fp(F)1955 1470 y Fs(1)2015 1456 y Fn(\001)g Fr(\()p Fp( )2154 1470 y Fs(1)2215 1456 y Fn(\000)f Fp( )2364 1470 y Fs(2)2404 1456 y Fr(\))p Fp(;)1435 1596 y(q)1476 1610 y Fs(2)1541 1596 y Fr(=)o(\001)p Fp( )1746 1610 y Fs(2)1806 1596 y Fn(\000)h Fp(F)1955 1610 y Fs(2)2015 1596 y Fn(\001)g Fr(\()p Fp( )2154 1610 y Fs(2)2215 1596 y Fn(\000)f Fp( )2364 1610 y Fs(1)2404 1596 y Fr(\))p Fp(;)456 1534 y Fr(\(20\))456 1768 y(in)32 b(the)j(class)e(of)i Fp(L)p Fr(-p)s(erio)s(dic)c (functions)i(with)f(initial)g(data)i Fp(q)s Fr(\()p Fp(x;)15 b(y)s(;)g Fr(0\))33 b(=)e Fp(q)3092 1782 y Fs(0)3131 1768 y Fr(\()p Fp(x;)15 b(y)s Fr(\))33 b Fn(\021)456 1876 y Fr(\()p Fp(q)532 1890 y Fs(01)606 1876 y Fr(\()p Fp(x;)15 b(y)s Fr(\))p Fp(;)g(q)897 1890 y Fs(02)973 1876 y Fr(\()p Fp(x;)g(y)s Fr(\)\))34 b Fn(2)e Fx(H)1427 1843 y Fh(\000)p Fs(1)1427 1898 y Fo(per)1533 1876 y Fr(,)k(where)e Fp(\021)k Fr(is)c(the)h(stationary)g(solution)e(to)j (\(13\))g(and)456 1995 y Fp(\030)496 2009 y Fs(1)565 1995 y Fr(and)30 b Fp(\030)782 2009 y Fs(2)851 1995 y Fr(are)h(solv)m(es)g(\(17\))h(in)d Fp(H)1643 1962 y Fs(2)1636 2017 y Fo(per)1742 1995 y Fr(.)456 2103 y(F)-8 b(or)35 b(the)f(rest)h(of)f(the)h(pap)s(er,)g(w)m(e)f(w)m(ork)h(on)f(this)f (coupled)h(system)g(of)h(random)f(par-)456 2211 y(tial)21 b(di\013eren)m(tial)g(equations)h(for)h(sto)s(c)m(hastically)f(forced)g (t)m(w)m(o-la)m(y)m(er)j(quasigeostrophic)456 2319 y(\015uid)j(system.) 456 2427 y(The)j(co)s(e\016cien)m(ts)i(of)g(coupled)e(system)h(\(19\),) i(\(20\))g(ha)m(v)m(e)g(similar)29 b(prop)s(erties)i(as)h(the)456 2535 y(co)s(e\016cien)m(ts)46 b(of)g(the)f(corresp)s(onding)f (deterministic)f(t)m(w)m(o-la)m(y)m(er)48 b(quasigeostrophic)456 2643 y(system)26 b(\(cf.)39 b([6)q(],)27 b(where)e Fp(N)10 b Fr(-la)m(y)m(er)27 b(mo)s(del)d(with)h(other)h(b)s(oundary)e (conditions)g(is)h(con-)456 2751 y(sidered\).)39 b(Therefore,)29 b(similar)e(to)j([6],)g(using)e(the)h(Galerkin)f(metho)s(d)h(and)f(the) h(com-)456 2858 y(pactness)36 b(argumen)m(t)h(w)m(e)f(can)h(pro)m(v)m (e)g(the)f(follo)m(wing)f(assertion)h(on)g(w)m(ell-p)s(osedness)456 2966 y(of)30 b(problem)f(\(19\),)j(\(20\))g(in)d(the)i(class)f(of)g Fp(L)p Fr(-p)s(erio)s(dic)e(functions.)456 3103 y Fx(Theorem)33 b(3.2.)41 b Fj(L)-5 b(et)32 b Fp(q)1299 3117 y Fs(0)1364 3103 y Fn(2)24 b Fx(H)1531 3071 y Fh(\000)p Fs(1)1531 3126 y Fo(per)1669 3103 y Fj(and)33 b Fp(f)i Fn(2)24 b Fp(L)2072 3071 y Fs(2)2072 3126 y Fo(per)2179 3103 y Fj(.)41 b(Then)32 b(for)h(al)5 b(l)32 b Fp(!)c Fn(2)d Fr(\012)32 b Fj(and)g(for)h(al)5 b(l)456 3215 y Fp(T)38 b(>)25 b Fr(0)p Fj(,)32 b(the)h(system)h(\(19\),)f(\(20\))h(has)g(a)f (unique)f(solution)h Fn(f)p Fp(q)s Fr(\()p Fp(t)p Fr(\))p Fp(;)15 b( )s Fr(\()p Fp(t)p Fr(\))p Fn(g)36 b Fj(such)c(that)904 3385 y Fp(q)c Fn(2)d Fp(C)7 b Fr(\([0)p Fp(;)15 b(T)e Fr(];)i Fx(H)1489 3348 y Fh(\000)p Fs(1)1489 3408 y Fo(per)1597 3385 y Fr(\))20 b Fn(\\)g Fp(L)1795 3348 y Fs(2)1834 3385 y Fr(\(0)p Fp(;)15 b(T)e Fr(;)i Fx(L)2123 3348 y Fs(2)2123 3408 y Fo(per)2231 3385 y Fr(\))20 b Fn(\\)g Fp(L)2429 3348 y Fs(2)2429 3408 y Fo(l)q(oc)2520 3385 y Fr(\(0)p Fp(;)15 b(T)e Fr(;)i Fx(H)2828 3348 y Fs(1)2828 3408 y Fo(per)2935 3385 y Fr(\))p Fp(:)456 3555 y Fj(The)32 b(function)h Fp( )j Fj(asso)-5 b(ciate)g(d)35 b(to)f Fp(q)h Fj(by)d(\(20\))i(satis\014es)886 3725 y Fp( )28 b Fn(2)d Fp(C)7 b Fr(\([0)p Fp(;)15 b(T)e Fr(];)i Fx(H)1489 3688 y Fs(1)1489 3748 y Fo(per)1596 3725 y Fr(\))21 b Fn(\\)f Fp(L)1795 3688 y Fs(2)1834 3725 y Fr(\(0)p Fp(;)15 b(T)e Fr(;)i Fx(H)2142 3688 y Fs(2)2142 3748 y Fo(per)2250 3725 y Fr(\))20 b Fn(\\)g Fp(L)2448 3688 y Fs(2)2448 3748 y Fo(l)q(oc)2538 3725 y Fr(\(0)p Fp(;)15 b(T)e Fr(;)i Fx(H)2846 3688 y Fs(3)2846 3748 y Fo(per)2954 3725 y Fr(\))p Fp(:)456 3903 y Fj(The)32 b(solution)i(dep)-5 b(ends)34 b(c)-5 b(ontinuously)35 b(on)e(the)g(initial)g(c)-5 b(ondition)34 b Fp(q)2836 3917 y Fs(0)2900 3903 y Fn(2)25 b Fx(H)3068 3870 y Fh(\000)p Fs(1)3068 3926 y Fo(per)3174 3903 y Fj(.)456 4044 y Fr(By)31 b(the)h(uniqueness)d(assertion)i(of)h (the)g(last)f(Theorem)g(the)h(solution)e Fp(t)d Fn(!)g Fp(q)s Fr(\()p Fp(t)p Fr(\))32 b(gen-)456 4152 y(erates)e(a)g(random)f (dynamical)f(system)i Fp(')g Fr(with)e(con)m(tin)m(uous)h Fx(H)2694 4119 y Fh(\000)p Fs(1)2694 4175 y Fo(per)2826 4152 y Fn(3)c Fp(q)j Fn(!)d Fp(')p Fr(\()p Fp(t;)15 b(!)s(;)g(q)s Fr(\))456 4272 y(on)30 b(the)g(phase)g(space)h Fx(H)1316 4239 y Fh(\000)p Fs(1)1316 4294 y Fo(per)1422 4272 y Fr(.)815 4514 y(4.)46 b Fq(Dissip)-6 b(a)g(tivity)34 b(of)g(the)g(random)f(d)n(ynamical)g(system)456 4676 y Fr(Dynamical)22 b(systems)h(generated)h(b)m(y)f(man)m(y)h(nonlinear)d (parab)s(olic)g(di\013eren)m(tial)h(equa-)456 4784 y(tions)j(ha)m(v)m (e)j(the)e(dissipativ)m(e)e(prop)s(ert)m(y)i(whic)m(h)f(means)h(that)h (there)g(exists)e(a)i(compact)456 4892 y(set)g(absorbing)e(the)i (states)h(of)f(the)f(system)h(in)f(\014nite)f(time.)39 b(Usually)25 b(one)i(can)g(c)m(ho)s(ose)456 5000 y(suc)m(h)j(a)g(set)h (whic)m(h)e(is)h(also)g(forw)m(ard)g(in)m(v)-5 b(arian)m(t.)456 5108 y(W)d(e)39 b(no)m(w)f(construct)h(an)f(absorbing)f(forw)m(ard)h (in)m(v)-5 b(arian)m(t)37 b(set)i(for)f(the)g(random)g(dy-)456 5216 y(namical)29 b(system)i(generated)g(b)m(y)f(\(19\).)42 b(This)29 b(set)i(will)d(b)s(e)h(a)i(random)f(set.)p eop %%Page: 11 11 11 10 bop 1043 251 a Fs(STOCHASTIC)31 b(TW)n(O-LA)-6 b(YER)30 b(GEOPHYSICAL)h(FLO)n(WS)549 b(11)456 450 y Fx(Theorem)34 b(4.1.)42 b Fj(Ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(omp)g(act)35 b(r)-5 b(andom)35 b(set)e Fp(B)5 b Fr(\()p Fp(!)s Fr(\))25 b Fn(\032)g Fx(H)2924 417 y Fh(\000)p Fs(1)2924 473 y Fo(per)3062 450 y Fj(such)33 b(that)1173 639 y Fp(')p Fr(\()p Fp(t;)15 b(!)s(;)g(B)5 b Fr(\()p Fp(!)s Fr(\)\))27 b Fn(\032)d Fp(B)5 b Fr(\()p Fp(\022)1953 653 y Fo(t)1983 639 y Fp(!)s Fr(\))93 b Fj(for)33 b Fp(t)25 b Fn(\025)g Fr(0)p Fp(;)1173 779 y(')p Fr(\()p Fp(t;)15 b(!)s(;)g(q)s Fr(\()p Fp(!)s Fr(\)\))27 b Fn(\032)e Fp(B)5 b Fr(\()p Fp(\022)1924 793 y Fo(t)1953 779 y Fp(!)s Fr(\))93 b Fj(for)33 b Fp(t)25 b Fn(\025)g Fp(t)2473 793 y Fs(0)2513 779 y Fr(\()p Fp(!)s(;)15 b(q)s Fr(\))456 717 y(\(21\))456 977 y Fj(wher)-5 b(e)33 b Fp(q)j Fj(is)c(a)h(r)-5 b(andom)35 b(variable)f(with)f(values)g(in)f Fx(H)2299 944 y Fh(\000)p Fs(1)2299 1000 y Fo(per)2406 977 y Fj(.)456 1124 y Fr(W)-8 b(e)34 b(no)m(w)g(divide)e(the)i(pro)s(of)f(of)g(this)g(theorem)h(in)e (some)i(lemmata.)52 b(W)-8 b(e)34 b(start)h(with)456 1232 y(the)30 b(follo)m(wing:)456 1378 y Fx(Lemma)e(4.2.)38 b Fj(L)-5 b(et)29 b Fp(q)s Fr(\()p Fp(t)p Fr(\))h Fj(b)-5 b(e)28 b(the)h(solution)i(of)e(\(19\).)41 b(Then)29 b Fp(q)s Fr(\()p Fp(t)p Fr(\))h Fj(satis\014es)f(the)h(fol)5 b(low-)456 1486 y(ing)32 b(ine)-5 b(quality)755 1653 y Fp(d)p 739 1693 81 4 v 739 1777 a(dt)829 1714 y Fn(k)p Fp(q)s Fr(\()p Fp(t)p Fr(\))p Fn(k)1066 1677 y Fs(2)1066 1737 y Fh(\003)1107 1714 y Fr(+)o Fp(\027)6 b Fr(\()p Fp(h)1315 1728 y Fs(1)1355 1714 y Fn(k)p Fr(\001)p Fp( )1535 1728 y Fs(1)1575 1714 y Fr(\()p Fp(t)p Fr(\))p Fn(k)1723 1677 y Fs(2)1723 1737 y(0)1784 1714 y Fr(+)20 b Fp(h)1927 1728 y Fs(2)1966 1714 y Fn(k)p Fr(\001)p Fp( )2146 1728 y Fs(2)2186 1714 y Fr(\()p Fp(t)p Fr(\))p Fn(k)2334 1677 y Fs(2)2334 1737 y(0)2375 1714 y Fr(\))1107 1892 y Fn(\024)o Fp(d)1224 1906 y Fs(0)1284 1892 y Fn(\001)h(k)p Fp(\021)s Fr(\()p Fp(\022)1501 1906 y Fo(t)1531 1892 y Fp(!)s Fr(\))p Fn(k)1671 1855 y Fs(2)1671 1915 y(0)1732 1892 y Fn(\001)f Fr(\()p Fp(h)1864 1906 y Fs(1)1904 1892 y Fn(kr)p Fp( )2084 1906 y Fs(1)2124 1892 y Fn(k)2169 1855 y Fs(2)2169 1915 y(0)2229 1892 y Fr(+)g Fp(h)2372 1906 y Fs(2)2412 1892 y Fn(kr)p Fp( )2592 1906 y Fs(2)2632 1892 y Fn(k)2677 1855 y Fs(2)2677 1915 y(0)2717 1892 y Fr(\))g(+)g Fp(m)p Fr(\()p Fp(\022)3021 1906 y Fo(t)3051 1892 y Fp(!)s Fr(\))p Fp(;)456 2078 y Fj(wher)-5 b(e)1234 2225 y Fp(m)p Fr(\()p Fp(!)s Fr(\))25 b(=)g Fp(d)1612 2239 y Fs(1)1652 2225 y Fn(k)p Fp(\021)s Fr(\()p Fp(!)s Fr(\))p Fn(k)1920 2187 y Fs(4)1920 2247 y(0)1982 2225 y Fr(+)19 b Fp(d)2119 2239 y Fs(2)2159 2225 y Fn(k)p Fp(\021)s Fr(\()p Fp(!)s Fr(\))p Fn(k)2427 2187 y Fs(2)2427 2247 y(0)2489 2225 y Fr(+)h Fp(d)2627 2239 y Fs(3)456 2391 y Fj(and)761 2614 y Fp(d)808 2628 y Fs(0)873 2614 y Fr(=)954 2553 y(6)p Fp(c)1038 2520 y Fs(2)1038 2577 y(0)p 954 2593 125 4 v 991 2677 a Fp(\027)1103 2486 y Fk(\022)1170 2614 y Fr(1)h(+)1556 2553 y Fp(p)1602 2520 y Fs(2)1641 2553 y Fp(\027)p 1337 2593 575 4 v 1337 2678 a(\025)1390 2647 y Fs(2)1390 2704 y(1)1444 2678 y Fr(min)o Fn(f)p Fp(h)1693 2692 y Fs(1)1733 2678 y Fp(;)15 b(h)1825 2692 y Fs(2)1865 2678 y Fn(g)1921 2486 y Fk(\023)2003 2614 y Fp(;)761 2877 y(d)808 2891 y Fs(1)873 2877 y Fr(=)954 2816 y(6)p Fp(c)1038 2783 y Fs(2)1038 2840 y(0)1078 2816 y Fp(h)1130 2830 y Fs(1)p 954 2856 217 4 v 990 2939 a Fp(\027)6 b(\025)1094 2953 y Fs(1)1180 2877 y Fp(;)761 3126 y(d)808 3140 y Fs(2)873 3126 y Fr(=9)1004 2997 y Fk(\022)1081 3064 y Fp(\014)1137 3031 y Fs(2)1177 3064 y Fr(\()p Fp(h)1264 3078 y Fs(1)1325 3064 y Fr(+)19 b Fp(h)1467 3078 y Fs(2)1507 3064 y Fr(\))p 1081 3105 462 4 v 1240 3190 a Fp(\027)6 b(\025)1344 3158 y Fs(3)1344 3216 y(1)1573 3126 y Fr(+)1674 3064 y Fp(\027)g(p)1771 3031 y Fs(2)p 1674 3105 136 4 v 1695 3190 a Fp(\025)1748 3158 y Fs(2)1748 3216 y(1)1835 2997 y Fk(\022)1935 3064 y Fr(1)p 1911 3105 92 4 v 1911 3188 a Fp(h)1963 3202 y Fs(1)2033 3126 y Fr(+)2180 3064 y(1)p 2134 3105 138 4 v 2134 3188 a(5)p Fp(h)2231 3202 y Fs(2)2282 2997 y Fk(\023)2369 3126 y Fr(+)2494 3064 y Fp(r)s(h)2590 3078 y Fs(2)p 2470 3105 184 4 v 2470 3188 a Fr(18)p Fp(\025)2613 3202 y Fs(1)2683 3126 y Fr(+)20 b Fp(k)2824 3088 y Fs(2)2864 3126 y Fp(\027)6 b(h)2967 3140 y Fs(1)3006 2997 y Fk(\023)3113 3126 y Fp(;)761 3376 y(d)808 3390 y Fs(3)873 3376 y Fr(=)957 3314 y(9)p Fp(h)1054 3328 y Fs(1)p 954 3355 144 4 v 954 3438 a Fp(\027)g(\025)1058 3452 y Fs(1)1107 3376 y Fn(k)p Fp(f)k Fn(k)1252 3338 y Fs(2)1252 3398 y Fh(\000)p Fs(1)1347 3376 y Fp(:)456 3630 y Fj(Pr)-5 b(o)g(of.)43 b Fr(Multiplying)18 b(the)23 b(equations)f(\(19\))i(b)m(y)e Fn(\000)p Fr(2)p Fp(h)2241 3644 y Fs(1)2281 3630 y Fp( )2340 3644 y Fs(1)2380 3630 y Fp(;)30 b Fn(\000)p Fr(2)p Fp(h)2603 3644 y Fs(2)2643 3630 y Fp( )2702 3644 y Fs(2)2742 3630 y Fr(,)24 b(resp)s(ectiv)m(ely) -8 b(,)24 b(and)456 3738 y(then)30 b(adding)f(together,)j(w)m(e)f (obtain)491 3905 y Fp(d)p 475 3945 81 4 v 475 4028 a(dt)565 3966 y Fn(k)p Fp(q)s Fr(\()p Fp(t)p Fr(\))p Fn(k)802 3928 y Fs(2)802 3988 y Fh(\003)843 3966 y Fr(+2)p Fp(\027)6 b Fr(\()p Fp(h)1097 3980 y Fs(1)1137 3966 y Fn(k)p Fr(\001)p Fp( )1317 3980 y Fs(1)1357 3966 y Fn(k)1402 3928 y Fs(2)1402 3988 y(0)1462 3966 y Fr(+)20 b Fp(h)1605 3980 y Fs(2)1645 3966 y Fn(k)p Fr(\001)p Fp( )1825 3980 y Fs(2)1864 3966 y Fn(k)1909 3928 y Fs(2)1909 3988 y(0)1949 3966 y Fr(\))h(+)f(2)p Fp(r)s(h)2237 3980 y Fs(2)2277 3966 y Fn(kr)p Fp( )2457 3980 y Fs(2)2496 3966 y Fn(k)2541 3928 y Fs(2)2541 3988 y(0)843 4132 y Fr(=2)p Fp(h)1011 4146 y Fs(1)1051 4132 y Fr(\()p Fp(J)9 b Fr(\()p Fp( )1239 4146 y Fs(1)1300 4132 y Fn(\000)19 b Fp(\030)1430 4146 y Fs(1)1470 4132 y Fp(;)c(q)1551 4146 y Fs(1)1610 4132 y Fr(+)20 b Fp(\021)k Fr(+)c Fp(\014)5 b(y)s Fr(\))p Fp(;)15 b( )2099 4146 y Fs(1)2139 4132 y Fr(\))2174 4146 y Fs(0)934 4272 y Fr(+)20 b(2)p Fp(h)1122 4286 y Fs(2)1162 4272 y Fr(\()p Fp(J)9 b Fr(\()p Fp( )1350 4286 y Fs(2)1411 4272 y Fn(\000)20 b Fp(\030)1542 4286 y Fs(2)1581 4272 y Fp(;)15 b(q)1662 4286 y Fs(2)1721 4272 y Fr(+)20 b Fp(\014)5 b(y)s Fr(\))p Fp(;)15 b( )2050 4286 y Fs(2)2091 4272 y Fr(\))2126 4286 y Fs(0)2186 4272 y Fn(\000)20 b Fr(2)p Fp(h)2374 4286 y Fs(1)2414 4272 y Fr(\()p Fp(f)5 b(;)15 b( )2598 4286 y Fs(1)2638 4272 y Fr(\))2673 4286 y Fs(0)2733 4272 y Fr(+)k(2)p Fp(h)2920 4286 y Fs(1)2961 4272 y Fp(\027)6 b(k)s Fr(\(\001)p Fp(\021)s(;)15 b( )3320 4286 y Fs(1)3360 4272 y Fr(\))3395 4286 y Fs(0)934 4412 y Fr(+)20 b(2)p Fp(\027)6 b(p)p Fr(\(\001)p Fp(\030)1318 4426 y Fs(1)1377 4412 y Fn(\000)20 b Fr(\001)p Fp(\030)1584 4426 y Fs(2)1623 4412 y Fp(;)15 b( )1722 4426 y Fs(1)1762 4412 y Fr(\))1797 4426 y Fs(0)1857 4412 y Fr(+)20 b(2)p Fp(\027)6 b(p)p Fr(\(\001)p Fp(\030)2241 4426 y Fs(2)2300 4412 y Fn(\000)20 b Fr(\001)p Fp(\030)2507 4426 y Fs(1)2546 4412 y Fp(;)15 b( )2645 4426 y Fs(2)2685 4412 y Fr(\))2720 4426 y Fs(0)934 4552 y Fn(\000)20 b Fr(2)p Fp(r)s(h)1166 4566 y Fs(2)1205 4552 y Fr(\(\001)p Fp(\030)1356 4566 y Fs(2)1396 4552 y Fp(;)15 b( )1495 4566 y Fs(2)1535 4552 y Fr(\))1570 4566 y Fs(0)1610 4552 y Fp(:)456 4737 y Fr(W)-8 b(e)31 b(ha)m(v)m(e)h(b)m(y)e(Lemma)h(3.1)1060 4922 y(\()p Fp(J)9 b Fr(\()p Fp( )1248 4936 y Fs(1)1289 4922 y Fp(;)15 b(q)1370 4936 y Fs(1)1429 4922 y Fr(+)20 b Fp(\021)s Fr(\))p Fp(;)15 b( )1702 4936 y Fs(1)1743 4922 y Fr(\))1778 4936 y Fs(0)1843 4922 y Fr(=)25 b(0)p Fp(;)41 b Fr(\()p Fp(J)9 b Fr(\()p Fp( )2238 4936 y Fs(2)2278 4922 y Fp(;)15 b(q)2359 4936 y Fs(2)2399 4922 y Fr(\))p Fp(;)g( )2533 4936 y Fs(2)2573 4922 y Fr(\))2608 4936 y Fs(0)2673 4922 y Fr(=)25 b(0)p Fp(:)456 5108 y Fr(A)32 b(simple)d(calculation)i(sho)m(ws)h(that)g(\()p Fp(J)9 b Fr(\()p Fp( )1950 5122 y Fo(i)1980 5108 y Fp(;)15 b(\014)5 b(y)s Fr(\))p Fp(;)15 b( )2258 5122 y Fo(i)2287 5108 y Fr(\))2322 5122 y Fs(0)2390 5108 y Fr(=)27 b(0.)46 b(W)-8 b(e)33 b(no)m(w)f(estimate)g(the)456 5216 y(righ)m(t)i(hand)g (side)g(terms.)55 b(On)34 b(accoun)m(t)j(of)e(\(20\),)j(\(10\),)g (\(12\),)g(the)e(bilinearit)m(y)c(of)j Fp(J)p eop %%Page: 12 12 12 11 bop 456 251 a Fs(12)327 b(IGOR)24 b(CHUESHO)n(V,)f(JINQIA)n(O)h (DUAN,)e(AND)h(BJ)2368 234 y(\177)2358 251 y(ORN)h(SCHMALFUSS)456 450 y Fr(and)29 b(\(18\))j(w)m(e)f(can)g(estimate)456 718 y Fn(\000)p Fr(2)635 604 y Fs(2)587 631 y Fk(X)595 827 y Fo(i)p Fs(=1)718 718 y Fp(h)770 732 y Fo(i)799 718 y Fr(\()p Fp(J)9 b Fr(\()p Fp(\030)968 732 y Fo(i)997 718 y Fp(;)15 b(q)1078 732 y Fo(i)1106 718 y Fr(\))p Fp(;)g( )1240 732 y Fo(i)1269 718 y Fr(\))1304 732 y Fs(0)1369 718 y Fr(=)25 b Fn(\000)p Fr(2)1644 604 y Fs(2)1596 631 y Fk(X)1605 827 y Fo(i)p Fs(=1)1743 718 y Fp(h)1795 732 y Fo(i)1823 718 y Fr(\()p Fp(J)9 b Fr(\()p Fp(\030)1992 732 y Fo(i)2021 718 y Fp(;)15 b Fr(\001)p Fp( )2196 732 y Fo(i)2225 718 y Fr(\))p Fp(;)g( )2359 732 y Fo(i)2388 718 y Fr(\))2423 732 y Fs(0)2483 718 y Fr(+)20 b(2)p Fp(p)p Fr(\()p Fp(J)9 b Fr(\()p Fp(\030)2834 732 y Fs(1)2894 718 y Fn(\000)20 b Fp(\030)3025 732 y Fs(2)3064 718 y Fp(;)15 b( )3163 732 y Fs(2)3203 718 y Fr(\))p Fp(;)g( )3337 732 y Fs(1)3378 718 y Fr(\))3413 732 y Fs(0)648 1037 y Fn(\024)o Fr(2)p Fp(c)802 1051 y Fs(0)906 923 y(2)858 950 y Fk(X)866 1146 y Fo(i)p Fs(=1)1004 1037 y Fp(h)1056 1051 y Fo(i)1085 1037 y Fn(k)p Fr(\001)p Fp(\030)1246 1051 y Fo(i)1274 1037 y Fn(k)1319 1051 y Fs(0)1359 1037 y Fn(k)p Fr(\001)p Fp( )1539 1051 y Fo(i)1567 1037 y Fn(k)1612 1051 y Fs(0)1652 1037 y Fn(kr)p Fp( )1832 1051 y Fo(i)1861 1037 y Fn(k)1906 1051 y Fs(0)1966 1037 y Fr(+)20 b(2)p Fp(c)2141 1051 y Fs(1)2181 1037 y Fp(p)p Fn(k)p Fr(\001\()p Fp(\030)2423 1051 y Fs(1)2483 1037 y Fn(\000)g Fp(\030)2614 1051 y Fs(2)2653 1037 y Fr(\))p Fn(k)2733 1051 y Fs(0)2773 1037 y Fn(kr)p Fp( )2953 1051 y Fs(1)2993 1037 y Fn(k)3038 1051 y Fs(0)3078 1037 y Fn(kr)p Fp( )3258 1051 y Fs(2)3298 1037 y Fn(k)3343 1051 y Fs(0)648 1327 y Fn(\024)728 1266 y Fp(\027)p 728 1306 51 4 v 731 1390 a Fr(6)789 1327 y(\()p Fp(h)876 1341 y Fs(1)916 1327 y Fn(k)p Fr(\001)p Fp( )1096 1341 y Fs(1)1136 1327 y Fn(k)1181 1290 y Fs(2)1181 1350 y(0)1241 1327 y Fr(+)g Fp(h)1384 1341 y Fs(2)1424 1327 y Fn(k)p Fr(\001)p Fp( )1604 1341 y Fs(2)1644 1327 y Fn(k)1689 1290 y Fs(2)1689 1350 y(0)1728 1327 y Fr(\))h(+)f(6)1930 1266 y Fp(c)1969 1233 y Fs(2)1969 1290 y(0)p 1930 1306 79 4 v 1944 1390 a Fp(\027)2019 1327 y Fn(k)p Fp(\021)s Fn(k)2157 1290 y Fs(2)2157 1350 y(0)2198 1327 y Fr(\()p Fp(h)2285 1341 y Fs(1)2325 1327 y Fn(kr)p Fp( )2505 1341 y Fs(1)2545 1327 y Fn(k)2590 1290 y Fs(2)2590 1350 y(0)2650 1327 y Fr(+)g Fp(h)2793 1341 y Fs(2)2833 1327 y Fn(kr)p Fp( )3013 1341 y Fs(2)3053 1327 y Fn(k)3098 1290 y Fs(2)3098 1350 y(0)3137 1327 y Fr(\))738 1555 y(+)921 1493 y Fp(c)960 1507 y Fs(1)1000 1493 y Fp(p)p Fn(k)p Fp(\021)s Fn(k)1184 1507 y Fs(0)p 839 1534 467 4 v 839 1617 a Fr(min)o Fn(f)p Fp(h)1088 1631 y Fs(1)1128 1617 y Fp(;)15 b(h)1220 1631 y Fs(2)1260 1617 y Fn(g)1316 1555 y Fr(\()p Fp(h)1403 1569 y Fs(1)1443 1555 y Fn(kr)p Fp( )1623 1569 y Fs(1)1663 1555 y Fn(k)1708 1517 y Fs(2)1708 1577 y(0)1768 1555 y Fr(+)20 b Fp(h)1911 1569 y Fs(2)1951 1555 y Fn(kr)p Fp( )2131 1569 y Fs(2)2170 1555 y Fn(k)2215 1517 y Fs(2)2215 1577 y(0)2255 1555 y Fr(\))648 1776 y Fn(\024)728 1714 y Fp(\027)p 728 1755 51 4 v 731 1838 a Fr(3)789 1776 y(\()p Fp(h)876 1790 y Fs(1)916 1776 y Fn(k)p Fr(\001)p Fp( )1096 1790 y Fs(1)1136 1776 y Fn(k)1181 1738 y Fs(2)1181 1798 y(0)1241 1776 y Fr(+)g Fp(h)1384 1790 y Fs(2)1424 1776 y Fn(k)p Fr(\001)p Fp( )1604 1790 y Fs(2)1644 1776 y Fn(k)1689 1738 y Fs(2)1689 1798 y(0)1728 1776 y Fr(\))h(+)f Fp(d)1922 1790 y Fs(0)1962 1776 y Fn(k)p Fp(\021)s Fn(k)2100 1738 y Fs(2)2100 1798 y(0)2140 1776 y Fr(\()p Fp(h)2227 1790 y Fs(1)2268 1776 y Fn(kr)p Fp( )2448 1790 y Fs(1)2487 1776 y Fn(k)2532 1738 y Fs(2)2532 1798 y(0)2593 1776 y Fr(+)f Fp(h)2735 1790 y Fs(2)2775 1776 y Fn(kr)p Fp( )2955 1790 y Fs(2)2995 1776 y Fn(k)3040 1738 y Fs(2)3040 1798 y(0)3080 1776 y Fr(\))p Fp(:)456 1986 y Fr(Similarly)-8 b(,)27 b(due)j(to)h(\(11\))h(and)d(\(18\))456 2225 y Fn(\000)p Fr(2)p Fp(h)624 2239 y Fs(1)664 2225 y Fr(\()p Fp(J)9 b Fr(\()p Fp(\030)833 2239 y Fs(1)873 2225 y Fp(;)15 b(\021)s Fr(\))p Fp(;)g( )1095 2239 y Fs(1)1136 2225 y Fr(\))1171 2239 y Fs(0)1236 2225 y Fn(\024)25 b Fr(2)p Fp(c)1416 2239 y Fs(1)1456 2225 y Fp(h)1508 2239 y Fs(1)1548 2225 y Fn(k)p Fr(\001)p Fp(\030)1709 2239 y Fs(1)1748 2225 y Fn(k)1793 2239 y Fs(0)1833 2225 y Fn(k)p Fp(\021)s Fn(k)1971 2239 y Fs(0)2012 2225 y Fn(k)p Fr(\001)p Fp( )2192 2239 y Fs(1)2232 2225 y Fn(k)2277 2239 y Fs(0)2342 2225 y Fn(\024)2448 2163 y Fr(6)p Fp(c)2532 2130 y Fs(2)2532 2188 y(1)2572 2163 y Fp(h)2624 2177 y Fs(1)2664 2163 y Fn(k)p Fp(\021)s Fn(k)2802 2130 y Fs(4)2802 2188 y(0)p 2448 2204 395 4 v 2620 2287 a Fp(\027)2867 2225 y Fr(+)2962 2163 y Fp(\027)6 b(h)3065 2177 y Fs(1)p 2962 2204 143 4 v 3010 2287 a Fr(6)3114 2225 y Fn(k)p Fr(\001)p Fp( )3294 2239 y Fs(1)3334 2225 y Fn(k)3379 2187 y Fs(2)3379 2247 y(0)3419 2225 y Fp(:)456 2430 y Fr(By)30 b(the)h(Cauc)m(h)m(y-Sc)m(h)m (w)m(arz)h(inequalit)m(y)744 2692 y Fn(\000)20 b Fr(2)944 2579 y Fs(2)896 2606 y Fk(X)904 2801 y Fo(i)p Fs(=1)1042 2692 y Fp(h)1094 2706 y Fo(i)1123 2692 y Fr(\()p Fp(J)9 b Fr(\()p Fp(\030)1292 2706 y Fo(i)1321 2692 y Fp(;)15 b(\014)5 b(y)s Fr(\))p Fp(;)15 b( )1599 2706 y Fo(i)1628 2692 y Fr(\))1663 2706 y Fs(0)1728 2692 y Fn(\024)25 b Fr(2)p Fp(\014)1989 2579 y Fs(2)1941 2606 y Fk(X)1950 2801 y Fo(i)p Fs(=1)2088 2692 y Fp(h)2140 2706 y Fo(i)2168 2692 y Fn(kr)p Fp(\030)2329 2706 y Fo(i)2357 2692 y Fn(k)2402 2706 y Fh(\000)p Fs(1)2497 2692 y Fn(kr)p Fp( )2677 2706 y Fo(i)2706 2692 y Fn(k)2751 2706 y Fs(0)749 2983 y Fn(\024)855 2921 y Fr(9)p Fp(\014)956 2888 y Fs(2)996 2921 y Fr(\()p Fp(h)1083 2935 y Fs(1)1144 2921 y Fr(+)20 b Fp(h)1287 2935 y Fs(2)1327 2921 y Fr(\))p 855 2962 507 4 v 1037 3047 a Fp(\027)6 b(\025)1141 3016 y Fs(3)1141 3073 y(1)1372 2983 y Fn(k)p Fp(\021)s Fn(k)1510 2945 y Fs(2)1510 3005 y(0)1571 2983 y Fr(+)1672 2921 y Fp(\027)p 1672 2962 51 4 v 1675 3045 a Fr(9)1732 2983 y(\()p Fp(h)1819 2997 y Fs(1)1859 2983 y Fn(k)p Fr(\001)p Fp( )2039 2997 y Fs(1)2079 2983 y Fn(k)2124 2945 y Fs(2)2124 3005 y(0)2184 2983 y Fr(+)20 b Fp(h)2327 2997 y Fs(2)2367 2983 y Fn(k)p Fr(\001)p Fp( )2547 2997 y Fs(2)2587 2983 y Fn(k)2632 2945 y Fs(2)2632 3005 y(0)2672 2983 y Fr(\))p Fp(;)744 3233 y Fn(\000)g Fr(2)p Fp(h)932 3247 y Fs(1)973 3233 y Fr(\()p Fp(f)5 b(;)15 b( )1157 3247 y Fs(1)1196 3233 y Fr(\))1231 3247 y Fs(0)1296 3233 y Fn(\024)25 b Fr(2)p Fp(h)1489 3247 y Fs(1)1529 3233 y Fn(k)p Fp(f)10 b Fn(k)1674 3247 y Fh(\000)p Fs(1)1769 3233 y Fn(kr)p Fp( )1949 3247 y Fs(1)1989 3233 y Fn(k)2034 3247 y Fs(0)2099 3233 y Fn(\024)2208 3172 y Fr(9)p Fp(h)2305 3186 y Fs(1)p 2205 3212 144 4 v 2205 3295 a Fp(\027)c(\025)2309 3309 y Fs(1)2358 3233 y Fn(k)p Fp(f)k Fn(k)2503 3196 y Fs(2)2503 3256 y Fh(\000)p Fs(1)2618 3233 y Fr(+)2719 3172 y Fp(\027)c(h)2822 3186 y Fs(1)p 2719 3212 143 4 v 2767 3295 a Fr(9)2871 3233 y Fn(k)p Fr(\001)p Fp( )3051 3247 y Fs(1)3091 3233 y Fn(k)3136 3196 y Fs(2)3136 3256 y(0)744 3469 y Fr(+)20 b(2)p Fp(h)932 3483 y Fs(1)973 3469 y Fp(\027)6 b Fr(\()p Fp(k)s Fr(\001)p Fp(\021)s(;)15 b( )1332 3483 y Fs(1)1372 3469 y Fr(\))1407 3483 y Fs(0)1472 3469 y Fn(\024)25 b Fr(9)p Fp(k)1663 3431 y Fs(2)1703 3469 y Fp(\027)6 b(h)1806 3483 y Fs(1)1846 3469 y Fn(k)p Fp(\021)s Fn(k)1984 3431 y Fs(2)1984 3491 y(0)2045 3469 y Fr(+)2145 3407 y Fp(\027)g(h)2248 3421 y Fs(1)p 2145 3448 V 2194 3531 a Fr(9)2298 3469 y Fn(k)p Fr(\001)p Fp( )2478 3483 y Fs(1)2518 3469 y Fn(k)2563 3431 y Fs(2)2563 3491 y(0)2603 3469 y Fp(:)456 3679 y Fr(Using)29 b(\(18\))j(once)f(more)983 3858 y(2)p Fp(\027)6 b(p)p Fr(\(\001\()p Fp(\030)1311 3872 y Fs(1)1370 3858 y Fn(\000)20 b Fp(\030)1501 3872 y Fs(2)1540 3858 y Fr(\))p Fp(;)15 b( )1674 3872 y Fs(1)1715 3858 y Fr(\))1750 3872 y Fs(0)1815 3858 y Fn(\024)25 b Fr(2)p Fp(\027)6 b(p)p Fn(k)p Fr(\001)p Fp(\030)2214 3872 y Fs(1)2273 3858 y Fn(\000)20 b Fr(\001)p Fp(\030)2480 3872 y Fs(2)2519 3858 y Fn(k)2564 3872 y Fh(\000)p Fs(2)2659 3858 y Fn(k)p Fr(\001)p Fp( )2839 3872 y Fs(1)2879 3858 y Fn(k)2924 3872 y Fs(0)1815 4067 y Fn(\024)1922 4005 y Fr(9)p Fp(\027)6 b(p)2064 3972 y Fs(2)p 1921 4046 185 4 v 1921 4131 a Fp(\025)1974 4100 y Fs(2)1974 4157 y(1)2013 4131 y Fp(h)2065 4145 y Fs(1)2115 4067 y Fn(k)p Fp(\021)s Fn(k)2253 4029 y Fs(2)2253 4089 y(0)2314 4067 y Fr(+)2415 4005 y Fp(\027)g(h)2518 4019 y Fs(1)p 2415 4046 143 4 v 2463 4129 a Fr(9)2567 4067 y Fn(k)p Fr(\001)p Fp( )2747 4081 y Fs(1)2787 4067 y Fn(k)2832 4029 y Fs(2)2832 4089 y(0)2872 4067 y Fp(;)912 4330 y Fn(\000)p Fr(2)p Fp(\027)g(p)p Fr(\(\001\()p Fp(\030)1311 4344 y Fs(1)1370 4330 y Fn(\000)20 b Fp(\030)1501 4344 y Fs(2)1540 4330 y Fr(\))p Fp(;)15 b( )1674 4344 y Fs(2)1715 4330 y Fr(\))1750 4344 y Fs(0)1815 4330 y Fn(\024)1945 4268 y Fr(9)p Fp(\027)6 b(p)2087 4235 y Fs(2)p 1921 4309 230 4 v 1921 4394 a Fr(5)p Fp(\025)2019 4362 y Fs(2)2019 4420 y(1)2059 4394 y Fp(h)2111 4408 y Fs(2)2160 4330 y Fn(k)p Fp(\021)s Fn(k)2298 4292 y Fs(2)2298 4352 y(0)2359 4330 y Fr(+)2460 4268 y(5)p Fp(\027)g(h)2608 4282 y Fs(2)p 2460 4309 188 4 v 2532 4392 a Fr(9)2658 4330 y Fn(k)p Fr(\001)p Fp( )2838 4344 y Fs(2)2878 4330 y Fn(k)2923 4292 y Fs(2)2923 4352 y(0)2963 4330 y Fp(;)1134 4580 y Fn(\000)p Fr(2)p Fp(r)s(h)1346 4594 y Fs(2)1385 4580 y Fr(\(\001)p Fp(\030)1536 4594 y Fs(2)1576 4580 y Fp(;)15 b( )1675 4594 y Fs(2)1715 4580 y Fr(\))1750 4594 y Fs(0)1815 4580 y Fn(\024)1922 4518 y Fp(r)s(h)2018 4532 y Fs(2)p 1921 4559 138 4 v 1921 4642 a Fr(2)p Fp(\025)2019 4656 y Fs(1)2069 4580 y Fn(k)p Fp(\021)s Fn(k)2207 4542 y Fs(2)2207 4602 y(0)2268 4580 y Fr(+)k(2)p Fp(r)s(h)2499 4594 y Fs(2)2539 4580 y Fn(kr)p Fp( )2719 4594 y Fs(2)2759 4580 y Fn(k)2804 4542 y Fs(2)2804 4602 y(0)2844 4580 y Fp(:)456 4798 y Fr(Adding)28 b(all)i(these)g(inequalities)e(together) k(w)m(e)f(obtain)f(the)g(conclusion.)393 b Fd(\003)456 5000 y Fr(W)-8 b(e)38 b(no)m(w)e(consider)g(the)h(random)f(v)-5 b(ariable)35 b Fp(\021)41 b Fr(de\014ned)35 b(in)h(Section)g(2.)60 b(Recall)37 b(that)456 5108 y Fp(\021)44 b Fr(dep)s(ends)38 b(on)j(the)f(con)m(trol)i(parameter)f Fp(k)s Fr(.)71 b(If)40 b Fp(k)k Fr(is)39 b(c)m(hosen)j(large)e(enough)h(then)456 5216 y(particular)28 b(momen)m(ts)j(of)g Fp(\021)i Fr(are)e(small.)39 b(Esp)s(ecially)28 b(w)m(e)j(can)g(form)m(ulate:)p eop %%Page: 13 13 13 12 bop 1043 251 a Fs(STOCHASTIC)31 b(TW)n(O-LA)-6 b(YER)30 b(GEOPHYSICAL)h(FLO)n(WS)549 b(13)456 450 y Fx(Lemma)42 b(4.3.)47 b Fj(L)-5 b(et)41 b Fp(W)53 b Fj(b)-5 b(e)41 b(a)g(Wiener)g(pr)-5 b(o)g(c)g(ess)42 b(in)f Fp(L)2386 417 y Fs(2)2386 473 y Fo(per)2533 450 y Fj(with)g(\014nite)g(tr)-5 b(ac)g(e)42 b(of)f(the)456 558 y(c)-5 b(ovarianc)g(e.)43 b(Then)33 b(under)g(assumptions)456 766 y Fr(\(22\))1256 704 y(2)p Fp(d)1348 718 y Fs(0)1388 704 y Fp(a)1436 718 y Fs(0)1476 704 y Fr(tr)1547 718 y Fs(0)1586 704 y Fp(Q)p 1227 745 461 4 v 1227 830 a(\025)1280 799 y Fs(2)1280 856 y(1)1319 830 y Fp(\027)1370 804 y Fs(2)1410 830 y Fr(\()p Fp(k)23 b Fr(+)d(1\))1722 766 y Fp(<)25 b Fr(1)p Fp(;)2077 704 y Fr(16)p Fp(d)2214 718 y Fs(0)2254 704 y Fr(tr)2325 718 y Fs(0)2365 704 y Fp(Q)p 2007 745 500 4 v 2007 830 a(\025)2060 799 y Fs(2)2060 856 y(1)2099 830 y Fp(\027)2150 804 y Fs(2)2190 830 y Fr(\()p Fp(k)e Fr(+)d(1\))2466 804 y Fs(2)2542 766 y Fp(<)25 b Fr(1)456 984 y Fj(the)33 b(r)-5 b(andom)35 b(variable)1024 1190 y Fp(R)1093 1204 y Fs(0)1132 1190 y Fr(\()p Fp(!)s Fr(\))26 b(:=)1409 1066 y Fk(Z)1500 1093 y Fs(0)1460 1272 y Fh(\0001)1605 1190 y Fp(e)1657 1112 y Fm(\027)s(\025\034)p 1657 1124 106 3 v 1676 1165 a(a)1710 1180 y Fl(0)1773 1139 y Fo(\034)8 b Fs(+)p Fo(d)1903 1148 y Fl(0)1949 1086 y Ff(R)1996 1106 y Fl(0)1982 1164 y Fm(\034)2043 1139 y Fh(k)p Fo(\021)r Fs(\()p Fo(\022)2175 1158 y Fm(\034)2210 1144 y Fe(0)2237 1139 y Fo(!)r Fs(\))p Fh(k)2345 1116 y Fl(2)2345 1160 y(0)2381 1139 y Fo(d\034)2456 1116 y Fe(0)2483 1190 y Fp(m)p Fr(\()p Fp(\022)2641 1204 y Fo(\034)2684 1190 y Fp(!)s Fr(\))p Fp(d\034)456 1408 y Fj(is)32 b(\014nite)h(and)g(temp) -5 b(er)g(e)g(d.)44 b(Mor)-5 b(e)g(over)848 1573 y Fk(\000)890 1646 y Fg(E)10 b Fp(R)1014 1609 y Fs(2)1013 1669 y(0)1060 1573 y Fk(\001)1101 1592 y Fs(1)p Fo(=)p Fs(2)1237 1646 y Fn(\024)25 b Fp(d)1380 1609 y Fs(2)1380 1669 y(4)1434 1518 y Fk(\022)1539 1585 y Fr(3)p Fp(a)1632 1599 y Fs(0)p 1511 1625 189 4 v 1511 1709 a Fr(2)p Fp(\027)6 b(\025)1660 1723 y Fs(1)1710 1518 y Fk(\023)1777 1541 y Fs(3)p Fo(=)p Fs(2)1907 1646 y Fn(\001)1952 1518 y Fk(\022)2029 1585 y Fr(2)p Fp(\025)2127 1599 y Fs(1)2167 1585 y Fp(\027)p 2029 1625 V 2080 1709 a(a)2128 1723 y Fs(0)2248 1646 y Fn(\000)2402 1585 y Fr(4)p Fp(d)2494 1599 y Fs(0)2534 1585 y Fr(tr)2605 1599 y Fs(0)2645 1585 y Fp(Q)p 2349 1625 421 4 v 2349 1709 a(\025)2402 1723 y Fs(1)2441 1709 y Fp(\027)g Fr(\()p Fp(k)24 b Fr(+)c(1\))2780 1518 y Fk(\023)2847 1541 y Fh(\000)p Fs(1)p Fo(=)p Fs(2)3027 1646 y Fp(;)456 1861 y Fj(wher)-5 b(e)1030 2026 y Fp(d)1077 2040 y Fs(4)1142 2026 y Fr(=)25 b Fp(C)1310 1978 y Fs(1)p Fo(=)p Fs(4)1303 2052 y(8)1490 1964 y Fp(d)1537 1978 y Fs(1)1576 1964 y Fr(\(tr)1683 1978 y Fs(0)1722 1964 y Fp(Q)p Fr(\))1829 1931 y Fs(2)p 1429 2005 500 4 v 1429 2090 a Fp(\025)1482 2059 y Fs(2)1482 2116 y(1)1522 2090 y Fp(\027)1573 2064 y Fs(2)1612 2090 y Fr(\()p Fp(k)f Fr(+)c(1\))1889 2064 y Fs(2)1959 2026 y Fr(+)g Fp(C)2122 1978 y Fs(1)p Fo(=)p Fs(4)2115 2052 y(4)2317 1964 y Fp(d)2364 1978 y Fs(2)2404 1964 y Fr(tr)2475 1978 y Fs(0)2514 1964 y Fp(Q)p 2241 2005 421 4 v 2241 2088 a(\025)2294 2102 y Fs(1)2334 2088 y Fp(\027)6 b Fr(\()p Fp(k)23 b Fr(+)d(1\))2692 2026 y(+)g Fp(d)2830 2040 y Fs(3)456 2245 y Fj(is)32 b(an)h(estimate)h(for)f Fr(\()p Fg(E)11 b Fp(m)1361 2212 y Fs(4)1406 2245 y Fr(\))1441 2212 y Fs(1)p Fo(=)p Fs(4)1584 2245 y Fj(\(the)33 b(c)-5 b(onstants)35 b Fp(C)2237 2259 y Fs(8)2276 2245 y Fp(;)c(C)2397 2259 y Fs(4)2469 2245 y Fj(ar)-5 b(e)33 b(de\014ne)-5 b(d)34 b(in)e(\(14\)\).)456 2382 y Fr(The)43 b(pro)s(of)g(of)h(this)f(lemma)g(can)h(b)s(e)f(found)g (in)g(Ch)m(uesho)m(v)g(et.)82 b(al.)f([13)q(])44 b(for)f(an)456 2490 y(Ornstein-Uhlen)m(b)s(ec)m(k)37 b(pro)s(cess)j(in)e(another)i (Hilb)s(ert)e(space.)69 b(Ho)m(w)m(ev)m(er)42 b(the)e(argu-)456 2598 y(men)m(t)30 b(giv)m(en)h(there)f(is)g(of)g(a)h(general)f(nature.) 456 2706 y(W)-8 b(e)31 b(no)m(w)g(construct)f(a)h(set)g(satisfying)e (\(21\).)456 2842 y Fx(Lemma)e(4.4.)39 b Fj(L)-5 b(et)28 b Fp(R)q Fr(\()p Fp(!)s Fr(\))e(:=)f Fp(aR)1638 2856 y Fs(0)1678 2842 y Fr(\()p Fp(!)s Fr(\))k Fj(for)g(some)g Fp(a)c(>)g Fr(1)k Fj(and)h Fp(R)2691 2856 y Fs(0)2758 2842 y Fj(as)g(in)e(L)-5 b(emma)30 b(4.3.)456 2968 y(Then)f(the)h (close)-5 b(d)31 b Fx(H)1181 2935 y Fh(\000)p Fs(1)1181 2991 y Fo(per)1287 2968 y Fj(-b)-5 b(al)5 b(l)29 b Fp(B)5 b Fr(\(0)p Fp(;)15 b(R)q Fr(\()p Fp(!)s Fr(\))1887 2905 y Fl(1)p 1888 2917 31 3 v 1888 2958 a(2)1933 2968 y Fr(\))29 b Fj(ful)5 b(l\014l)g(ls)30 b(\(21\))g(pr)-5 b(ovide)g(d)32 b(c)-5 b(onditions)31 b(\(22\))456 3076 y(hold.)456 3259 y(Pr)-5 b(o)g(of.)43 b Fr(Using)29 b(Lemma)i(4.2)g(and)f(relation)f (\(7\))j(w)m(e)f(ha)m(v)m(e)839 3418 y Fp(d)p 823 3458 81 4 v 823 3542 a(dt)913 3479 y Fn(k)p Fp(q)s Fr(\()p Fp(t)p Fr(\))p Fn(k)1150 3442 y Fs(2)1150 3502 y Fh(\003)1216 3479 y Fn(\024)1312 3351 y Fk(\022)1379 3479 y Fn(\000)1460 3418 y Fp(\027)6 b(\025)1564 3432 y Fs(1)p 1460 3458 144 4 v 1487 3542 a Fp(a)1535 3556 y Fs(0)1633 3479 y Fr(+)20 b Fp(d)1771 3493 y Fs(0)1831 3479 y Fn(\001)g(k)p Fp(\021)s Fr(\()p Fp(\022)2047 3493 y Fo(t)2078 3479 y Fp(!)s Fr(\))p Fn(k)2218 3442 y Fs(2)2218 3502 y(0)2258 3351 y Fk(\023)2345 3479 y Fn(\001)g(k)p Fp(q)s Fr(\()p Fp(t)p Fr(\))p Fn(k)2627 3442 y Fs(2)2627 3502 y Fh(\003)2688 3479 y Fr(+)g Fp(m)p Fr(\()p Fp(\022)2937 3493 y Fo(t)2967 3479 y Fp(!)s Fr(\))p Fp(:)456 3707 y Fr(Let)33 b Fp(q)662 3721 y Fs(0)729 3707 y Fr(=)c Fp(q)s Fr(\(0\))k(and)f Fp(\032)p Fr(\()p Fp(t;)15 b(!)s(;)g Fn(k)p Fp(q)1541 3721 y Fs(0)1581 3707 y Fn(k)1626 3674 y Fs(2)1626 3729 y Fh(\003)1666 3707 y Fr(\))33 b(b)s(e)f(the)g(solution)f(of)i(one)g (dimensional)c(random)456 3815 y(a\016ne)h(equation)456 4027 y(\(23\))755 3966 y Fp(d\032)p Fr(\()p Fp(t)p Fr(\))p 755 4006 198 4 v 814 4090 a Fp(dt)983 4027 y Fr(+)1084 3966 y Fp(\027)6 b(\025)1188 3980 y Fs(1)p 1084 4006 144 4 v 1112 4090 a Fp(a)1160 4104 y Fs(0)1237 4027 y Fp(\032)25 b Fr(=)g Fp(d)1452 4041 y Fs(0)1492 4027 y Fn(k)p Fp(\021)s Fr(\()p Fp(\022)1663 4041 y Fo(t)1693 4027 y Fp(!)s Fr(\))p Fn(k)1833 3990 y Fs(2)1833 4050 y(0)1874 4027 y Fp(\032)20 b Fr(+)g Fp(m)p Fr(\()p Fp(\022)2190 4041 y Fo(t)2219 4027 y Fp(!)s Fr(\))p Fp(;)107 b(\032)p Fr(\(0)p Fp(;)15 b(!)s(;)g Fn(k)p Fp(q)2799 4041 y Fs(0)2840 4027 y Fn(k)2885 3990 y Fs(2)2885 4050 y Fh(\003)2924 4027 y Fr(\))26 b(=)f Fn(k)p Fp(q)3167 4041 y Fs(0)3207 4027 y Fn(k)3252 3990 y Fs(2)3252 4050 y Fh(\003)3291 4027 y Fp(:)456 4232 y Fr(A)30 b(comparison)g(argumen)m(t)g(giv)m(es)h (that)1170 4397 y Fn(k)p Fp(')p Fr(\()p Fp(t;)15 b(!)s(;)g(q)1523 4411 y Fs(0)1563 4397 y Fr(\))p Fn(k)1643 4359 y Fs(2)1643 4419 y Fh(\003)1709 4397 y Fn(\021)25 b(k)p Fp(q)s Fr(\()p Fp(t)p Fr(\))p Fn(k)2042 4359 y Fs(2)2042 4419 y Fh(\003)2108 4397 y Fn(\024)g Fp(\032)p Fr(\()p Fp(t;)15 b(!)s(;)g Fn(k)p Fp(q)2545 4411 y Fs(0)2585 4397 y Fn(k)2630 4359 y Fs(2)2630 4419 y Fh(\003)2670 4397 y Fr(\))p Fp(:)456 4562 y Fr(Here)33 b Fp(')f Fr(is)g(the)h(dynamical)e(system)h(in)m(tro) s(duced)f(in)g(Section)i(3:)45 b Fp(')p Fr(\()p Fp(t;)15 b(!)s(;)g(q)3068 4576 y Fs(0)3108 4562 y Fr(\))29 b(=)g Fp(q)s Fr(\()p Fp(t)p Fr(\),)456 4670 y(where)d Fp(q)s Fr(\()p Fp(t)p Fr(\))i(is)e(the)i(solution)d(to)j(\(19\))h(with)d(the)h (initial)e(data)j Fp(q)2631 4684 y Fs(0)2670 4670 y Fr(.)39 b(Equation)27 b(\(23\))i(has)456 4778 y(the)h(stationary)h(solution)e (giv)m(en)h(b)m(y)g Fp(t)25 b Fn(!)h Fp(R)1996 4792 y Fs(0)2035 4778 y Fr(\()p Fp(\022)2113 4792 y Fo(t)2143 4778 y Fp(!)s Fr(\):)1456 4943 y Fp(\032)p Fr(\()p Fp(t;)15 b(!)s(;)g(R)1780 4957 y Fs(0)1820 4943 y Fr(\()p Fp(!)s Fr(\)\))26 b(=)f Fp(R)2176 4957 y Fs(0)2216 4943 y Fr(\()p Fp(\022)2294 4957 y Fo(t)2324 4943 y Fp(!)s Fr(\))p Fp(:)456 5108 y Fr(This)36 b(can)i(b)s(e)f(c)m(hec)m(k)m(ed)j(b)m(y)d(the)h(v)-5 b(ariation)37 b(of)h(constan)m(ts)h(form)m(ula.)63 b(This)36 b(solution)456 5216 y(is)43 b(exp)s(onen)m(tially)f(attracting)k(whic)m (h)d(follo)m(ws)g(from)h(the)g(v)-5 b(ariation)44 b(of)g(constan)m(ts)p eop %%Page: 14 14 14 13 bop 456 251 a Fs(14)327 b(IGOR)24 b(CHUESHO)n(V,)f(JINQIA)n(O)h (DUAN,)e(AND)h(BJ)2368 234 y(\177)2358 251 y(ORN)h(SCHMALFUSS)456 450 y Fr(form)m(ula)29 b(again:)668 635 y Fn(j)p Fp(R)762 649 y Fs(0)802 635 y Fr(\()p Fp(\022)880 649 y Fo(t)909 635 y Fp(!)s Fr(\))21 b Fn(\000)f Fp(\032)p Fr(\()p Fp(t;)15 b(!)s(;)g Fn(k)p Fp(q)1457 649 y Fs(0)1497 635 y Fn(k)1542 597 y Fs(2)1542 657 y Fh(\003)1582 635 y Fr(\))p Fn(j)26 b Fr(=)f Fn(j)p Fp(\032)p 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Fp(")d(>)f Fr(0,)1004 1251 y Fk(Z)1095 1277 y Fo(t)1055 1457 y Fs(0)1140 1375 y Fp(d)1187 1389 y Fs(0)1227 1375 y Fn(k)p Fp(\021)s Fr(\()p Fp(\022)1398 1389 y Fo(\034)1442 1375 y Fp(!)s Fr(\))p Fn(k)1582 1337 y Fs(2)1582 1397 y(0)1622 1375 y Fp(d\034)36 b(<)1851 1313 y(\027)6 b(\025)1955 1327 y Fs(1)2014 1313 y Fn(\000)20 b Fp(")p 1851 1354 297 4 v 1955 1437 a(a)2003 1451 y Fs(0)2157 1375 y Fp(t;)2422 1313 y(\027)6 b(\025)2526 1327 y Fs(1)2586 1313 y Fn(\000)19 b Fp(")p 2422 1354 V 2527 1437 a(a)2575 1451 y Fs(0)2754 1375 y Fp(>)25 b Fr(0)456 1610 y(for)30 b(large)g Fp(t)25 b(>)g Fr(0)31 b(under)e(conditions)g(\(22\).)1449 b Fd(\003)456 1819 y Fr(It)30 b(remains)f(to)i(pro)m(v)m(e)h(the)e(existence)h(of)g (a)f(compact)i(set)f Fp(B)k Fr(satisfying)29 b(\(23\).)456 1966 y Fx(Lemma)35 b(4.5.)43 b Fj(Supp)-5 b(ose)36 b(that)g(the)f(r)-5 b(andom)37 b(variable)f Fp(R)q Fr(\()p Fp(!)s Fr(\))e Fj(is)h(de\014ne)-5 b(d)36 b(in)e(L)-5 b(emma)456 2073 y(4.4.)42 b(The)33 b(set)1237 2252 y Fp(B)5 b Fr(\()p Fp(!)s Fr(\))25 b(:=)p 1588 2146 1076 4 v 26 w Fp(')p Fr(\(1)p Fp(;)15 b(\022)1810 2266 y Fh(\000)p Fs(1)1905 2252 y Fp(!)s(;)g(B)5 b Fr(\(0)p Fp(;)15 b(R)q Fr(\()p Fp(\022)2347 2266 y Fh(\000)p Fs(1)2442 2252 y Fp(!)s Fr(\))2547 2189 y Fl(1)p 2548 2201 31 3 v 2548 2242 a(2)2592 2252 y Fr(\)\))456 2417 y Fj(is)32 b(a)h(c)-5 b(omp)g(act)35 b(absorbing)e(forwar)-5 b(d)36 b(invariant)d(r)-5 b(andom)35 b(set.)42 b(Mor)-5 b(e)g(over)456 2602 y Fr(\(24\))388 b Fp(!)28 b Fn(7!)d Fr(sup)1357 2528 y Fk(\010)1410 2602 y Fn(k)p Fr(\001)p Fp( )1590 2616 y Fs(1)1630 2602 y Fn(k)1675 2564 y Fs(2)1675 2624 y(0)1735 2602 y Fr(+)20 b Fn(k)p Fr(\001)p Fp( )2006 2616 y Fs(1)2046 2602 y Fn(k)2091 2564 y Fs(2)2091 2624 y(0)2181 2602 y Fr(:)51 b(\()p Fp(q)2333 2616 y Fs(1)2372 2602 y Fp(;)15 b(q)2453 2616 y Fs(2)2493 2602 y Fr(\))25 b Fn(2)g Fp(B)5 b Fr(\()p Fp(!)s Fr(\))2843 2528 y Fk(\011)456 2787 y Fj(is)32 b(a)h(temp)-5 b(er)g(e)g(d)35 b(r)-5 b(andom)35 b(variable)e(\()p Fp( )1779 2801 y Fs(1)1852 2787 y Fj(and)g Fp( )2087 2801 y Fs(2)2160 2787 y Fj(ar)-5 b(e)33 b(de\014ne)-5 b(d)34 b(by)e(\(20\)\).)456 2996 y(Pr)-5 b(o)g(of.)43 b Fr(Since)30 b Fp(')p Fr(\()p Fp(t;)15 b(!)s(;)g Fn(\001)p Fr(\))34 b(is)c(completely)i(con)m(tin)m(uous)f(for)g Fp(t)d(>)f Fr(0)32 b(\(see)g(the)g(regularit)m(y)456 3104 y(assertion)e(of)i(Theorem)f(3.2\))h(the)g(sets)f Fp(B)5 b Fr(\()p Fp(!)s Fr(\))31 b(are)h(compact.)44 b(Since)31 b Fp(R)g Fr(is)g(a)g(random)456 3230 y(v)-5 b(ariable)23 b(the)h(ball)f Fp(B)5 b Fr(\(0)p Fp(;)15 b(R)1386 3167 y Fl(1)p 1387 3179 V 1387 3220 a(2)1431 3230 y Fr(\))25 b(is)e(a)i(random)f(set.)39 b(The)24 b(con)m(tin)m(uit)m(y)g(of)h Fp(')p Fr(\()p Fp(t;)15 b(!)s(;)g Fn(\001)p Fr(\))26 b(allo)m(ws)456 3338 y(us)k(to)h(conclude) f(that)i Fp(B)j Fr(is)30 b(a)h(random)f(set.)43 b(The)30 b(construction)h(of)g Fp(B)k Fr(ensures)30 b(that)456 3446 y(that)i(set)g(in)f(absorbing)f(and)h(forw)m(ard)h(in)m(v)-5 b(arian)m(t.)44 b(The)31 b(temp)s(eredness)g(of)h(\(24\))h(can)456 3554 y(b)s(e)23 b(pro)m(v)m(ed)h(in)f(the)i(same)f(w)m(a)m(y)h(as)f(in) f([13)q(])i(for)f(the)g(2)p Fp(D)j Fr(Na)m(vier-Stok)m(es)f(equations.) 91 b Fd(\003)456 3763 y Fr(Hence)42 b(w)m(e)h(ha)m(v)m(e)g(sho)m(wn)f (that)g(the)g(t)m(w)m(o-la)m(y)m(er)j(quasigeostrophic)c(\015o)m(w)h (system)g(is)456 3871 y(dissipativ)m(e)37 b(in)h(the)h(probabilistic)c (sense.)67 b(In)39 b(the)g(next)g(section)g(w)m(e)h(consider)e(as-)456 3979 y(ymptotic)30 b(probabilistic)d(determining)h(functionals.)1280 4274 y(5.)46 b Fq(Determinging)35 b(functionals)456 4435 y Fr(In)23 b(this)g(section,)i(w)m(e)g(compare)f(t)m(w)m(o)h(solutions) e(as)h(a)g(w)m(a)m(y)h(to)g(understand)d(asymptotic)456 4543 y(dynamics)j(in)h(the)h(probabilistic)c(sense.)39 b(Consider)25 b(the)i(di\013erence)g(of)g(t)m(w)m(o)h(solutions)462 4656 y(^)-51 b Fp(q)33 b Fr(and)j(\026)-51 b Fp(q)r Fr(,)31 b(corresp)s(onding)d(to)j(the)g(stream)g(functions)2367 4633 y(^)2349 4656 y Fp( )s Fr(,)2485 4633 y(\026)2467 4656 y Fp( )i Fr(.)41 b(W)-8 b(e)32 b(set)1559 4841 y Fp(q)c Fr(=)d(\()p Fp(q)1800 4855 y Fs(1)1840 4841 y Fp(;)15 b(q)1921 4855 y Fs(2)1960 4841 y Fr(\))26 b(=)31 b(^)-51 b Fp(q)23 b Fn(\000)j Fr(\026)-51 b Fp(q)r(;)456 5026 y Fr(corresp)s(onding)28 b(to)j(the)g(stream)f(function)1727 5216 y Fp( )f Fr(=)1930 5192 y(^)1911 5216 y Fp( )24 b Fn(\000)2103 5192 y Fr(\026)2085 5216 y Fp( )s(:)p eop %%Page: 15 15 15 14 bop 1043 251 a Fs(STOCHASTIC)31 b(TW)n(O-LA)-6 b(YER)30 b(GEOPHYSICAL)h(FLO)n(WS)549 b(15)456 450 y Fr(W)-8 b(e)32 b(get)g(random)e(partial)f(di\013eren)m(tial)h (equations)g(for)h(the)g(solution)e(di\013erence)i Fp(q)e Fr(=)462 558 y(^)-51 b Fp(q)23 b Fn(\000)j Fr(\026)-51 b Fp(q)33 b Fr(from)d(\(19\):)822 712 y Fp(q)863 726 y Fs(1)p Fo(t)953 712 y Fr(=)o Fp(\027)6 b Fr(\001)1150 675 y Fs(2)1189 712 y Fp( )1248 726 y Fs(1)1044 859 y Fn(\000)19 b Fp(J)9 b Fr(\()p Fp( )1287 873 y Fs(1)1328 859 y Fp(;)22 b Fr(^)-52 b Fp(q)1409 873 y Fs(1)1468 859 y Fr(+)20 b Fp(\014)5 b(y)s Fr(\))21 b Fn(\000)f Fp(J)9 b Fr(\()1923 835 y(\026)1904 859 y Fp( )1963 873 y Fs(1)2003 859 y Fp(;)15 b(q)2084 873 y Fs(1)2123 859 y Fr(\))21 b Fn(\000)f Fp(J)9 b Fr(\()p Fp( )2423 873 y Fs(1)2463 859 y Fp(;)15 b(\021)s Fr(\))21 b(+)f Fp(J)9 b Fr(\()p Fp(\030)2832 873 y Fs(1)2872 859 y Fp(;)15 b(q)2953 873 y Fs(1)2992 859 y Fr(\))26 b Fp(;)-2622 b Fr(\(25\))822 1012 y Fp(q)863 1026 y Fs(2)p Fo(t)953 1012 y Fr(=)o Fp(\027)6 b Fr(\001)1150 974 y Fs(2)1189 1012 y Fp( )1248 1026 y Fs(2)1308 1012 y Fn(\000)20 b Fp(r)s Fr(\001)p Fp( )1578 1026 y Fs(2)1044 1159 y Fn(\000)f Fp(J)9 b Fr(\()p Fp( )1287 1173 y Fs(2)1328 1159 y Fp(;)22 b Fr(^)-52 b Fp(q)1409 1173 y Fs(2)1468 1159 y Fr(+)20 b Fp(\014)5 b(y)s Fr(\))21 b Fn(\000)f Fp(J)9 b Fr(\()1923 1135 y(\026)1904 1159 y Fp( )1963 1173 y Fs(2)2003 1159 y Fp(;)15 b(q)2084 1173 y Fs(2)2123 1159 y Fr(\))21 b(+)f Fp(J)9 b Fr(\()p Fp(\030)2404 1173 y Fs(2)2443 1159 y Fp(;)15 b(q)2524 1173 y Fs(2)2564 1159 y Fr(\))25 b Fp(;)-2193 b Fr(\(26\))456 1315 y(Multiplying)23 b(\(25\))28 b(b)m(y)f Fn(\000)p Fp(h)1381 1329 y Fs(1)1421 1315 y Fp( )1480 1329 y Fs(1)1519 1315 y Fr(,)h(and)e(\(26\))i(b)m(y)f Fn(\000)p Fp(h)2179 1329 y Fs(2)2219 1315 y Fp( )2278 1329 y Fs(2)2344 1315 y Fr(and)f(adding)g(together,)j(w)m(e)e(get)794 1452 y(1)p 794 1492 46 4 v 794 1575 a(2)876 1452 y Fp(d)p 859 1492 81 4 v 859 1575 a(dt)949 1513 y Fn(k)p Fp(q)s Fn(k)1083 1476 y Fs(2)1083 1536 y Fh(\003)1149 1513 y Fr(=)20 b Fn(\000)f Fp(\027)6 b Fr([)p Fp(h)1458 1527 y Fs(1)1498 1513 y Fn(k)p Fr(\001)p Fp( )1678 1527 y Fs(1)1718 1513 y Fn(k)1763 1476 y Fs(2)1763 1536 y(0)1823 1513 y Fr(+)20 b Fp(h)1966 1527 y Fs(2)2006 1513 y Fn(k)p Fr(\001)p Fp( )2186 1527 y Fs(2)2226 1513 y Fn(k)2271 1476 y Fs(2)2271 1536 y(0)2311 1513 y Fr(])g Fn(\000)g Fp(r)s(h)2543 1527 y Fs(2)2582 1513 y Fn(kr)p Fp( )2762 1527 y Fs(2)2802 1513 y Fn(k)2847 1476 y Fs(2)2847 1536 y(0)1149 1686 y Fr(+)o Fp(h)1271 1700 y Fs(1)1311 1686 y Fr(\()p Fp(J)9 b Fr(\()1460 1662 y(\026)1440 1686 y Fp( )1499 1700 y Fs(1)1560 1686 y Fn(\000)20 b Fp(\030)1691 1700 y Fs(1)1730 1686 y Fp(;)15 b(q)1811 1700 y Fs(1)1850 1686 y Fr(\))p Fp(;)g( )1984 1700 y Fs(1)2025 1686 y Fr(\))2060 1700 y Fs(0)2120 1686 y Fr(+)20 b Fp(h)2263 1700 y Fs(2)2303 1686 y Fr(\()p Fp(J)9 b Fr(\()2451 1662 y(\026)2432 1686 y Fp( )2491 1700 y Fs(2)2551 1686 y Fn(\000)20 b Fp(\030)2682 1700 y Fs(2)2721 1686 y Fp(;)15 b(q)2802 1700 y Fs(2)2842 1686 y Fr(\))p Fp(;)g( )2976 1700 y Fs(2)3016 1686 y Fr(\))3051 1700 y Fs(0)3091 1686 y Fp(:)456 1579 y Fr(\(27\))456 1837 y(W)-8 b(e)31 b(start)g(with)1302 1950 y Fp(h)1354 1964 y Fs(1)1393 1950 y Fr(\()p Fp(J)9 b Fr(\()1542 1926 y(\026)1522 1950 y Fp( )1581 1964 y Fs(1)1642 1950 y Fn(\000)20 b Fp(\030)1773 1964 y Fs(1)1812 1950 y Fp(;)15 b(q)1893 1964 y Fs(1)1933 1950 y Fr(\))p Fp(;)g( )2067 1964 y Fs(1)2107 1950 y Fr(\))2142 1964 y Fs(0)2207 1950 y Fn(\021)25 b Fp(I)2343 1964 y Fs(1)2403 1950 y Fr(+)20 b Fp(I)2534 1964 y Fs(2)2573 1950 y Fp(;)456 2081 y Fr(where)757 2232 y Fp(I)797 2246 y Fs(1)862 2232 y Fr(=)25 b Fp(h)1010 2246 y Fs(1)1050 2232 y Fr(\()p Fp(J)9 b Fr(\()1198 2208 y(\026)1179 2232 y Fp( )1238 2246 y Fs(1)1299 2232 y Fn(\000)19 b Fp(\030)1429 2246 y Fs(1)1469 2232 y Fp(;)c Fr(\001)p Fp( )1644 2246 y Fs(1)1683 2232 y Fr(\))p Fp(;)g( )1817 2246 y Fs(1)1858 2232 y Fr(\))1893 2246 y Fs(0)1933 2232 y Fp(;)106 b(I)2104 2246 y Fs(2)2169 2232 y Fr(=)25 b Fp(p)p Fr(\()p Fp(J)9 b Fr(\()2459 2208 y(\026)2440 2232 y Fp( )2499 2246 y Fs(1)2559 2232 y Fn(\000)20 b Fp(\030)2690 2246 y Fs(1)2729 2232 y Fp(;)15 b( )2828 2246 y Fs(2)2868 2232 y Fr(\))p Fp(;)g( )3002 2246 y Fs(1)3043 2232 y Fr(\))3078 2246 y Fs(0)3117 2232 y Fp(:)456 2383 y Fr(By)30 b(\(10\))i(w)m(e)f(ha)m(v)m (e)1212 2533 y Fn(j)p Fp(I)1277 2547 y Fs(1)1317 2533 y Fn(j)25 b(\024)g Fp(c)1502 2547 y Fs(0)1542 2533 y Fp(h)1594 2547 y Fs(1)1634 2533 y Fn(k)1698 2509 y Fr(\026)1679 2533 y Fp( )1738 2547 y Fs(1)1798 2533 y Fn(\000)20 b Fp(\030)1929 2547 y Fs(1)1968 2533 y Fn(k)2013 2547 y Fs(2)2053 2533 y Fn(k)p Fr(\001)p Fp( )2233 2547 y Fs(1)2273 2533 y Fn(k)2318 2547 y Fs(0)2358 2533 y Fn(kr)p Fp( )2538 2547 y Fs(1)2578 2533 y Fn(k)2623 2547 y Fs(0)2663 2533 y Fp(:)456 2684 y Fr(Using)29 b(\(12\))j(w)m(e)f(obtain)1235 2834 y Fn(j)p Fp(I)1300 2848 y Fs(2)1340 2834 y Fn(j)25 b(\024)g Fp(c)1525 2848 y Fs(1)1565 2834 y Fp(p)p Fn(k)1675 2810 y Fr(\026)1656 2834 y Fp( )1715 2848 y Fs(1)1775 2834 y Fn(\000)20 b Fp(\030)1906 2848 y Fs(1)1945 2834 y Fn(k)1990 2848 y Fs(2)2030 2834 y Fn(kr)p Fp( )2210 2848 y Fs(2)2250 2834 y Fn(k)2295 2848 y Fs(0)2335 2834 y Fn(kr)p Fp( )2515 2848 y Fs(1)2555 2834 y Fn(k)2600 2848 y Fs(0)2640 2834 y Fp(:)456 2985 y Fr(Th)m(us)1154 3115 y Fn(j)p Fp(I)1219 3129 y Fs(2)1259 3115 y Fn(j)25 b(\024)g Fp(c)1444 3129 y Fs(1)1484 3115 y Fp(p\025)1583 3057 y Fh(\000)1648 3030 y Fl(1)p 1647 3042 31 3 v 1647 3083 a(2)1583 3141 y Fs(1)1692 3115 y Fn(k)1756 3091 y Fr(\026)1737 3115 y Fp( )1796 3129 y Fs(1)1856 3115 y Fn(\000)20 b Fp(\030)1987 3129 y Fs(1)2026 3115 y Fn(k)2071 3129 y Fs(2)2111 3115 y Fn(k)p Fr(\001)p Fp( )2291 3129 y Fs(2)2331 3115 y Fn(k)2376 3129 y Fs(0)2416 3115 y Fn(kr)p Fp( )2596 3129 y Fs(1)2636 3115 y Fn(k)2681 3129 y Fs(0)2721 3115 y Fp(:)456 3252 y Fr(Using)29 b(the)i(inequalit)m(y)d Fp(ab)e Fn(\024)f Fp(\017a)1583 3219 y Fs(2)1642 3252 y Fr(+)1757 3216 y Fs(1)p 1743 3231 64 4 v 1743 3283 a(4)p Fo(\017)1817 3252 y Fp(b)1856 3219 y Fs(2)1895 3252 y Fr(,)31 b(w)m(e)g(get)871 3434 y Fp(h)923 3448 y Fs(1)963 3434 y Fr(\()p Fp(J)9 b Fr(\()1112 3410 y(\026)1092 3434 y Fp( )1151 3448 y Fs(1)1212 3434 y Fn(\000)20 b Fp(\030)1343 3448 y Fs(1)1382 3434 y Fp(;q)1448 3448 y Fs(1)1487 3434 y Fr(\))p Fp(;)15 b( )1621 3448 y Fs(1)1662 3434 y Fr(\))1697 3448 y Fs(0)1762 3434 y Fn(\024)1868 3373 y Fp(\027)p 1868 3414 51 4 v 1871 3497 a Fr(4)1943 3361 y Fk(\000)1985 3434 y Fp(h)2037 3448 y Fs(1)2077 3434 y Fn(k)p Fr(\001)p Fp( )2257 3448 y Fs(1)2297 3434 y Fn(k)2342 3397 y Fs(2)2342 3457 y(0)2402 3434 y Fr(+)20 b Fp(h)2545 3448 y Fs(2)2585 3434 y Fn(k)p Fr(\001)p Fp( )2765 3448 y Fs(2)2804 3434 y Fn(k)2849 3397 y Fs(2)2849 3457 y(0)2889 3361 y Fk(\001)1427 3669 y Fr(+)1528 3608 y Fp(h)1580 3622 y Fs(1)1620 3608 y Fp(c)1659 3575 y Fs(2)1659 3632 y(0)p 1528 3648 171 4 v 1588 3732 a Fp(\027)1709 3669 y Fr(\(1)h(+)f Fp(F)1959 3683 y Fs(1)1999 3669 y Fp(F)2057 3683 y Fs(2)2097 3669 y Fp(\025)2150 3631 y Fh(\000)p Fs(2)2150 3695 y(1)2244 3669 y Fr(\))p Fn(k)2343 3645 y Fr(\026)2324 3669 y Fp( )2383 3683 y Fs(1)2444 3669 y Fn(\000)g Fp(\030)2575 3683 y Fs(1)2614 3669 y Fn(k)2659 3632 y Fs(2)2659 3692 y(2)2699 3669 y Fn(kr)p Fp( )2879 3683 y Fs(1)2918 3669 y Fn(k)2963 3632 y Fs(2)2963 3692 y(0)3003 3669 y Fp(:)456 3556 y Fr(\(28\))456 3846 y(In)29 b(a)i(similar)d(w)m(a)m(y)j(w)m(e)g(ha)m(v)m(e)884 4015 y Fp(h)936 4029 y Fs(2)976 4015 y Fr(\()p Fp(J)9 b Fr(\()1124 3991 y(\026)1105 4015 y Fp( )1164 4029 y Fs(2)1225 4015 y Fn(\000)19 b Fp(\030)1355 4029 y Fs(2)1395 4015 y Fp(;)c(q)1476 4029 y Fs(2)1515 4015 y Fr(\))p Fp(;)g( )1649 4029 y Fs(2)1689 4015 y Fr(\))1724 4029 y Fs(0)1790 4015 y Fn(\024)1895 3954 y Fp(\027)p 1895 3994 51 4 v 1898 4077 a Fr(4)1971 3941 y Fk(\000)2013 4015 y Fp(h)2065 4029 y Fs(1)2105 4015 y Fn(k)p Fr(\001)p Fp( )2285 4029 y Fs(1)2325 4015 y Fn(k)2370 3978 y Fs(2)2370 4038 y(0)2430 4015 y Fr(+)20 b Fp(h)2573 4029 y Fs(2)2612 4015 y Fn(k)p Fr(\001)p Fp( )2792 4029 y Fs(2)2832 4015 y Fn(k)2877 3978 y Fs(2)2877 4038 y(0)2917 3941 y Fk(\001)1415 4250 y Fr(+)1516 4189 y Fp(h)1568 4203 y Fs(2)1608 4189 y Fp(c)1647 4156 y Fs(2)1647 4213 y(0)p 1516 4229 171 4 v 1576 4312 a Fp(\027)1696 4250 y Fr(\(1)h(+)f Fp(F)1946 4264 y Fs(1)1986 4250 y Fp(F)2044 4264 y Fs(2)2084 4250 y Fp(\025)2137 4212 y Fh(\000)p Fs(2)2137 4276 y(1)2231 4250 y Fr(\))p Fn(k)2331 4226 y Fr(\026)2311 4250 y Fp( )2370 4264 y Fs(2)2431 4250 y Fn(\000)g Fp(\030)2562 4264 y Fs(2)2601 4250 y Fn(k)2646 4213 y Fs(2)2646 4273 y(2)2686 4250 y Fn(kr)p Fp( )2866 4264 y Fs(2)2906 4250 y Fn(k)2951 4213 y Fs(2)2951 4273 y(0)2991 4250 y Fp(:)456 4137 y Fr(\(29\))456 4432 y(Using)29 b(\(27\),)j(\(28\))g(and)e(\(29\))i(w)m (e)f(obtain)824 4568 y Fp(d)p 807 4609 81 4 v 807 4692 a(dt)897 4630 y Fn(k)p Fp(q)s Fn(k)1031 4592 y Fs(2)1031 4652 y Fh(\003)1097 4630 y Fn(\024)20 b(\000)g Fp(\027)6 b Fr([)p Fp(h)1407 4644 y Fs(1)1446 4630 y Fn(k)p Fr(\001)p Fp( )1626 4644 y Fs(1)1666 4630 y Fn(k)1711 4592 y Fs(2)1711 4652 y(0)1771 4630 y Fr(+)20 b Fp(h)1914 4644 y Fs(2)1954 4630 y Fn(k)p Fr(\001)p Fp( )2134 4644 y Fs(2)2174 4630 y Fn(k)2219 4592 y Fs(2)2219 4652 y(0)2259 4630 y Fr(])g Fn(\000)g Fr(2)p Fp(r)s(h)2536 4644 y Fs(2)2576 4630 y Fn(kr)p Fp( )2756 4644 y Fs(2)2796 4630 y Fn(k)2841 4592 y Fs(2)2841 4652 y(0)1188 4808 y Fr(+)g Fp(b)1318 4822 y Fs(0)1372 4734 y Fk(\000)1414 4808 y Fp(h)1466 4822 y Fs(1)1506 4808 y Fn(k)1570 4784 y Fr(\026)1551 4808 y Fp( )1610 4822 y Fs(1)1670 4808 y Fn(\000)g Fp(\030)1801 4822 y Fs(1)1840 4808 y Fn(k)1885 4770 y Fs(2)1885 4830 y(2)1925 4808 y Fn(kr)p Fp( )2105 4822 y Fs(1)2145 4808 y Fn(k)2190 4770 y Fs(2)2190 4830 y(0)2250 4808 y Fr(+)g Fp(h)2393 4822 y Fs(2)2433 4808 y Fn(k)2497 4784 y Fr(\026)2478 4808 y Fp( )2537 4822 y Fs(2)2597 4808 y Fn(\000)g Fp(\030)2728 4822 y Fs(2)2767 4808 y Fn(k)2812 4770 y Fs(2)2812 4830 y(2)2852 4808 y Fn(kr)p Fp( )3032 4822 y Fs(2)3072 4808 y Fn(k)3117 4770 y Fs(2)3117 4830 y(0)3157 4734 y Fk(\001)3239 4808 y Fp(;)456 4698 y Fr(\(30\))456 5000 y(where)33 b Fp(b)761 5014 y Fs(0)831 5000 y Fr(=)943 4951 y Fs(2)p Fo(c)1009 4927 y Fl(2)1009 4972 y(0)p 943 4979 101 4 v 973 5031 a Fo(\027)1053 5000 y Fr(\(1)23 b(+)g Fp(F)1308 5014 y Fs(1)1347 5000 y Fp(F)1405 5014 y Fs(2)1445 5000 y Fp(\025)1498 4961 y Fh(\000)p Fs(2)1498 5026 y(1)1593 5000 y Fr(\).)51 b(This)32 b(estimate)i(is)f(the)h(main)f(p)s(oin)m(t)f (in)h(the)h(con-)456 5108 y(struction)41 b(of)g(determining)f (functionals.)73 b(It)42 b(is)f(our)g(aim)g(to)i(obtain)e(an)h(in)m (tegral)456 5216 y(equation)c(lik)m(e)h(\(5\).)67 b(T)-8 b(o)39 b(this)f(end)g(w)m(e)i(\014rst)e(consider)g(the)h(case)h(when)e (parameter)p eop %%Page: 16 16 16 15 bop 456 251 a Fs(16)327 b(IGOR)24 b(CHUESHO)n(V,)f(JINQIA)n(O)h (DUAN,)e(AND)h(BJ)2368 234 y(\177)2358 251 y(ORN)h(SCHMALFUSS)456 454 y Fp(r)k(>)d Fr(0)32 b(is)d(arbitrary)-8 b(.)41 b(Let)31 b Fn(L)g Fr(=)f Fn(f)p Fp(l)1635 468 y Fo(j)1672 454 y Fn(g)1718 412 y Fo(N)1718 485 y Fs(1)1815 454 y Fr(b)s(e)g(a)i(set)f (of)g(linearly)d(indep)s(enden)m(t)g(b)s(ounded)456 569 y(linear)k(functionals)h(on)h(the)g(space)h Fx(H)1798 536 y Fs(2)1798 592 y Fo(per)1904 569 y Fr(.)52 b(Assume)34 b(that)g(the)h(set)g Fn(L)e Fr(p)s(ossesses)h(the)456 677 y(prop)s(ert)m(y)456 856 y(\(31\))683 b Fn(k)p Fp( )s Fn(k)1451 870 y Fs(1)1517 856 y Fn(\024)25 b Fp(C)1678 870 y Fh(L)1745 856 y Fr(max)1814 915 y Fo(j)1930 856 y Fn(j)p Fp(l)1982 870 y Fo(j)2019 856 y Fr(\()p Fp( )s Fr(\))p Fn(j)c Fr(+)f Fp(")2330 870 y Fh(L)2383 856 y Fn(k)p Fp( )s Fn(k)2535 870 y Fs(2)2576 856 y Fp(;)456 1092 y Fr(for)38 b(an)m(y)g Fp( )k Fn(2)c Fx(H)1064 1059 y Fs(2)1064 1114 y Fo(per)1209 1092 y Fr(with)f(some)h(p)s(ositiv)m(e)g (constan)m(t)h Fp(C)2443 1106 y Fh(L)2534 1092 y Fr(and)f Fp(")2761 1106 y Fh(L)2814 1092 y Fr(.)64 b(W)-8 b(e)40 b(note)f(that)456 1199 y(the)26 b(b)s(est)g(p)s(ossible)e(v)-5 b(alue)26 b(of)h(the)g(parameter)g Fp(")2094 1213 y Fh(L)2173 1199 y Fr(is)f(called)f(the)i(completeness)g(defect)456 1307 y(of)36 b(the)g(family)f Fn(L)h Fr(with)f(resp)s(ect)h(of)g(the)h (pair)e(of)h(the)h(spaces)f Fx(H)2743 1274 y Fs(1)2743 1330 y Fo(per)2886 1307 y Fr(and)f Fx(H)3150 1274 y Fs(2)3150 1330 y Fo(per)3292 1307 y Fr(\(see)456 1419 y([10)q(,)30 b(11)q(]\))h(and)f(the)g(smallness)e(of)j(the)f(parameter)h Fp(")2279 1433 y Fh(L)2362 1419 y Fr(is)f(crucial)f(in)g(the)h (subsequen)m(t)456 1527 y(considerations.)55 b(W)-8 b(e)37 b(refer)f(to)h([10)q(,)f(11)q(])g(for)f(general)h(prop)s(erties)e(of)i (completeness)456 1635 y(defect)c(and)e(for)h(estimates)h(of)f Fp(")1586 1649 y Fh(L)1670 1635 y Fr(for)g(sev)m(eral)h(families)d(of)i (functionals)e(on)i(Sob)s(olev)456 1743 y(spaces.)456 1851 y(F)-8 b(rom)30 b(\(31\))i(for)f Fp( )d Fr(=)d(\()1244 1780 y Fn(p)p 1320 1780 92 4 v 71 x Fp(h)1372 1865 y Fs(1)1412 1851 y Fp( )1471 1865 y Fs(1)1511 1851 y Fp(;)1551 1780 y Fn(p)p 1627 1780 V 71 x Fp(h)1679 1865 y Fs(2)1719 1851 y Fp( )1778 1865 y Fs(2)1817 1851 y Fr(\))31 b(w)m(e)g(ha)m(v)m(e) 680 2004 y Fk(\000)722 2077 y Fp(h)774 2091 y Fs(1)813 2077 y Fn(k)p Fr(\001)p Fp( )993 2091 y Fs(1)1033 2077 y Fn(k)1078 2040 y Fs(2)1078 2100 y(0)1138 2077 y Fr(+)20 b Fp(h)1281 2091 y Fs(2)1321 2077 y Fn(k)p Fr(\001)p Fp( )1501 2091 y Fs(2)1541 2077 y Fn(k)1586 2040 y Fs(2)1586 2100 y(0)1626 2004 y Fk(\001)1693 2077 y Fn(\025)1773 2016 y Fr(1)h Fn(\000)f Fp(\016)p 1773 2056 201 4 v 1826 2141 a(")1868 2110 y Fs(2)1868 2170 y Fh(L)1999 2004 y Fk(\000)2041 2077 y Fp(h)2093 2091 y Fs(1)2132 2077 y Fn(kr)p Fp( )2312 2091 y Fs(1)2352 2077 y Fn(k)2397 2040 y Fs(2)2397 2100 y(0)2457 2077 y Fr(+)g Fn(k)p Fp(h)2645 2091 y Fs(2)2686 2077 y Fn(r)p Fp( )2821 2091 y Fs(2)2860 2077 y Fn(k)2905 2040 y Fs(2)2905 2100 y(0)2945 2004 y Fk(\001)1784 2297 y Fn(\000)g Fp(C)1940 2312 y Fo(\016)o(;)p Fh(L)2120 2297 y Fr(max)2057 2357 y Fo(j)t Fs(=1)p Fo(;)p Fh(\001\001\001)10 b Fo(;N)2367 2297 y Fn(j)p Fp(l)2419 2311 y Fo(j)2456 2297 y Fr(\()2491 2213 y Fk(p)p 2583 2213 92 4 v 2583 2297 a Fp(h)2635 2311 y Fs(1)2674 2297 y Fp( )2733 2311 y Fs(1)2773 2297 y Fp(;)2813 2213 y Fk(p)p 2904 2213 V 84 x Fp(h)2956 2311 y Fs(2)2996 2297 y Fp( )3055 2311 y Fs(2)3095 2297 y Fr(\))p Fn(j)3155 2259 y Fs(2)3195 2297 y Fp(:)456 2521 y Fr(for)32 b(an)m(y)h(0)d Fp(<)e(\016)33 b(<)c Fr(1)k(with)e(appropriate)h(p)s(ositiv)m(e)g (constan)m(t)i Fp(C)2671 2536 y Fo(\016)o(;)p Fh(L)2773 2521 y Fr(.)47 b(Therefore)33 b(from)456 2629 y(\(30\))f(and)d(\(7\))j (w)m(e)f(obtain)456 2856 y(\(32\))866 2794 y Fp(d)p 849 2835 81 4 v 849 2918 a(dt)939 2856 y Fn(k)p Fp(q)s Fn(k)1073 2818 y Fs(2)1073 2878 y Fh(\003)1139 2856 y Fn(\024)25 b Fp(l)1264 2808 y Fs(\(1\))1262 2884 y Fh(L)1358 2856 y Fr(\()1412 2832 y(\026)1393 2856 y Fp( )t(;)15 b(!)s Fr(\))p Fn(k)p Fp(q)s Fn(k)1725 2818 y Fs(2)1725 2878 y Fh(\003)1786 2856 y Fr(+)20 b Fp(C)1942 2871 y Fo(\016)o(;)p Fh(L)2122 2856 y Fr(max)2059 2916 y Fo(j)t Fs(=1)p Fo(;)p Fh(\001\001\001)10 b Fo(;N)2370 2856 y Fn(j)p Fp(l)2422 2870 y Fo(j)2458 2856 y Fr(\()2493 2772 y Fk(p)p 2585 2772 92 4 v 2585 2856 a Fp(h)2637 2870 y Fs(1)2677 2856 y Fp( )2736 2870 y Fs(1)2775 2856 y Fp(;)2815 2772 y Fk(p)p 2907 2772 V 2907 2856 a Fp(h)2959 2870 y Fs(2)2998 2856 y Fp( )3057 2870 y Fs(2)3097 2856 y Fr(\))p Fn(j)3157 2818 y Fs(2)3197 2856 y Fp(;)456 3080 y Fr(where)846 3284 y Fp(l)875 3236 y Fs(\(1\))873 3313 y Fh(L)969 3284 y Fr(\()1023 3260 y(\026)1004 3284 y Fp( )t(;)15 b(!)s Fr(\))26 b(=)f Fn(\000)1405 3222 y Fr(1)20 b Fn(\000)g Fp(\016)p 1405 3263 201 4 v 1414 3348 a(a)1462 3362 y Fs(0)1501 3348 y Fp(")1543 3317 y Fs(2)1543 3377 y Fh(L)1635 3284 y Fr(+)g Fp(b)1765 3298 y Fs(0)1819 3284 y Fr(max)2004 3210 y Fk(\010)2057 3284 y Fn(k)2121 3260 y Fr(\026)2102 3284 y Fp( )2161 3298 y Fs(1)2221 3284 y Fn(\000)g Fp(\030)2352 3298 y Fs(1)2391 3284 y Fn(k)2436 3246 y Fs(2)2436 3306 y(2)2501 3284 y Fp(;)15 b Fn(k)2606 3260 y Fr(\026)2586 3284 y Fp( )2645 3298 y Fs(2)2706 3284 y Fn(\000)20 b Fp(\030)2837 3298 y Fs(2)2876 3284 y Fn(k)2921 3246 y Fs(2)2921 3306 y(2)2961 3210 y Fk(\011)3029 3284 y Fp(:)456 3519 y Fr(The)29 b(application)g(of)i(Theorem)f(2.2)h(giv)m(es)g(the)g (follo)m(wing)d(assertion.)456 3662 y Fx(Theorem)50 b(5.1.)g Fj(L)-5 b(et)46 b Fn(L)k Fr(=)f Fn(f)p Fp(l)1603 3676 y Fo(j)1689 3662 y Fr(:)h Fp(j)55 b Fr(=)49 b(1)p Fp(;)15 b Fn(\001)g(\001)g(\001)32 b Fp(;)15 b(N)10 b Fn(g)47 b Fj(b)-5 b(e)45 b(a)i(\014nite)f(set)g(of)g(line)-5 b(arly)456 3773 y(indep)g(endent)31 b(c)-5 b(ontinuous)31 b(functionals)g(on)g(the)f(sp)-5 b(ac)g(e)31 b Fx(H)2466 3740 y Fs(2)2466 3795 y Fo(per)2572 3773 y Fj(.)41 b(We)30 b(assume)g(that)i(this)456 3885 y(set)g(satis\014es)i(\(31\).)43 b(L)-5 b(et)33 b(\(22\))h(hold.)43 b(If)1515 4135 y Fp(")1557 4149 y Fh(L)1636 4135 y Fp(<)1741 4071 y(\027)1792 3994 y Fk(p)p 1883 3994 467 4 v 77 x Fr(min)o Fn(f)p Fp(h)2132 4085 y Fs(1)2172 4071 y Fp(;)15 b(h)2264 4085 y Fs(2)2304 4071 y Fn(g)p 1741 4114 609 4 v 1869 4132 a(p)p 1945 4132 278 4 v 70 x Fr(2)p Fp(a)2038 4216 y Fs(0)2078 4202 y Fp(b)2117 4216 y Fs(0)2156 4202 y Fr(\006)2360 4135 y Fp(;)456 4361 y Fj(wher)-5 b(e)456 4540 y Fr(\(33\))300 b(\006)25 b(=)g Fp(d)1150 4554 y Fs(0)1190 4540 y Fr(\()p Fg(E)11 b Fn(k)p Fp(\021)t Fn(k)1419 4502 y Fs(4)1419 4562 y(0)1465 4540 y Fr(\))1510 4475 y Fl(1)p 1510 4487 31 3 v 1510 4528 a(2)1554 4540 y Fr(\()p Fg(E)g Fp(R)1714 4502 y Fs(2)1760 4540 y Fr(\))1805 4475 y Fl(1)p 1805 4487 V 1805 4528 a(2)1870 4540 y Fr(+)20 b Fg(E)10 b Fp(m)26 b Fr(+)20 b Fp(\027)h Fr(min)n Fn(f)p Fp(h)2526 4554 y Fs(1)2567 4540 y Fp(;)15 b(h)2659 4554 y Fs(2)2699 4540 y Fn(g)p Fg(E)c Fn(k)p Fp(\021)t Fn(k)2938 4502 y Fs(2)2938 4562 y(0)456 4718 y Fj(and)40 b Fp(m)p Fj(,)h(involving)f (the)g(Wiener)g(pr)-5 b(o)g(c)g(ess)42 b(thr)-5 b(ough)41 b(the)g(Ornstein-Uhlenb)-5 b(e)g(ck)39 b(pr)-5 b(o-)456 4835 y(c)g(ess)45 b Fp(\021)s Fj(,)j(is)c(de\014ne)-5 b(d)46 b(in)f(L)-5 b(emma)46 b(4.2,)i(then)e Fn(L)2138 4802 y Fo(h)2179 4811 y Fl(1)2213 4802 y Fo(;h)2274 4811 y Fl(2)2359 4835 y Fr(=)h Fn(f)p Fp(l)2551 4790 y Fo(h)2592 4799 y Fl(1)2627 4790 y Fo(;h)2688 4799 y Fl(2)2549 4862 y Fo(j)2773 4835 y Fr(:)g Fp(j)53 b Fr(=)47 b(1)p Fp(;)15 b Fn(\001)g(\001)g(\001)33 b Fp(;)15 b(N)10 b Fn(g)456 4973 y Fj(wher)-5 b(e)36 b Fp(l)744 4929 y Fo(h)785 4938 y Fl(1)819 4929 y Fo(;h)880 4938 y Fl(2)742 5001 y Fo(j)918 4973 y Fr(\()p Fp( )1012 4987 y Fs(1)1052 4973 y Fp(;)15 b( )1151 4987 y Fs(2)1191 4973 y Fr(\))31 b(:=)g Fp(l)1411 4987 y Fo(j)1447 4973 y Fr(\()1482 4903 y Fn(p)p 1559 4903 92 4 v 1559 4973 a Fp(h)1611 4987 y Fs(1)1650 4973 y Fp( )1709 4987 y Fs(1)1749 4973 y Fp(;)1789 4903 y Fn(p)p 1865 4903 V 70 x Fp(h)1917 4987 y Fs(2)1957 4973 y Fp( )2016 4987 y Fs(2)2056 4973 y Fr(\))p Fp(;)46 b(l)2189 4987 y Fo(j)2256 4973 y Fn(2)30 b(L)35 b Fj(is)h(a)g(set)f(of)h (asymptotic)-5 b(al)5 b(ly)456 5085 y(determining)33 b(functionals)g(in)f(pr)-5 b(ob)g(ability)35 b(for)e(the)f(sto)-5 b(chastic)g(al)5 b(ly)36 b(for)-5 b(c)g(e)g(d)34 b(two-layer)456 5193 y(quasige)-5 b(ostr)g(ophic)35 b(\015uid)e(system)g(\(19\),)h (\(20\).)p eop %%Page: 17 17 17 16 bop 1043 251 a Fs(STOCHASTIC)31 b(TW)n(O-LA)-6 b(YER)30 b(GEOPHYSICAL)h(FLO)n(WS)549 b(17)456 450 y Fj(Pr)-5 b(o)g(of.)43 b Fr(W)-8 b(e)36 b(in)m(tegrate)h(\(32\).)58 b(The)35 b(conditions)f(of)h(Theorem)h(\(2.2\))h(are)f(satis\014ed)e (if)456 558 y Fg(E)10 b Fp(l)537 584 y Fh(L)586 564 y Fl(\(1\))704 558 y Fp(<)25 b Fr(0.)41 b Fp(l)938 584 y Fh(L)987 564 y Fl(\(1\))1104 558 y Fr(can)30 b(b)s(e)g(estimated)h(b) m(y)740 773 y Fn(\000)p Fp(\027)872 712 y Fr(1)20 b Fn(\000)g Fp(\016)p 872 752 201 4 v 881 837 a(a)929 851 y Fs(0)968 837 y Fp(")1010 806 y Fs(2)1010 866 y Fh(L)1102 773 y Fr(+)1382 712 y(2)p Fp(\027)6 b(b)1517 726 y Fs(0)p 1203 752 533 4 v 1203 835 a Fp(\027)21 b Fr(min)o Fn(f)p Fp(h)1518 849 y Fs(1)1558 835 y Fp(;)15 b(h)1650 849 y Fs(2)1690 835 y Fn(g)1746 773 y Fr(\()p Fp(h)1833 787 y Fs(1)1873 773 y Fn(k)p Fr(\001)2013 749 y(\026)1994 773 y Fp( )2053 787 y Fs(1)2093 773 y Fn(k)2138 736 y Fs(2)2138 796 y(0)2198 773 y Fr(+)20 b Fp(h)2341 787 y Fs(2)2380 773 y Fn(k)p Fr(\001)2520 749 y(\026)2501 773 y Fp( )2560 787 y Fs(2)2600 773 y Fn(k)2645 736 y Fs(2)2645 796 y(0)2685 773 y Fr(\))h(+)f(2)p Fp(b)2916 787 y Fs(0)2956 773 y Fn(k)p Fp(\021)s Fn(k)3094 736 y Fs(2)3094 796 y(0)3134 773 y Fp(:)456 994 y Fr(On)29 b(accoun)m(t)j(of)f(Lemma)f(4.2)904 1197 y(sup)850 1279 y Fo(x)p Fh(2)p Fo(B)s Fs(\()p Fo(!)r Fs(\))1119 1135 y Fr(1)p 1119 1176 46 4 v 1125 1259 a Fp(t)1175 1197 y(\027)1241 1073 y Fk(Z)1332 1099 y Fo(t)1291 1279 y Fs(0)1361 1197 y Fr(\()p Fp(h)1448 1211 y Fs(1)1488 1197 y Fn(k)p Fr(\001)1628 1173 y(\026)1609 1197 y Fp( )1668 1211 y Fs(1)1708 1197 y Fr(\()p Fp(\034)5 b(;)15 b(!)s(;)g(x)p Fr(\))p Fn(k)2060 1159 y Fs(2)2060 1219 y(0)2122 1197 y Fr(+)20 b Fp(h)2265 1211 y Fs(1)2304 1197 y Fn(k)p Fr(\001)2444 1173 y(\026)2425 1197 y Fp( )2484 1211 y Fs(2)2524 1197 y Fr(\()p Fp(\034)5 b(;)15 b(!)s(;)g(x)p Fr(\))p Fn(k)2876 1159 y Fs(2)2876 1219 y(0)2917 1197 y Fr(\))p Fp(d\034)456 1433 y Fr(has)30 b(the)g(b)s(ound)965 1575 y(1)p 965 1615 V 971 1699 a Fp(t)1036 1508 y Fk(\022)1103 1636 y Fp(R)q Fr(\()p Fp(!)s Fr(\))20 b(+)1414 1512 y Fk(Z)1505 1539 y Fo(t)1465 1719 y Fs(0)1535 1636 y Fr(\()p Fp(d)1617 1650 y Fs(0)1657 1636 y Fn(k)p Fp(\021)s Fr(\()p Fp(\022)1828 1650 y Fo(\034)1872 1636 y Fp(!)s Fr(\))p Fn(k)2012 1599 y Fs(2)2012 1659 y(0)2052 1636 y Fp(R)q Fr(\()p Fp(\022)2200 1650 y Fo(\034)2243 1636 y Fp(!)s Fr(\))g(+)g Fp(m)p Fr(\()p Fp(\022)2607 1650 y Fo(\034)2650 1636 y Fp(!)s Fr(\)\))p Fp(d\034)2877 1508 y Fk(\023)456 1856 y Fr(since)27 b Fp(B)33 b Fr(is)27 b(forw)m(ard)h(in)m(v)-5 b(arian)m(t.)39 b(Note)29 b(that)g(the)f(exp)s(ectation)h(of)f Fp(R)q(=t)g Fr(can)h(b)s(e)e(made)456 1963 y(arbitrarily)e(small)h(if)h Fp(t)h Fr(is)f(large.)40 b(Calculation)27 b(the)h(exp)s(ectation)g(of)g (the)h(last)e(expres-)456 2071 y(sion)i(and)h(c)m(ho)s(osing)g Fp(t)g Fr(su\016cien)m(tly)f(large)i(yields)d(the)j(conclusion.)596 b Fd(\003)456 2258 y Fx(Remark)42 b(5.2.)k Fr(i\))36 b(It)i(is)e(easily)g(seen)i(that)g(there)f(exists)g(a)g(set)2785 2235 y(^)2764 2258 y Fn(L)2827 2225 y Fo(h)2868 2234 y Fl(1)2902 2225 y Fo(;h)2963 2234 y Fl(2)3038 2258 y Fr(of)h(linearly)456 2376 y(indep)s(enden)m(t)33 b(linear)h(b)s(ounded) f(functionals)h Fn(f)2122 2352 y Fr(^)2123 2376 y Fp(l)2150 2390 y Fo(j)2187 2376 y Fn(g)2232 2343 y Fo(N)2232 2400 y Fs(1)2335 2376 y Fr(whic)m(h)g(are)i(determining)e(with)456 2484 y(resp)s(ect)d(to)h Fp(q)s Fr(.)44 b(Indeed,)31 b Fp(q)j Fr(and)d Fp( )k Fr(are)c(connected)i(b)m(y)e(a)h(linear)e (homeomorphism)f(\003)456 2600 y(from)35 b Fx(L)739 2567 y Fs(2)739 2623 y Fo(per)881 2600 y Fr(to)i Fx(H)1080 2567 y Fs(2)1080 2623 y Fo(per)1222 2600 y Fr(de\014ned)d(b)m(y)i (\(20\))i(suc)m(h)d(that)i(w)m(e)f(can)g(set)2741 2576 y(^)2742 2600 y Fp(l)2771 2556 y Fo(h)2812 2565 y Fl(1)2846 2556 y Fo(;h)2907 2565 y Fl(2)2769 2627 y Fo(j)2980 2600 y Fr(=)e Fp(l)3114 2556 y Fo(h)3155 2565 y Fl(1)3189 2556 y Fo(;h)3250 2565 y Fl(2)3112 2627 y Fo(j)3312 2600 y Fn(\016)24 b Fr(\003)456 2735 y(where)747 2712 y(^)726 2735 y Fn(L)37 b Fr(=)g Fn(f)p Fp(l)1006 2749 y Fo(j)1043 2735 y Fn(g)1088 2702 y Fo(N)1088 2759 y Fs(1)1194 2735 y Fr(de\014nes)g(the)h(set)g(of)g(determining)d(functionals)h(in)m(tro) s(duced)h(in)456 2843 y(Theorem)30 b(5.1.)456 2951 y(ii\))h(If)h Fp(h)719 2965 y Fs(1)788 2951 y Fr(=)d Fp(h)940 2965 y Fs(2)1012 2951 y Fr(then)j(the)h(set)g Fn(L)g Fr(itself)e(is)h (determining)e(in)i(the)g(sense)h(of)g(De\014nition)456 3059 y(2.1.)456 3167 y(iii\))28 b(Assumption)h(\(22\))i(holds,)f(if)456 3379 y(\(34\))1370 3317 y(4)p Fp(d)1462 3331 y Fs(0)1502 3317 y Fp(a)1550 3331 y Fs(0)1589 3317 y Fr(tr)1660 3331 y Fs(0)1700 3317 y Fp(Q)p 1341 3358 461 4 v 1341 3443 a(\025)1394 3411 y Fs(2)1394 3469 y(1)1433 3443 y Fp(\027)1484 3417 y Fs(2)1523 3443 y Fr(\()p Fp(k)24 b Fr(+)c(1\))1836 3379 y Fp(<)25 b Fr(1)p Fp(;)107 b(k)23 b Fr(+)d(1)26 b Fp(>)2468 3317 y Fr(4)p 2447 3358 88 4 v 2447 3441 a Fp(a)2495 3455 y Fs(0)2544 3379 y Fp(;)456 3655 y Fr(for)31 b(example.)43 b(In)31 b(this)g(case)h(it)f(is)f(easy)j(to)f(see)g(that) 2321 3581 y Fk(\000)2363 3655 y Fg(E)10 b Fp(R)2487 3622 y Fs(2)2486 3679 y(0)2532 3581 y Fk(\001)2574 3604 y Fs(1)p Fo(=)p Fs(2)2711 3655 y Fn(\024)27 b Fr(4)p Fp(d)2901 3622 y Fs(2)2901 3679 y(4)2956 3554 y Fk(\020)3042 3617 y Fs(3)p Fo(a)3114 3626 y Fl(0)p 3020 3634 151 4 v 3020 3686 a Fs(2)p Fo(\027)t(\025)3135 3695 y Fl(1)3180 3554 y Fk(\021)3235 3576 y Fs(2)3306 3655 y Fr(\(w)m(e)456 3790 y(c)m(ho)s(ose)k Fp(a)25 b Fr(=)g(4)p Fp(=)p Fr(3)32 b(in)d(Lemma)h(4.4\).)43 b(Therefore)30 b(using)f(\(14\))j(w)m(e)e (obtain)746 4037 y(\006)25 b Fn(\024)g Fp(C)1005 3989 y Fs(1)p Fo(=)p Fs(2)998 4063 y(4)1197 3975 y Fr(tr)1268 3989 y Fs(0)1307 3975 y Fp(Q)p 1124 4016 329 4 v 1124 4099 a(\027)6 b Fr(\()p Fp(k)24 b Fr(+)19 b(1\))1482 4037 y Fn(\001)1528 3881 y Fk( )1600 4037 y Fr(4)p Fp(d)1692 4051 y Fs(0)1732 4037 y Fp(d)1779 3999 y Fs(2)1779 4059 y(4)1834 3909 y Fk(\022)1939 3975 y Fr(3)p Fp(a)2032 3989 y Fs(0)p 1911 4016 189 4 v 1911 4099 a Fr(2)p Fp(\027)6 b(\025)2060 4113 y Fs(1)2109 3909 y Fk(\023)2176 3931 y Fs(2)2236 4037 y Fr(+)20 b Fp(\027)h Fr(min)n Fn(f)p Fp(h)2641 4051 y Fs(1)2681 4037 y Fp(;)15 b(h)2773 4051 y Fs(2)2814 4037 y Fn(g)2859 3881 y Fk(!)2951 4037 y Fr(+)20 b Fp(d)3089 4051 y Fs(4)3129 4037 y Fp(:)456 4283 y Fr(Using)29 b(\(34\))j(again)f(w)m(e)f(ha)m(v)m(e)i(the)f (estimate)1083 4503 y Fp(d)1130 4517 y Fs(4)1195 4503 y Fn(\024)1313 4441 y Fr(tr)1384 4455 y Fs(0)1424 4441 y Fp(Q)p 1301 4482 207 4 v 1301 4565 a(k)23 b Fr(+)d(1)1538 4503 y Fn(\001)1583 4375 y Fk(\022)1650 4503 y Fp(C)1722 4455 y Fs(1)p Fo(=)p Fs(4)1715 4529 y(8)1908 4441 y Fp(d)1955 4455 y Fs(1)p 1842 4482 220 4 v 1842 4565 a Fr(4)p Fp(d)1934 4579 y Fs(0)1974 4565 y Fp(a)2022 4579 y Fs(0)2092 4503 y Fr(+)f Fp(C)2254 4455 y Fs(1)p Fo(=)p Fs(4)2247 4529 y(4)2402 4441 y Fp(d)2449 4455 y Fs(2)p 2374 4482 144 4 v 2374 4565 a Fp(\025)2427 4579 y Fs(1)2466 4565 y Fp(\027)2527 4375 y Fk(\023)2614 4503 y Fr(+)h Fp(d)2752 4517 y Fs(3)2792 4503 y Fp(:)456 4722 y Fr(Therefore)43 b(under)f(conditions)g(\(34\))j(in)e(the)g(deterministic)f(limit)f(tr) 2955 4736 y Fs(0)2995 4722 y Fp(Q)47 b Fn(!)g Fr(0)d(w)m(e)456 4830 y(obtain)29 b(estimate)456 5158 y(\(35\))47 b Fp(")705 5172 y Fh(L)783 5158 y Fp(<)889 5095 y(\027)940 5062 y Fs(2)979 5095 y Fp(\025)1032 5109 y Fs(1)1072 5017 y Fk(p)p 1163 5017 467 4 v 78 x Fr(min)n Fn(f)p Fp(h)1411 5109 y Fs(1)1451 5095 y Fp(;)15 b(h)1543 5109 y Fs(2)1584 5095 y Fn(g)p 889 5137 740 4 v 993 5226 a Fr(6)p Fn(k)p Fp(f)10 b Fn(k)1183 5240 y Fh(\000)p Fs(1)1278 5226 y Fp(c)1317 5240 y Fs(0)1357 5155 y Fn(p)p 1433 5155 92 4 v 71 x Fp(h)1485 5240 y Fs(1)1659 5158 y Fn(\001)1705 5030 y Fk(\032)1773 5158 y Fr(\()p Fp(\025)1861 5172 y Fs(1)1921 5158 y Fr(+)20 b(2)15 b(max)q Fn(f)p Fp(F)2345 5172 y Fs(1)2385 5158 y Fp(;)g(F)2483 5172 y Fs(2)2523 5158 y Fn(g)p Fr(\))2619 5030 y Fk(\022)2686 5158 y Fr(1)21 b(+)2853 5097 y Fp(F)2911 5111 y Fs(1)2951 5097 y Fp(F)3009 5111 y Fs(2)p 2853 5137 196 4 v 2904 5222 a Fp(\025)2957 5191 y Fs(2)2957 5248 y(1)3058 5030 y Fk(\023\033)3193 5053 y Fh(\000)p Fs(1)p Fo(=)p Fs(2)3374 5158 y Fp(:)p eop %%Page: 18 18 18 17 bop 456 251 a Fs(18)327 b(IGOR)24 b(CHUESHO)n(V,)f(JINQIA)n(O)h (DUAN,)e(AND)h(BJ)2368 234 y(\177)2358 251 y(ORN)h(SCHMALFUSS)456 457 y Fr(If)e(this)f(estimate)i(holds,)g(then)f(functionals)f Fn(f)p Fp(l)2049 413 y Fo(h)2090 422 y Fl(1)2125 413 y Fo(;h)2186 422 y Fl(2)2047 485 y Fo(j)2224 457 y Fn(g)i Fr(are)g(determining)d(for)j(the)f Fj(deter-)456 574 y(ministic)33 b Fr(t)m(w)m(o-la)m(y)m(er)j(mo)s(del.)48 b(Moreo)m(v)m(er)36 b(under)31 b(condition)h(\(35\))j(these)f (functionals)456 682 y(are)26 b(also)g(determining)e(for)i(our)g(sto)s (c)m(hastic)h(t)m(w)m(o-la)m(y)m(er)h(mo)s(del)d(\(1\))i(and)f(\(2\))h (pro)m(vided)456 790 y(the)j(noise)g(parameter)h(tr)1346 804 y Fs(0)1385 790 y Fp(Q)f Fr(is)g(small)f(enough.)456 926 y(A)m(t)37 b(the)f(end)f(of)i(this)e(section,)j(w)m(e)e(will)e (discuss)g(appropriate)h(v)-5 b(alues)35 b(for)h(\015uid)e(pa-)456 1034 y(rameters.)41 b(W)-8 b(e)32 b(will)c(estimate)j(the)g(su\016cien) m(t)f(condition)f(in)g(Theorem)h(5.1,)i(namely)456 1142 y(the)e(inequalit)m(y)f(\(35\),)j(in)d(terms)h(of)h(\015uid)d (parameters.)456 1250 y(W)-8 b(e)31 b(further)e(sho)m(w)i(that)g(the)g (long-time)f(dynamics)f(of)i(the)f(t)m(w)m(o-la)m(y)m(er)j (geophsysical)456 1358 y(\015uid)22 b(system)j(is)f(determined)f(b)m(y) i(the)f(long)h(time)f(dynamics)f(of)i(the)g(top)g(la)m(y)m(er)g(alone)g (,)456 1466 y(when)g(the)h(\015uid)e(parameters)j(and)f(the)g(Wiener)g (pro)s(cess)g(satisfy)g(certain)g(conditons.)456 1573 y(In)i(this)h(case)h(the)g(long)f(time)g(dynamics)g(will)e(also)i(b)s (e)g(determined)f(b)m(y)h(only)g(\014nitely)456 1681 y(man)m(y)h(functionals.)456 1789 y(Let)f Fn(L)c Fr(=)g Fn(f)p Fp(l)873 1803 y Fo(j)936 1789 y Fr(:)g Fp(j)31 b Fr(=)25 b(1)p Fp(;)15 b(:::;)g(N)10 b Fn(g)31 b Fr(b)s(e)e(a)h (\014nite)e(set)h(of)h(linearly)d(indep)s(enden)m(t)g(con)m(tin)m(uous) 456 1900 y(functionals)h(on)i(the)h(space)g Fp(H)1527 1867 y Fs(2)1520 1922 y Fo(per)1656 1900 y Fr(of)g(the)f(top)h(stream)g (functions.)39 b(W)-8 b(e)31 b(assume)g(that)456 2080 y(\(36\))176 b Fn(kr)p Fp( )972 2094 y Fs(1)1012 2080 y Fn(k)1057 2094 y Fs(0)1122 2080 y Fn(\024)25 b Fp(")1260 2094 y Fh(L)1333 2080 y Fn(\001)c(k)p Fr(\001)p Fp( )1559 2094 y Fs(1)1598 2080 y Fn(k)1643 2094 y Fs(0)1704 2080 y Fr(+)e Fp(C)1859 2094 y Fh(L)1932 2080 y Fn(\001)78 b Fr(max)1978 2141 y Fo(j)t Fs(=1)p Fo(;:::)n(;N)2277 2080 y Fn(j)p Fp(l)2329 2094 y Fo(j)2365 2080 y Fr(\()p Fp( )2459 2094 y Fs(1)2499 2080 y Fr(\))p Fn(j)p Fp(;)107 b( )2750 2094 y Fs(1)2815 2080 y Fn(2)25 b Fp(H)2984 2043 y Fs(2)2977 2103 y Fo(per)3083 2080 y Fp(;)456 2290 y Fr(where)k Fp(C)783 2304 y Fh(L)861 2290 y Fp(>)c Fr(0)31 b(is)f(a)g(constan)m(t)i(dep)s(ending)c(on)i Fn(L)p Fr(.)40 b(As)31 b(ab)s(o)m(v)m(e)g(w)m(e)g(ha)m(v)m(e)861 2497 y Fp(h)913 2511 y Fs(1)953 2497 y Fn(k)p Fr(\001)p Fp( )1133 2511 y Fs(1)1172 2497 y Fn(k)1217 2460 y Fs(2)1217 2520 y(0)1283 2497 y Fn(\025)1388 2436 y Fr(1)21 b Fn(\000)f Fp(\016)p 1388 2476 201 4 v 1441 2561 a(")1483 2530 y Fs(2)1483 2590 y Fh(L)1599 2497 y Fp(h)1651 2511 y Fs(1)1691 2497 y Fn(kr)p Fp( )1871 2511 y Fs(1)1911 2497 y Fn(k)1956 2460 y Fs(2)1956 2520 y(0)2016 2497 y Fn(\000)g Fp(C)2172 2512 y Fo(\016)o(;)p Fh(L)2274 2497 y Fp(h)2326 2511 y Fs(1)2444 2497 y Fr(max)2381 2558 y Fo(j)t Fs(=1)p Fo(;)p Fh(\001\001\001)9 b Fo(;N)2691 2497 y Fn(j)p Fp(l)2743 2511 y Fo(j)2780 2497 y Fr(\()p Fp( )2874 2511 y Fs(1)2914 2497 y Fr(\))p Fn(j)2974 2460 y Fs(2)3014 2497 y Fp(:)456 2718 y Fr(for)30 b(an)m(y)g(0)c Fp(<)f(\016)k(<)c Fr(1)31 b(with)e(appropriate)g(p)s(ositiv)m(e)h(constan)m(t)h Fp(C)2639 2733 y Fo(\016)o(;)p Fh(L)2742 2718 y Fr(.)456 2829 y(Th)m(us)19 b(under)g(the)i(condition)e(\(36\))j(and)e(using)f (the)i(inequalit)m(y)e Fn(kr)p Fp( )2776 2843 y Fs(2)2816 2829 y Fn(k)2861 2796 y Fs(2)2861 2854 y(0)2926 2829 y Fn(\024)25 b Fp(\025)3075 2843 y Fs(1)3114 2829 y Fn(k)p Fr(\001)p Fp( )3294 2843 y Fs(2)3334 2829 y Fn(k)3379 2796 y Fs(2)3379 2854 y(0)3419 2829 y Fr(,)456 2937 y(the)30 b(estimate)h(\(30\))h(for)e(the)h(solution)e(di\013erence)36 b(^)-51 b Fp(q)23 b Fn(\000)j Fr(\026)-51 b Fp(q)33 b Fr(b)s(ecomes)518 3088 y Fp(d)p 501 3128 81 4 v 501 3212 a(dt)592 3149 y Fn(k)p Fp(q)s Fn(k)726 3112 y Fs(2)726 3172 y Fh(\003)786 3149 y Fr(+)20 b Fp(\027)6 b(h)980 3163 y Fs(1)1029 3088 y Fr(1)21 b Fn(\000)f Fp(\016)p 1029 3128 201 4 v 1082 3213 a(")1124 3182 y Fs(2)1124 3242 y Fh(L)1240 3149 y Fn(kr)p Fp( )1420 3163 y Fs(1)1460 3149 y Fn(k)1505 3112 y Fs(2)1505 3172 y(0)1565 3149 y Fr(+)g(\()p Fp(\027)6 b(\025)1795 3163 y Fs(1)1854 3149 y Fr(+)20 b(2)p Fp(r)s Fr(\))p Fp(h)2121 3163 y Fs(2)2161 3149 y Fn(kr)p Fp( )2341 3163 y Fs(2)2381 3149 y Fn(k)2426 3112 y Fs(2)2426 3172 y(0)791 3358 y Fn(\024)25 b Fp(b)926 3372 y Fs(0)981 3358 y Fr(max)1165 3284 y Fk(\010)1218 3358 y Fn(k)1282 3334 y Fr(\026)1263 3358 y Fp( )1322 3372 y Fs(1)1382 3358 y Fn(\000)20 b Fp(\030)1513 3372 y Fs(1)1552 3358 y Fn(k)1597 3320 y Fs(2)1597 3380 y(2)1663 3358 y Fp(;)15 b Fn(k)1767 3334 y Fr(\026)1748 3358 y Fp( )1807 3372 y Fs(2)1867 3358 y Fn(\000)20 b Fp(\030)1998 3372 y Fs(2)2037 3358 y Fn(k)2082 3320 y Fs(2)2082 3380 y(2)2122 3284 y Fk(\011)2190 3358 y Fn(k)p Fp(q)s Fn(k)2324 3320 y Fs(2)2324 3380 y Fh(\003)2385 3358 y Fr(+)g Fp(C)2541 3373 y Fo(\016)o(;)p Fh(L)2721 3358 y Fr(max)2658 3418 y Fo(j)t Fs(=1)p Fo(;)p Fh(\001\001\001)10 b Fo(;N)2968 3358 y Fn(j)p Fp(l)3020 3372 y Fo(j)3057 3358 y Fr(\()p Fp(h)3144 3372 y Fs(1)3185 3358 y Fp( )3244 3372 y Fs(1)3283 3358 y Fr(\))p Fn(j)3343 3320 y Fs(2)3383 3358 y Fp(:)456 3568 y Fr(Th)m(us)29 b(w)m(e)i(obtain)991 3695 y Fp(d)p 975 3736 81 4 v 975 3819 a(dt)1065 3757 y Fn(k)p Fp(q)s Fn(k)1199 3719 y Fs(2)1199 3779 y Fh(\003)1264 3757 y Fn(\024)25 b Fp(l)1389 3709 y Fs(\(2\))1387 3786 y Fh(L)1484 3757 y Fr(\()1538 3733 y(\026)1519 3757 y Fp( )s(;)15 b(!)s Fr(\))p Fn(k)p Fp(q)s Fn(k)1850 3719 y Fs(2)1850 3779 y Fh(\003)1911 3757 y Fr(+)20 b Fp(C)2067 3772 y Fo(\016)o(;)p Fh(L)2248 3757 y Fr(max)2185 3818 y Fo(j)t Fs(=1)p Fo(;)p Fh(\001\001\001)9 b Fo(;N)2495 3757 y Fn(j)p Fp(l)2547 3771 y Fo(j)2584 3757 y Fr(\()p Fp(h)2671 3771 y Fs(1)2711 3757 y Fp( )2770 3771 y Fs(1)2810 3757 y Fr(\))p Fn(j)2870 3719 y Fs(2)2910 3757 y Fp(;)456 3967 y Fr(where)1004 4164 y Fp(l)1033 4116 y Fs(\(2\))1031 4192 y Fh(L)1127 4164 y Fr(\()1181 4140 y(\026)1162 4164 y Fp( )t(;)15 b(!)s Fr(\))26 b(=)20 b Fn(\000)g Fr(min)1734 4035 y Fk(\032)1802 4164 y Fp(\027)1863 4102 y Fr(1)h Fn(\000)f 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Fj(b)-5 b(e)41 b(a)h(\014nite)f(set)h(of)f(line)-5 b(arly)43 b(in-)456 4783 y(dep)-5 b(endent)39 b(c)-5 b(ontinuous)39 b(functionals)f(on)g(the)g(sp)-5 b(ac)g(e)40 b Fp(H)2435 4750 y Fs(2)2428 4805 y Fo(per)2533 4783 y Fj(.)57 b(We)37 b(assume)i(that)g(this)456 4891 y(set)h(satis\014es)h (the)f(fol)5 b(lowing)41 b(c)-5 b(ondition)42 b(involving)e(only)h (with)f(the)h(top)g(\015uid)f(layer)456 4999 y(dynamic)-5 b(al)34 b(variable,)g(i.e.,)d(the)i(str)-5 b(e)g(am)35 b(function)e Fp( )2299 5013 y Fs(1)2338 4999 y Fr(\()p Fp(x;)15 b(y)s(;)g(t)p Fr(\))456 5166 y(\(37\))175 b Fn(kr)p Fp( )971 5180 y Fs(1)1011 5166 y Fn(k)1056 5180 y Fs(0)1121 5166 y Fn(\024)25 b Fp(")1259 5180 y Fh(L)1332 5166 y Fn(\001)c(k)p Fr(\001)p Fp( )1558 5180 y Fs(1)1597 5166 y Fn(k)1642 5180 y Fs(0)1703 5166 y Fr(+)e Fp(C)1858 5180 y Fh(L)1931 5166 y Fn(\001)78 b Fr(max)1977 5227 y Fo(j)t Fs(=1)p Fo(;:::)n(;N)2276 5166 y Fn(j)p Fp(l)2328 5180 y Fo(j)2364 5166 y Fr(\()p Fp( )2458 5180 y Fs(1)2498 5166 y Fr(\))p Fn(j)p Fp(;)109 b( )2751 5180 y Fs(1)2816 5166 y Fn(2)25 b Fp(H)2985 5129 y Fs(2)2978 5189 y Fo(per)3084 5166 y Fp(:)p eop %%Page: 19 19 19 18 bop 1043 251 a Fs(STOCHASTIC)31 b(TW)n(O-LA)-6 b(YER)30 b(GEOPHYSICAL)h(FLO)n(WS)549 b(19)456 450 y Fj(If)456 663 y Fr(\(38\))483 b(\006)25 b Fp(<)g Fr(min)1452 534 y Fk(\032)1553 601 y Fp(\027)p 1530 642 96 4 v 1530 727 a(")1572 695 y Fs(2)1572 755 y Fh(L)1661 663 y Fp(;)15 b Fr(\()q Fp(\027)6 b(\025)1841 677 y Fs(1)1900 663 y Fr(+)20 b(2)p Fp(r)s Fr(\))2115 534 y Fk(\033)2208 601 y Fp(\027)h Fr(min)o Fn(f)p Fp(h)2523 615 y Fs(1)2563 601 y Fp(;)15 b(h)2655 615 y Fs(2)2695 601 y Fn(g)p 2208 642 533 4 v 2369 725 a Fr(2)p Fp(a)2462 739 y Fs(0)2502 725 y Fp(b)2541 739 y Fs(0)2776 663 y Fp(;)456 886 y Fj(wher)-5 b(e)28 b Fr(\006)f Fj(is)h(given)e(by)i(\(33\),)h(then)f Fn(L)f Fj(is)h(a)f(set)h(of)g(asymptotic)-5 b(al)5 b(ly)30 b(determining)e(func-)456 994 y(tionals)34 b(in)f(pr)-5 b(ob)g(ability)35 b(for)e(the)h(sto)-5 b(chastic)g(al)5 b(ly)36 b(for)-5 b(c)g(e)g(d)34 b(two-layer)h(quasige)-5 b(ostr)g(ophic)456 1102 y(\015uid)33 b(system)g(\(19\),)h(\(20\).)456 1238 y Fr(The)29 b(pro)s(of)h(of)h(this)e(theorem)i(is)e(the)i(same)f (as)h(the)g(pro)s(of)e(of)i(Theorem)f(5.1.)456 1508 y(Relation)g (\(38\))h(holds)e(if)456 1694 y(\(39\))1692 1633 y Fp(\027)p 1670 1674 96 4 v 1670 1759 a(")1712 1727 y Fs(2)1712 1787 y Fh(L)1801 1694 y Fn(\025)24 b Fp(\027)6 b(\025)2000 1708 y Fs(1)2060 1694 y Fr(+)20 b(2)p Fp(r)456 1913 y Fr(and)456 2106 y(\(40\))725 b(\006)25 b Fp(<)g Fr(\()p Fp(\027)6 b(\025)1667 2120 y Fs(1)1726 2106 y Fr(+)20 b(2)p Fp(r)s Fr(\))1967 2044 y Fp(\027)g Fr(min)o Fn(f)p Fp(h)2281 2058 y Fs(1)2321 2044 y Fp(;)15 b(h)2413 2058 y Fs(2)2453 2044 y Fn(g)p 1967 2085 533 4 v 2127 2168 a Fr(2)p Fp(b)2211 2182 y Fs(0)2251 2168 y Fp(a)2299 2182 y Fs(0)2534 2106 y Fp(:)456 2308 y Fr(The)38 b(parameter)h(\006)g (dep)s(ends)d(on)j Fp(r)i Fr(via)e Fp(d)1980 2322 y Fs(2)2020 2308 y Fr(.)65 b(Therefore)39 b(it)f(is)g(not)h(clear)g(whether)456 2416 y(\(40\))32 b(holds)e(for)h(some)h Fp(r)s Fr(.)43 b(Ho)m(w)m(ev)m(er)33 b(as)f(in)e(Remark)h(5.2)i(in)d(the)h (deterministic)f(limit)456 2524 y(tr)527 2538 y Fs(0)566 2524 y Fp(Q)25 b Fn(!)g Fr(0)31 b(the)g(estimate)g(\(40\))h(turns)d(in) m(to)h(the)h(relation)456 2752 y(\(41\))57 b Fp(\027)6 b(\025)777 2766 y Fs(1)836 2752 y Fr(+)20 b(2)p Fp(r)29 b(>)1229 2685 y Fr(36)p Fp(c)1358 2652 y Fs(2)1358 2710 y(0)1399 2685 y Fn(k)p Fp(f)10 b Fn(k)1544 2652 y Fs(2)1544 2710 y Fh(\000)p Fs(1)1638 2685 y Fp(h)1690 2699 y Fs(1)p 1148 2731 665 4 v 1148 2814 a Fp(\027)1199 2788 y Fs(3)1238 2814 y Fp(\025)1291 2828 y Fs(1)1345 2814 y Fr(min)o Fn(f)p Fp(h)1594 2828 y Fs(1)1634 2814 y Fp(;)15 b(h)1726 2828 y Fs(2)1766 2814 y Fn(g)1842 2752 y(\001)1887 2624 y Fk(\022)1954 2752 y Fr(1)21 b(+)2144 2691 y(2)p 2121 2731 93 4 v 2121 2814 a Fp(\025)2174 2828 y Fs(1)2238 2752 y Fr(max)q Fn(f)p Fp(F)2511 2766 y Fs(1)2551 2752 y Fp(;)15 b(F)2649 2766 y Fs(2)2689 2752 y Fn(g)2734 2624 y Fk(\023)h(\022)2884 2752 y Fr(1)k(+)3050 2691 y Fp(F)3108 2705 y Fs(1)3148 2691 y Fp(F)3206 2705 y Fs(2)p 3050 2731 196 4 v 3102 2816 a Fp(\025)3155 2785 y Fs(2)3155 2842 y(1)3256 2624 y Fk(\023)3363 2752 y Fp(:)456 2968 y Fr(This)28 b(observ)-5 b(ation)30 b(leads)g(to)h(the)g (follo)m(wing)e(assertion.)456 3104 y Fx(Corollary)43 b(5.4.)i Fj(Assume)37 b(that)j(\(41\))f(holds.)59 b(L)-5 b(et)38 b Fn(L)c Fr(=)h Fn(f)p Fp(l)2599 3118 y Fo(j)2670 3104 y Fr(:)g Fp(j)41 b Fr(=)34 b(1)p Fp(;)15 b(:::;)g(N)10 b Fn(g)40 b Fj(b)-5 b(e)38 b(a)456 3214 y(\014nite)d(set)g(of)h(line)-5 b(arly)37 b(indep)-5 b(endent)37 b(c)-5 b(ontinuous)36 b(functionals)g(on)g(the)g(sp)-5 b(ac)g(e)37 b Fp(H)3318 3181 y Fs(2)3311 3237 y Fo(per)3417 3214 y Fj(.)456 3326 y(We)28 b(assume)i(that)h(this)f(set)f(satis\014es)h(\(37\))g(with)g (the)g(p)-5 b(ar)g(ameter)32 b Fp(")2766 3340 y Fh(L)2848 3326 y Fj(satisfying)e(\(39\).)456 3434 y(Then)41 b(ther)-5 b(e)42 b(exists)g Fp(\015)j(>)c Fr(0)g Fj(such)g(that)i Fn(L)e Fj(is)g(a)g(set)g(of)h(asymptotic)-5 b(al)5 b(ly)44 b(determin-)456 3542 y(ing)d(functionals)i(in)f(pr)-5 b(ob)g(ability)44 b(for)e(the)g(sto)-5 b(chastic)g(al)5 b(ly)45 b(for)-5 b(c)g(e)g(d)43 b(two-layer)h(quasi-)456 3650 y(ge)-5 b(ostr)g(ophic)35 b(\015uid)e(system)g(\(19\),)h(\(20\))g (pr)-5 b(ovide)g(d)35 b Fr(tr)2288 3664 y Fs(0)2327 3650 y Fp(Q)25 b Fn(\024)g Fp(\015)5 b Fj(.)456 3758 y(Conse)-5 b(quenc)g(ely,)46 b(the)d(asymptotic)i(pr)-5 b(ob)g(abilistic)45 b(dynamics)f(of)f(the)g(sto)-5 b(chastic)g(al)5 b(ly)456 3866 y(for)-5 b(c)g(e)g(d)34 b(two-layer)g(quasige)-5 b(ostr)g(ophic)35 b(\015uid)e(system)g(is)g(determine)-5 b(d)34 b(only)f(by)f(the)h(top)456 3974 y(\015uid)g(layer.)456 4109 y Fx(Remark)26 b(5.5.)37 b Fr(Note)25 b(that)f(the)g(main)f(task)h (is)f(to)h(pro)m(v)m(e)g(the)g(existence)g(of)g(determin-)456 4217 y(ing)33 b(functionals)f(of)i(the)g(main)f(equation)h(\(1\).)53 b(It)34 b(is)f(easily)g(seen)h(b)m(y)g(the)g(structure)456 4325 y(the)26 b(transformations)f(\(15\))j(a)f(set)f Fn(L)g Fr(is)f(determining)g(in)f(probabilit)m(y)g(for)i(\(19\))i(if)d (and)456 4433 y(only)k(if)g(the)i(same)g(set)g Fn(L)f Fr(is)f(determining)f(in)h(probabilit)m(y)f(for)j(\(1\);)g(see)g([13)q (].)456 4568 y(As)k(Theorems)g(5.1)h(and)f(5.3)h(sho)m(w)f(the)h (problem)d(of)j(describing)d(of)i(\014nite)f(families)456 4676 y(of)d(determining)f(functionals)f(is)i(reduced)f(to)j(the)e (study)g(of)g(sets)h(of)g(functionals)e(for)456 4784 y(whic)m(h)38 b(the)i(estimate)h(\(31\))h(\(or)e(\(36\)\))i(holds)c (with)h Fp(")2366 4798 y Fh(L)2459 4784 y Fr(small)f(enough.)69 b(It)40 b(is)f(also)456 4892 y(imp)s(ortan)m(t)29 b(to)h(calculate)g (the)g(b)s(est)g(p)s(ossible)d(v)-5 b(alue)29 b(for)h Fp(")2480 4906 y Fh(L)2563 4892 y Fr(for)f(the)h(giv)m(en)g(family)e (of)456 5000 y(functional.)46 b(F)-8 b(or)33 b(this)f(parameter)h (there)g(is)e(the)i(estimate)h(from)e(b)s(elo)m(w)g(dep)s(ending)456 5108 y(only)23 b(on)h(a)g(n)m(um)m(b)s(er)f(of)h(functionals)e([10)q(]) i(and)g(this)e(estimate)j(coinside)e(with)f Fp(")3160 5122 y Fh(L)3237 5108 y Fr(when)456 5216 y(functionals)g(are)i(mo)s (des)f(\(see)i(Example)e(1)h(b)s(elo)m(w\).)38 b(W)-8 b(e)25 b(refer)f(to)g([10)q(,)g(11)q(])g(for)g(further)p eop %%Page: 20 20 20 19 bop 456 251 a Fs(20)327 b(IGOR)24 b(CHUESHO)n(V,)f(JINQIA)n(O)h (DUAN,)e(AND)h(BJ)2368 234 y(\177)2358 251 y(ORN)h(SCHMALFUSS)456 450 y Fr(discussion)f(concerning)j(an)g(optimal)f(c)m(hoice)i(of)f (families)e(of)j(functionals)d(with)h(prop-)456 558 y(erties)30 b(lik)m(e)g(\(31\))j(or)d(\(36\).)44 b(W)-8 b(e)32 b(also)f(note)g (that)g(the)g(deterministic)e(coun)m(terparts)i(of)456 666 y(Theorems)f(5.1)h(and)f(5.3)h(w)m(ere)g(pro)m(v)m(ed)g(in)e([7)q (])h(for)g(other)h(b)s(oundary)d(conditions.)456 837 y Fj(Example)47 b(1)f(\(Determining)h(mo)-5 b(des\))47 b Fr(Let)e Fn(f)p Fp(e)2097 851 y Fo(i)2126 837 y Fr(\()p Fp(x;)15 b(y)s Fr(\))p Fn(g)2381 804 y Fh(1)2381 863 y Fo(i)p Fs(=1)2546 837 y Fr(b)s(e)44 b(the)i(basis)e(of)h(eigen-)456 945 y(functions)30 b(of)j Fn(\000)p Fr(\001)f(with)f(the)h(p)s(erio)s (dic)e(b)s(oundary)g(conditions)h(in)g Fp(O)k Fr(suc)m(h)d(that)h(the) 456 1053 y(corresp)s(onding)28 b(eigen)m(v)-5 b(alues)30 b(p)s(ossesses)g(the)h(prop)s(ert)m(y)456 1224 y(\(42\))964 b(0)25 b Fp(<)g(\025)1799 1238 y Fs(1)1864 1224 y Fn(\024)g Fp(\025)2013 1238 y Fs(2)2078 1224 y Fn(\024)g Fp(:)15 b(:)g(:)h(:)456 1397 y Fr(W)-8 b(e)46 b(note)h(that)f(ev)m(ery)g (eigenfunction)e(has)i(the)f(form)g Fp(L)2526 1364 y Fh(\000)p Fs(1)2636 1397 y Fr(exp)p Fn(f)2830 1361 y Fs(2)p Fo(\031)p 2830 1376 79 4 v 2845 1428 a(L)2918 1397 y Fr(\()p Fp(j)2990 1411 y Fs(1)3030 1397 y Fp(x)31 b Fr(+)f Fp(j)3251 1411 y Fs(2)3291 1397 y Fp(y)s Fr(\))p Fn(g)p Fr(.)456 1505 y(Ho)m(w)m(ev)m(er)46 b(w)m(e)e(n)m(umerate)h (them)f(suc)m(h)g(that)g(\(42\))i(holds.)81 b(Supp)s(ose)42 b Fn(L)3028 1519 y Fo(N)3143 1505 y Fr(=)48 b Fn(f)p Fp(l)3334 1519 y Fo(j)3419 1505 y Fr(:)456 1616 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b(p)s(ossible)e(v)-5 b(alue)30 b Fp(")1268 3934 y Fh(L)1317 3945 y Fm(N)1408 3920 y Fr(for)g(families)d(of)j (functionals)e(consisting)h(of)h Fp(N)40 b Fr(elemen)m(ts.)456 4136 y(W)-8 b(e)44 b(also)g(note)g(that)g(the)g(estimates)g(for)g(the)f (completeness)h(defect)h Fp(")3032 4150 y Fh(L)3128 4136 y Fr(and)e(the)456 4244 y(Ekman)i(constan)m(t)h Fp(r)i Fr(giv)m(en)e(in)e(Theorems)h(5.1)i(and)e(5.3)h(are)g(rather)g(crude.) 85 b(In)456 4352 y(the)44 b(t)m(w)m(o-la)m(y)m(er)j(quasigeostrophic)c (mo)s(del)g(considered)g(ab)s(o)m(v)m(e,)50 b(the)44 b(parameter)h Fp(\027)456 4460 y Fr(is)d(the)h(molecular)f(viscosit)m (y)-8 b(.)79 b(F)-8 b(or)44 b(o)s(cean)f(w)m(ater,)48 b Fp(\027)k Fr(=)46 b(10)2628 4427 y Fh(\000)p Fs(6)2723 4460 y Fp(m)2803 4427 y Fs(2)2842 4460 y Fp(s)2885 4427 y Fh(\000)p Fs(1)2979 4460 y Fr(.)79 b(Ho)m(w)m(ev)m(er,)456 4568 y(\015uid)30 b(turbulence)i(at)i(small)d(scales)j(can)f(act)h(as)g (an)f(extra)h(dissipativ)m(e)d(mec)m(hanism,)456 4676 y(th)m(us)d(calling)g(for)h(the)g(substitution)e(of)i(the)h(molecular)e (viscosit)m(y)h(b)m(y)g(a)g(m)m(uc)m(h)h(larger)456 4784 y(eddy)35 b(viscosit)m(y)-8 b(.)57 b(F)-8 b(or)36 b(example,)h(in)e (the)h(t)m(w)m(o-la)m(y)m(er)i(quasigeostrophic)d(mo)s(del)f(\015o)m(w) 456 4892 y(sim)m(ulation)29 b(in)h([28)q(],)i(the)f(eddy)f(viscosit)m (y)h(\(w)m(e)h(still)e(use)g(the)i(same)f(notation)h(as)f(the)456 5000 y(molecular)21 b(viscosit)m(y\))i(is)f(tak)m(en)i(as)f Fp(\027)31 b Fr(=)25 b(50)p Fp(m)2027 4967 y Fs(2)2067 5000 y Fp(s)2110 4967 y Fh(\000)p Fs(1)2204 5000 y Fr(.)38 b(The)22 b(gra)m(vitational)h(acceleration)456 5108 y Fp(g)28 b Fr(=)d(9)p Fp(:)p Fr(81)p Fp(ms)906 5075 y Fh(\000)p Fs(2)1002 5108 y Fr(.)456 5216 y(A)m(t)f(mid-latitude)d(\(45) 26 b(degree)q Fp(N)10 b Fr(\),)25 b Fp(f)1710 5230 y Fs(0)1774 5216 y Fr(=)g(8)6 b Fn(\002)g Fr(10)2088 5183 y Fh(\000)p Fs(5)2183 5216 y Fp(s)2226 5183 y Fh(\000)p Fs(1)2343 5216 y Fr(and)22 b Fp(\014)30 b Fr(=)25 b(2)p Fp(:)p Fr(3)6 b Fn(\002)g Fr(10)2977 5183 y Fh(\000)p Fs(11)3108 5216 y Fp(m)3188 5183 y Fh(\000)p Fs(1)3282 5216 y Fp(s)3325 5183 y Fh(\000)p Fs(1)3419 5216 y Fr(.)p eop %%Page: 21 21 21 20 bop 1043 251 a Fs(STOCHASTIC)31 b(TW)n(O-LA)-6 b(YER)30 b(GEOPHYSICAL)h(FLO)n(WS)549 b(21)456 450 y Fr(F)-8 b(or)25 b(large-scale)g(\015o)m(ws)f(at)h(mid-latitude,)f(suc)m (h)g(as)g(the)h(Gulf)e(Stream)h(in)f(the)i(A)m(tlan)m(tic)456 558 y(o)s(cean,)35 b(the)f(horizon)m(tal)g(spatial)f(scale)h Fp(L)f Fr(is)g(at)i(the)f(order)f(of)h(1000)p Fp(k)s(m)p Fr(.)53 b(Moreo)m(v)m(er,)456 666 y(la)m(y)m(er)37 b(depth)g Fp(h)999 680 y Fs(1)1076 666 y Fr(and)g Fp(h)1312 680 y Fs(2)1389 666 y Fr(are)h(at)g(the)f(order)g(of)g(500)p Fp(m)i Fr(eac)m(h)f(for)f(large)h(scale)f(\015o)m(ws)456 774 y(suc)m(h)31 b(as)g(the)h(Gulf)e(Stream)i(at)g(the)f(mid-latitude)f (in)g(the)h(A)m(tlan)m(tic)h(o)s(cean.)45 b(F)-8 b(or)32 b(the)456 882 y(eddy)g(viscosit)m(y)h(50)p Fp(m)1218 849 y Fs(2)1258 882 y Fp(s)1301 849 y Fh(\000)p Fs(1)1429 882 y Fr(in)f([28)q(],)i(the)g(Ekman)f(constan)m(t)h Fp(r)i Fr(is)c(then)h(at)h(the)g(order)456 990 y(of)g(10)653 957 y Fh(\000)p Fs(5)749 990 y Fp(s)792 957 y Fh(\000)p Fs(1)886 990 y Fr(.)53 b(The)34 b(n)m(um)m(b)s(er)f Fp(r)k Fr(is)d(large)h(when)e(the)i(eddy)f(viscosit)m(y)g(is)g(tak)m(en)i(to)f (b)s(e)456 1098 y(large.)456 1206 y(The)f(o)s(cean)i(w)m(ater)f(mean)g (densit)m(y)f Fp(\032)1773 1220 y Fs(0)1848 1206 y Fr(is)g(ab)s(out)g (1025)p Fp(k)s(g)s(m)2564 1173 y Fh(\000)p Fs(3)2696 1206 y Fr(or)h(1)p Fp(:)p Fr(025)p Fp(g)s(cm)3182 1173 y Fh(\000)p Fs(3)3279 1206 y Fr(.)54 b Fp(\032)3405 1220 y Fs(1)456 1314 y Fr(and)29 b Fp(\032)679 1328 y Fs(2)749 1314 y Fr(are)i(in)e(the)h(vicinit)m(y)f(of)i Fp(\032)1640 1328 y Fs(0)1679 1314 y Fr(.)41 b(The)30 b(densit)m(y)f(di\013erence)h Fp(\032)2699 1328 y Fs(2)2759 1314 y Fn(\000)20 b Fp(\032)2897 1328 y Fs(1)2961 1314 y Fn(\031)25 b Fr(25)p Fp(k)s(g)s(m)3323 1281 y Fh(\000)p Fs(3)3419 1314 y Fr(.)456 1421 y(But)33 b(it)g(is)f(this)h(small)f(densit)m(y)g(di\013erence)h(that)h(in)e (turn)g(determines)h(pressure)f(dif-)456 1529 y(ferences)e(and)g (thereb)m(y)h(driv)m(e)e(the)i(o)s(cean)g(circulation)d([34)r(].)456 1637 y(The)f(mean)g(wind)e(forcing)i Fp(f)10 b Fr(,)28 b(i.e.,)g(the)g(deterministic)d(part)j(of)f(the)h(curl)e(of)i(the)f (wind)456 1745 y(stree)34 b(on)g(the)g(top)g(\015uid)d(la)m(y)m(er)j (is)f(usually)f(tak)m(en)j(as)f(a)g(stationary)-8 b(,)35 b(or)f(b)s(eing)e(time-)456 1853 y(a)m(v)m(eraged)j(and)e(ev)m(en)h (also)g(zonally)f(a)m(v)m(eraged,)k(sin)m(usoidal)30 b(function.)49 b(F)-8 b(or)35 b(example)456 1961 y([16)q(],)1564 2154 y Fp(f)g Fr(=)1780 2092 y(2)p Fp(\031)s(\034)1920 2106 y Fs(0)p 1750 2133 241 4 v 1750 2216 a Fp(\032)1797 2230 y Fs(0)1836 2216 y Fp(h)1888 2230 y Fs(1)1928 2216 y Fp(L)2015 2154 y Fr(sin)2152 2092 y(2)p Fp(\031)s(y)p 2152 2133 149 4 v 2195 2216 a(L)2310 2154 y(;)456 2378 y Fr(where)j(the)i(wind)d(tension)h Fp(\034)1486 2392 y Fs(0)1565 2378 y Fr(is)g(of)i(order)e(1)i Fp(dy)s(ne=cm)2465 2345 y Fs(2)2544 2378 y Fr(or)g(of)f(order)g(0.1)h Fp(N)r(=m)3379 2345 y Fs(2)3419 2378 y Fr(.)456 2486 y(With)33 b(this)g(mean)i(wind)d (forcing)h(and)h(ph)m(ysical)f(parameters)h(sp)s(eci\014ed)f(ab)s(o)m (v)m(e,)j(the)456 2594 y(inequalit)m(y)d(\(35\))j(turns)d(in)m(to)i (the)g(estimate)g Fp(")2080 2608 y Fh(L)2166 2594 y Fn(\024)d Fr(1)p Fp(:)p Fr(35)24 b Fn(\001)g Fr(10)2592 2561 y Fh(\000)p Fs(3)2687 2594 y Fp(m)34 b Fr(and)h(therefore)g(in)456 2702 y(the)c(case)h(of)f(Examples)f(1)i(or)f(2)g(for)g(the)h(n)m(um)m (b)s(er)d Fp(N)41 b Fr(of)32 b(functionals)d(w)m(e)j(obtain)e(the)456 2810 y(estimate)41 b Fp(N)53 b Fn(\031)43 b Fr(10)1160 2777 y Fs(18)1236 2810 y Fr(.)73 b(Th)m(us)40 b(Theorem)g(5.1)j(should) c(b)s(e)h(only)g(considered)g(as)h(an)456 2917 y(qualitativ)m(e)30 b(assertion)g(ab)s(out)h(\014nite-dimensionalit)m(y)c(of)k(the)f (long-time)h(b)s(eha)m(viour)456 3025 y(of)i(the)g(sto)s(c)m (hastically)g(forced)g(t)m(w)m(o-la)m(y)m(er)i(quasigeostrophic)e (\015uid)e(system.)49 b(As)33 b(for)456 3133 y(Theorem)42 b(5.3,)k(a)d(similar)c(calculations)j(sho)m(w)g(that)g(the)h(condition) e(\(41\))i(can)g(b)s(e)456 3241 y(v)-5 b(alid)28 b(under)g(some)j(sp)s (ecial)d(c)m(hoice)j(of)f(parameters.)41 b(Therefore)30 b(Theorem)f(5.3)i(only)456 3349 y(predicts)g(the)h(p)s(ossibilit)m(y)d (of)j(a)h(situation)e(when)h(the)g(b)s(ottom)h(\015uid)d(la)m(y)m(er)j (is)e(sla)m(v)m(ed)456 3457 y(b)m(y)f(the)g(top)h(la)m(y)m(er.)1688 3759 y(6.)46 b Fq(Summar)-6 b(y)456 3920 y Fr(W)e(e)30 b(ha)m(v)m(e)h(considered)d(asymptotic)i(probabilistic)c(dynamics)i(of) i(the)g(sto)s(c)m(hastically)456 4028 y(forced)40 b(t)m(w)m(o-la)m(y)m (er)i(quasigeostrophic)e(\015uid)d(system.)70 b(W)-8 b(e)42 b(\014rst)d(transformed)g(this)456 4136 y(system)c(in)m(to)h(a)g (coupled)f(system)h(of)g(random)f(partial)f(di\013eren)m(tial)g (equations)i(and)456 4244 y(then)21 b(sho)m(w)g(that)h(the)f (asymptotic)h(probabilistic)17 b(dynamics)j(of)i(this)e(system)h(dep)s (ends)456 4352 y(only)31 b(on)h(the)g(top)g(\015uid)e(la)m(y)m(er,)j (pro)m(vided)e(that)i(the)f(Wiener)f(pro)s(cess)h(and)f(the)i(\015uid) 456 4460 y(parameters)26 b(satisfy)g(a)h(certain)f(condition,)g(i.e.,)i (the)e(inequalit)m(y)f(\(38\);)30 b(see)d(Theorem)456 4568 y(5.3.)69 b(In)39 b(particular,)i(this)d(latter)i(condition)f(is)f (satis\014ed)h(when)g(the)h(trace)g(of)g(the)456 4676 y(co)m(v)-5 b(ariance)37 b(op)s(erator)g(of)f(the)h(Wiener)f(pro)s (cess)g(is)f(con)m(troled)i(b)m(y)f(a)h(certain)f(upp)s(er)456 4784 y(b)s(ound)j(\(see)j(Corollary)d(5.4\))k(and)d(the)i(Ekman)e (constan)m(t)j Fp(r)g Fr(is)d(su\016cien)m(tly)g(large)456 4892 y(\(see)c(the)g(inequalit)m(y)e(\(41\)\).)58 b(Note)37 b(that)g(the)f(generalized)f(time)g(deriv)-5 b(ativ)m(e)35 b(of)h(the)456 5000 y(Wiener)25 b(pro)s(cess)h(mo)s(dels)f(the)i (\015uctuating)f(part)g(of)g(the)h(wind)d(stress)i(forcing)g(on)g(the) 456 5108 y(top)37 b(\015uid)e(la)m(y)m(er,)k(and)e(the)g(Ekman)g (constan)m(t)h Fp(r)i Fr(measures)d(the)g(rate)h(for)f(v)m(orticit)m(y) 456 5216 y(deca)m(y)31 b(due)f(to)h(the)g(friction)d(in)h(the)i(b)s (ottom)g(Ekman)f(la)m(y)m(er.)p eop %%Page: 22 22 22 21 bop 456 251 a Fs(22)327 b(IGOR)24 b(CHUESHO)n(V,)f(JINQIA)n(O)h (DUAN,)e(AND)h(BJ)2368 234 y(\177)2358 251 y(ORN)h(SCHMALFUSS)456 450 y Fx(Ac)m(kno)m(wledgemen)m(t.)75 b Fr(A)42 b(part)f(of)h(this)f(w) m(ork)h(w)m(as)h(done)e(at)i(the)f(Ob)s(erw)m(olfac)m(h)456 558 y(Mathematical)22 b(Researc)m(h)h(Institute,)g(German)m(y)-8 b(,)25 b(while)20 b(J.)h(Duan)h(and)f(B.)h(Sc)m(hmalfu\031)456 666 y(w)m(ere)29 b(Researc)m(h)g(in)f(P)m(airs)g(F)-8 b(ello)m(ws,)29 b(supp)s(orted)e(b)m(y)i Fj(V)-7 b(olkswagen)32 b(Stiftung)p Fr(.)40 b(J.)28 b(Duan)456 774 y(w)m(ould)f(lik)m(e)h(to)h (thank)f(T)-8 b(ama)m(y)30 b(Ozgokmen,)f(Univ)m(ersit)m(y)f(of)h (Miami,)f(for)g(helpful)d(dis-)456 882 y(cussions.)38 b(This)25 b(w)m(ork)i(w)m(as)g(partly)f(supp)s(orted)f(b)m(y)h(the)h (NSF)g(Gran)m(t)g(DMS-9973204.)1685 1143 y Fq(References)494 1289 y Fw([1])39 b(L.)26 b(Arnold,)g Ft(Random)i(Dynamic)l(al)f (Systems)p Fw(.)h(Springer,)e(Berlin,)g(1998.)494 1380 y([2])39 b(L.)32 b(Arnold,)h(Hasselmann's)f(programm)f(visited:)47 b(The)31 b(analysis)i(of)g(sto)r(c)n(hasticit)n(y)f(in)g(deter-)613 1471 y(ministic)g(climate)g(mo)r(dels.)g(Rep)r(ort)g(450,)i(Univ)n (ersit\177)-38 b(at)32 b(Bremen,)h(Institut)e(f)r(\177)-41 b(ur)33 b(Dynamisc)n(he)613 1563 y(Systeme,)25 b(2000.)494 1654 y([3])39 b(E.)34 b(S.)f(Benilo)n(v,)j(On)d(the)g(stabilit)n(y)g (of)h(large-amplitude)f(geostrophic)i(\015o)n(ws)f(in)f(a)g(t)n(w)n (o-la)n(y)n(er)613 1745 y(\015uid:)h(the)26 b(case)g(of)h("strong")g(b) r(eta-e\013ect.)e Ft(J.)j(Fluid)f(Me)l(ch.)e Fb(284)i Fw(\(1995\),)g(137{158.)494 1837 y([4])39 b(P)-6 b(.)20 b(Berlo\013)h(and)f(S.)g(P)-6 b(.)20 b(Meac)n(ham,)h(On)e(the)h (stabilit)n(y)g(of)h(equiv)l(alen)n(t-barotropic)e(and)h(baro)r(clinic) 613 1928 y(mo)r(dels)26 b(of)g(the)g(wind-driv)n(en)e(circulation,)k (preprin)n(t,)d(1999.)494 2019 y([5])39 b(P)-6 b(.)21 b(Berlo\013)g(and)e(J.)i(C.)f(McWilliams,)j(Large-scale,)g(lo)n (w-frequency)d(v)l(ariabilit)n(y)h(in)e(wind-driv)n(en)613 2111 y(o)r(cean)27 b(gyres,)f Ft(J.)i(Phys.)f(Oc)l(e)l(ano)l(gr.)g Fb(29)f Fw(\(1999\),)h(1925{1949.)494 2202 y([6])39 b(C.)j(Bernier,)k Ft(Existenc)l(e)e(of)d(attr)l(actor)k(for)d(the)h(quasi-ge)l(ostr)l (ophic)i(appr)l(oximation)e(of)e(the)613 2293 y(Navier-Stokes)d(e)l (quations)f(and)e(estimate)i(of)d(its)i(dimension.)d Fw(Adv)l(ances)g(in)g(Mathematical)613 2385 y(Sciences)27 b(and)e(Applications,)h Fb(4)g Fw(\(2\),)g(pp)f(465-489)j(\(1994\).)494 2476 y([7])39 b(Ch.)30 b(Bernier-Kazan)n(tsev)f(and)f(I.D.)h(Ch)n (uesho)n(v,)g(The)g(\014niteness)g(of)g(determining)f(degrees)i(of)613 2567 y(freedom)35 b(for)g(the)f(quasi-geostrophic)h(m)n(ulti-la)n(y)n (er)f(o)r(cean)h(mo)r(del,)h Ft(Nonline)l(ar)g(A)n(nalysis)f Fb(42)613 2658 y Fw(\(2000\),)28 b(1499-1512.)494 2750 y([8])39 b(L.)27 b(Berselli)g(and)f(F.)g(Flandoli.)h(Remarks)e(on)h (determining)f(pro)t(jections)i(for)g(sto)r(c)n(hastic)g(dissi-)613 2841 y(pativ)n(e)f(equations.)g Ft(Discr)l(ete)j(and)f(Continuous)h (Dynamic)l(al)e(Systems)p Fw(,)h(5\(8\):197{214,)h(1999.)494 2932 y([9])39 b(J.)28 b(Brannan,)h(J.)e(Duan)g(and)g(T.)h(W)-6 b(anner,)27 b(Dissipativ)n(e)h(Quasigeostrophic)h(Dynamics)d(under)613 3024 y(Random)f(F)-6 b(orcing)26 b(,)g Ft(J.)h(Math.)h(A)n(nal.)f (Appl.)e Fb(228)h Fw(\(1998\),)h(221{233.)456 3115 y([10])39 b(I.)27 b(Ch)n(uesho)n(v,)g(Theory)f(of)i(functionals)f(that)f (uniquely)g(determine)f(asymptotic)h(dynamics)g(of)613 3206 y(in\014nite-dimensional)h(dissipativ)n(e)g(systems.)f Ft(Usp)l(ekhi)k(Mat.)e(Nauk)p Fw(,)g(53\(4\):77{124,)j(1998.)e(\(in)613 3298 y(Russian\).)d(English)h(translation)g(in)e Ft(R)n(ussian)k (Mathematic)l(al)f(Surveys)f Fw(53:731{776,)j(1998.)456 3389 y([11])39 b(I.)24 b(Ch)n(uesho)n(v,)g Ft(Intr)l(o)l(duction)k(to)f (the)f(The)l(ory)h(of)f(In\014nite-Dimensional)g(Dissip)l(ative)g (Systems)p Fw(.)613 3480 y(Acta,)g(Khark)n(o)n(v,)f(1999.)j(\(in)d (Russian\).)456 3572 y([12])39 b(I.)c(Ch)n(uesho)n(v,)i(On)d (determining)g(functionals)i(for)f(sto)r(c)n(hastic)h(Na)n(vier)f(-)f (Stok)n(es)h(equations.)613 3663 y Ft(Sto)l(chastics)c(and)c(Sto)l (chastics)k(R)l(ep)l(orts)p Fw(,)c(68:45{64,)j(1999.)456 3754 y([13])39 b(I.)26 b(Ch)n(uesho)n(v,)f(J.)h(Duan)e(and)h(B.)h(Sc)n (hmalfu\031.)f(Determining)f(functionals)i(for)g(random)e(partial)613 3846 y(di\013eren)n(tial)36 b(equations,)h(T)-6 b(o)35 b(app)r(ear)g(in)f Ft(Nonline)l(ar)j(Di\013er)l(ential)f(Equations)h (and)f(Applic)l(a-)613 3937 y(tions)p Fw(,)27 b(2001.)456 4028 y([14])39 b(P)-6 b(.)47 b(Constan)n(tin)g(and)g(C.)g(F)-6 b(oias,)53 b Ft(Navier-Stokes)d(Equations)p Fw(,)j(Univ.)46 b(of)i(Chicago)g(Press,)613 4120 y(Chicago,)28 b(1988.)456 4211 y([15])39 b(T.)32 b(DelSole)f(and)f(B.)h(F.)g(F)-6 b(arrell,)33 b(A)d(sto)r(c)n(hastically)j(excited)d(linear)i(system)d (as)i(a)g(mo)r(del)f(for)613 4302 y(quasigeostrophic)24 b(turbulence:)32 b(Analytic)22 b(results)h(for)g(one-)e(and)h(t)n(w)n (o-la)n(y)n(er)g(\015uids,)h Ft(J.)h(A)n(tmos.)613 4394 y(Sci.)i Fb(52)g Fw(\(1995\))g(2531-2547.)456 4485 y([16])39 b(V.)22 b(Dymnik)n(o)n(v)e(and)i(E.)g(Kazan)n(tsev,)h(On)f(the)f (genetic)i(\\memory")e(of)i(c)n(haotic)g(attractor)g(of)g(the)613 4576 y(barotropic)f(o)r(cean)g(mo)r(del.)e(In)h(Pro)r(ceedings)h(of)g (the)e(third)g(bilateral)j(conference)e(\\Predictabil-)613 4668 y(it)n(y)32 b(of)g(atmospheric)g(and)f(o)r(ceanic)i(circulations)h (")e(of)g(the)g(F)-6 b(renc)n(h-Russian)30 b(A.M.Liapuno)n(v)613 4759 y(Institute)35 b(in)h(Computer)f(Science)h(and)f(Applied)g (Mathematics)h(\(INRIA)e(-)h(Mosco)n(w)i(State)613 4850 y(Univ)n(ersit)n(y\).)25 b(Nancy)-6 b(,)25 b(April,)h(1997.Edition)i (MSU,)e(1997,)h(pp.)e(25-36.)456 4942 y([17])39 b(F.)21 b(Flandoli)f(and)g(J.)g(A.)g(Langa.)g(On)f(determining)g(mo)r(des)g (for)i(dissipativ)n(e)f(random)f(dynamical)613 5033 y(systems.)26 b Ft(Sto)l(chastics)k(and)e(Sto)l(chastics)i(R)l(ep)l(orts)p Fw(,)e(66:1{25,)g(1999.)456 5124 y([18])39 b(C.)28 b(F)-6 b(oias,)29 b(O.)d(Manley)-6 b(,)28 b(R.T)-6 b(emam)25 b(and)i(Y.M.)g(T)-6 b(rev)n(e,)27 b(Asymptotic)f(analysis)i(of)f(the)g (Na)n(vier-)613 5216 y(Stok)n(es)f(equations,)g Ft(Physic)l(a)i(D)e Fb(9)g Fw(\(1983\),)h(157{188.)p eop %%Page: 23 23 23 22 bop 1043 251 a Fs(STOCHASTIC)31 b(TW)n(O-LA)-6 b(YER)30 b(GEOPHYSICAL)h(FLO)n(WS)549 b(23)456 450 y Fw([19])39 b(C.)34 b(F)-6 b(oias)34 b(and)f(G.)g(Pro)r(di.)h(Sur)e(le)h (comp)r(ortemen)n(t)f(global)i(des)f(solutions)h(nonstationnaires)613 541 y(des)26 b(\023)-36 b(equations)28 b(de)g(Na)n(vier-Stok)n(es)f(en) g(dimension)g(deux.)g Ft(R)l(end.)j(Sem.)f(Mat.)g(Univ.)g(Padova)p Fw(,)613 633 y(39:1{34,)g(1967.)456 724 y([20])39 b(C.)31 b(F)-6 b(oias,)33 b(E.S.)e(Titi,)h(Determining)d(no)r(des,)j(\014nite)e (di\013erence)g(sc)n(hemes)g(and)f(inertial)i(mani-)613 815 y(folds.)d Ft(Nonline)l(arity)p Fw(,)e Fb(4)p Fw(,)g(pp)f(135-153)j (\(1991\).)456 907 y([21])39 b(A.)31 b(Gri\013a)f(and)g(S.)g (Castellari,)35 b(Nonlinear)30 b(general)i(circulation)f(of)g(an)f(o)r (cean)h(mo)r(del)f(driv)n(en)613 998 y(b)n(y)25 b(wind)h(with)g(a)g (sto)r(c)n(hastic)h(comp)r(onen)n(t,)e Ft(J.)i(Marine)h(R)l(ese)l(ar)l (ch)f Fb(49)f Fw(\(1991\),)i(53-73.)456 1089 y([22])39 b(K.)26 b(Hasselmann.)g(Sto)r(c)n(hastic)g(climate)g(mo)r(dels,)g(Part) g(I.)g Ft(T)-6 b(el)t(lus)p Fw(,)26 b(28:473{485,)j(1976.)456 1181 y([23])39 b(G.)24 b(Hollo)n(w)n(a)n(y)-6 b(,)24 b(Ocean)f(circulation:)34 b(Flo)n(w)24 b(in)f(probabilit)n(y)g(under)f (statistical)j(dynamical)d(forc-)613 1272 y(ing,)34 b(in)e Ft(Sto)l(chastic)j(Mo)l(dels)f(in)e(Ge)l(osystems)p Fw(,)k(S.)c(Molc)n (hano)n(v)g(and)g(W.)f(W)-6 b(o)n(yczynski)31 b(\(eds.\),)613 1363 y(Springer,)26 b(1996.)456 1455 y([24])39 b(R.)27 b(X.)g(Huang)f(and)g(H.)h(Stommel,)f(Cross)i(sections)g(of)f(a)g(t)n(w) n(o-la)n(y)n(er)g(inertial)h(Gulf)f(Stream,)g Ft(J.)613 1546 y(Phys.)h(Oc)l(e)l(ano)l(gr)l(aphy)h Fb(20)d Fw(\(1990\),)h (907-901.)456 1637 y([25])39 b(D.A.)c(Jones)h(and)e(E.S.)i(Titi,)i (Determining)c(\014nite)h(v)n(olume)f(elemen)n(ts)g(for)i(the)e(2D)h (Na)n(vier-)613 1729 y(Stok)n(es)26 b(equations,)g Ft(Physic)l(a)i(D)e Fb(60)g Fw(\(1992\),)h(165-174.)456 1820 y([26])39 b(O.)31 b(Ladyzhensk)l(a)n(y)n(a.)e(A)h(dynamical)f(system)h(generated)g(b)n(y) g(the)g(Na)n(vier{Stok)n(es)g(equations.)613 1911 y Ft(Journal)f(of)e (Soviet)h(Mathematics)p Fw(,)g(3:458{479,)h(1975.)456 2003 y([27])39 b(P)-6 b(.)30 b(M)r(\177)-41 b(uller,)32 b(Sto)r(c)n(hastic)e(forcing)h(of)f(quasi-geostrophic)g(eddies,)h(in)e Ft(Sto)l(chastic)k(Mo)l(del)t(ling)e(in)613 2094 y(Physic)l(al)h(Oc)l (e)l(ano)l(gr)l(aphy)p Fw(,)i(R.)c(J.)g(Adler,)h(P)-6 b(.)30 b(M)r(\177)-41 b(uller)31 b(and)f(B.)g(Rozo)n(vskii)g(\(eds.\),) i(Birkh\177)-38 b(auser,)613 2185 y(1996.)456 2277 y([28])39 b(T.)h(Ozgokmen,)i(Emergence)c(of)i(inertial)g(gyres)f(in)g(a)g(t)n(w)n (o-la)n(y)n(er)g(quasigeostrophic)i(o)r(cean)613 2368 y(mo)r(del,)26 b Ft(J.)h(Phys.)h(Oc)l(e)l(ano)l(gr)l(aphy)g Fb(28)e Fw(\(1998\),)i(461-484.)456 2459 y([29])39 b(J.)27 b(P)n(edlosky)-6 b(.)25 b Ft(Ge)l(ophysic)l(al)k(Fluid)e(Dynamics)p Fw(.)g(Springer)e(V)-6 b(erlag,)27 b(New-Y)-6 b(ork,)25 b(Berlin,)i(1987.)456 2551 y([30])39 b(J.)27 b(P)n(edlosky)-6 b(,)25 b Ft(Oc)l(e)l(an)k(Cir)l(culation)f(The)l(ory)p Fw(.)f(Springer{V)-6 b(erlag,)27 b(Berlin,)f(1996.)456 2642 y([31])39 b(R.)34 b(Salmon.)f Ft(L)l(e)l(ctur)l(es)k(on)e(Ge)l (ophysic)l(al)h(Fluid)e(Dynamics)p Fw(.)g(Oxford)g(Univ.)f(Press,)j (Oxford,)613 2733 y(1998.)456 2825 y([32])j(R.)22 b(Salmon.)f (Generalized)h(t)n(w)n(o-la)n(y)n(er)g(mo)r(dels)f(of)h(o)r(cean)g (circulation,)i Ft(J.)f(Marine)h(R)l(ese)l(ar)l(ch)f Fb(52)613 2916 y Fw(\(1994\),)28 b(865-908.)456 3007 y([33])39 b(R.)26 b(M.)g(Samelson,)f(Sto)r(c)n(hastically)i(forced)f (curren)n(t)e(\015uctuations)i(in)f(v)n(ertical)h(shear)g(and)f(o)n(v)n (er)613 3099 y(top)r(ograph)n(y)-6 b(,)26 b Ft(J.)h(Ge)l(ophys.)i(R)l (es.)d Fb(94)g Fw(\(1989\))h(8207-8215.)456 3190 y([34])39 b(W.)26 b(M.)g(W)-6 b(ashington)25 b(and)g(C.)h(L.)f(P)n(arkinson,)h Ft(A)n(n)h(Intr)l(o)l(duction)i(to)f(Thr)l(e)l(e-Dimensional)g(Cli-)613 3281 y(mate)h(Mo)l(deling)p Fw(,)d(Oxford)f(Univ.)g(Press,)i(1986.)456 3452 y(\(Igor)32 b(Ch)n(uesho)n(v\))f Fv(Institute)j(f)1465 3446 y(\177)1463 3452 y(ur)h(D)n(ynamische)e(Systeme,)i(FB3,)h (Universit)3012 3446 y(\177)3010 3452 y(at)d(Bremen,)456 3544 y(D-28334)21 b(Bremen,)f(Germany.)33 b(On)20 b(lea)-7 b(ve)20 b(fr)n(om)h(Dep)-5 b(ar)g(tment)18 b(of)j(Mechanics)e(andMa)-5 b(th-)456 3635 y(ema)g(tics,)27 b(Khark)n(o)n(v)i(University,)e(310077) j(Khark)n(o)n(v,)f(Ukraine)456 3726 y Ft(E-mail)e(addr)l(ess)p Fw(,)h(Igor)e(D.)f(Ch)n(uesho)n(v)q(:)34 b Fa (chueshov@math.uni-bremen.de)456 3882 y Fw(\(Jinqiao)22 b(Duan\))f Fv(Dep)-5 b(ar)g(tment)24 b(of)h(Applied)d(Ma)-5 b(thema)g(tics,)25 b(Illinois)g(Institute)f(of)h(Tech-)456 3973 y(nology,)k(Chica)n(go,)f(IL)g(60616,)i(USA.)456 4064 y Ft(E-mail)d(addr)l(ess)p Fw(,)h(Jinqiao)e(Duan:)34 b Fa(duan@iit.edu)456 4220 y Fw(\(Bj\177)-38 b(orn)30 b(Sc)n(hmalfu\031\))f Fv(Dep)-5 b(ar)g(tment)31 b(of)h(Applied)f (Sciences,)h(University)e(of)j(Technology)456 4311 y(and)27 b(Applied)g(Sciences,)g(Geusaer)g(Strasse,)h(D{06217)i(Mersebur)n(g,)e (Germany,)456 4402 y Ft(E-mail)f(addr)l(ess)p Fw(,)h(Bj\177)-38 b(orn)26 b(Sc)n(hmalfu\031:)34 b Fa(schmalfuss@in.fh-merseburg.d)q(e)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0104251443200--