%%%% How to reproduce a hard copy of this paper: %%%% %%%% This paper is in PostScript format. %%%% Remove lines above the "cut here" line; save this file %%%% as stable.ps, and use "lpr stable.ps" %%%% (or some command like it) to print. %%%% %%%%%%%%%%%%%%%%%%% cut here %%%%%%%%%%%%%%%% %!PS-Adobe-2.0 %%Creator: dvipsk 5.55a Copyright 1986, 1994 Radical Eye Software %%Title: stable.dvi %%Pages: 11 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips stable -o %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1995.10.18:2013 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if 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Fp(\()p Fk(h)p Fp(\))31 b(can)g(b)s(e)f(found)f(b)m(y)h(solving)1303 870 y(0)c(=)f([)p Fk(F)s(;)15 b(C)7 b Fp(\()p Fk(h)p Fp(\)])26 b(=)f(\()1982 808 y Fk(\016)s(F)p 1982 849 115 4 v 1991 932 a(\016)s(h)2108 870 y(;)15 b Ff(D)2230 808 y Fk(\016)s(C)p 2230 849 116 4 v 2240 932 a(\016)s(h)2356 870 y Fp(\))p Fk(;)834 b Fp(\(2.7\))448 1107 y(for)30 b(all)g Fk(F)13 b Fp(\()p Fk(h)p Fp(\))26 b Ff(2)f Fk(L)1081 1074 y Fg(2)1120 1107 y Fp(\(\012\).)41 b(Hence)32 b(w)m(e)e(ha)m(v)m (e)1585 1354 y Fk(J)9 b Fp(\()p Fk(h;)1782 1293 y(\016)s(C)p 1782 1333 V 1792 1417 a(\016)s(h)1907 1354 y Fp(\))26 b(=)f(0)p Fk(;)1116 b Fp(\(2.8\))448 1598 y(whic)m(h)29 b(means)i(that)1194 1563 y Fe(\016)r(C)p 1194 1578 89 4 v 1201 1630 a(\016)r(h)1323 1598 y Fp(needs)f(to)h(b)s(e)e(an)i (arbitrary)e(\(smo)s(oth\))i(function)e(of)h Fk(h)p Fp(,)h(or)1422 1844 y Fk(C)7 b Fp(\()p Fk(h)p Fp(\))25 b(=)1737 1729 y Fd(Z)53 b(Z)1934 1844 y Fk(g)s Fp(\()p Fk(h)p Fp(\))p Fk(dxdy)957 b Fp(\(2.9\))448 2082 y(for)30 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y Fd(Z)52 b(Z)1804 2517 y Fp(\()1839 2402 y Fd(Z)1923 2428 y Fe(h)1983 2517 y Fk(f)10 b Fp(\()p Fk(\030)t Fp(\))p Fk(d\030)t Fp(\))p Fk(dxdy)s(:)753 b Fp(\(3.4\))448 2754 y(With)30 b(this)f(c)m(hosen)i Fk(C)7 b Fp(\()p Fk(h)p Fp(\),)31 b(w)m(e)g(calculate)521 2992 y Fk(\016)564 2954 y Fg(2)628 2969 y Fp(~)605 2992 y Fk(H)7 b Fp(\()p Fk(h)775 3006 y Fg(0)815 2992 y Fp(\))25 b(=)971 2877 y Fd(Z)52 b(Z)1152 2992 y Fp([\()p Ff(4)p Fk(h)1345 3006 y Fg(0)1406 2992 y Fp(+)20 b Fk(\013y)j Ff(\000)d Fk(f)1769 2954 y Fc(0)1792 2992 y Fp(\()p Fk(h)1879 3006 y Fg(0)1919 2992 y Fp(\)\)\()p Fk(\016)s(h)p Fp(\))2154 2954 y Fg(2)2216 2992 y Ff(\000)g Fk(h)2359 3006 y Fg(0)2399 2992 y Ff(r)p Fp(\()p Fk(\016)s(h)2605 3006 y Fg(0)2646 2992 y Fp(\))g Ff(\001)h Fp(\()p Fk(\016)s(h)2877 3006 y Fg(0)2918 2992 y Fp(\)])p Fk(dxdy)s(:)53 b Fp(\(3.5\))448 3231 y(W)-8 b(e)32 b(note)f(that)g Fk(f)1059 3198 y Fc(0)1082 3231 y Fp(\()p Fk(h)1169 3245 y Fg(0)1209 3231 y Fp(\))g(can)f(b)s(e)g (obtained)g(b)m(y)g(di\013eren)m(tiating)g(the)g(equation)h(\(2.11\).) 448 3344 y(Hence)h(w)m(e)e(ha)m(v)m(e)473 3568 y Fk(\016)516 3530 y Fg(2)580 3545 y Fp(~)556 3568 y Fk(H)7 b Fp(\()p Fk(h)726 3582 y Fg(0)766 3568 y Fp(\))26 b(=)f Ff(\000)1009 3453 y Fd(Z)52 b(Z)1205 3568 y Fk(h)1257 3582 y Fg(0)1297 3568 y Fk(h)1349 3530 y Fg(2)1349 3590 y(0)p Fe(y)1426 3568 y Ff(r)p Fp(\()1547 3506 y Fk(\016)s(h)1642 3520 y Fg(0)p 1547 3547 136 4 v 1550 3630 a Fk(h)1602 3644 y Fg(0)p Fe(y)1693 3568 y Fp(\))20 b Ff(\001)h(r)p Fp(\()1915 3506 y Fk(\016)s(h)2010 3520 y Fg(0)p 1915 3547 V 1918 3630 a Fk(h)1970 3644 y Fg(0)p Fe(y)2060 3568 y Fp(\))p Fk(dxdy)j Ff(\000)c Fk(\013)2475 3453 y Fd(Z)52 b(Z)2700 3506 y Fk(h)2752 3520 y Fg(0)p 2681 3547 130 4 v 2681 3630 a Fk(h)2733 3644 y Fg(0)p Fe(y)2820 3568 y Fp(\()p Fk(\016)s(h)p Fp(\))2985 3530 y Fg(2)3027 3568 y Fk(dxdy)s(:)t Fp(\(3.6\))448 3820 y(W)-8 b(e)42 b(kno)m(w)e(that)h Fk(h)1121 3834 y Fg(0)1203 3820 y Fk(>)h Fp(0)e(on)h(the)f(domain)g (\012.)70 b(Therefore,)43 b(for)d(meridionally)d(in-)448 3933 y(creasing)d(fron)m(ts)g(\()p Fk(h)1150 3947 y Fg(0)p Fe(y)1259 3933 y Fk(>)c Fp(0\),)36 b Fk(\016)1544 3900 y Fg(2)1608 3910 y Fp(~)1585 3933 y Fk(H)6 b Fp(\()p Fk(h)1754 3947 y Fg(0)1795 3933 y Fp(\))34 b(is)f(negativ)m (e-de\014nite.)52 b(Hence)34 b(w)m(e)h(ha)m(v)m(e)g(the)448 4045 y(follo)m(wing)29 b(result:)448 4258 y Fl(Theorem)34 b(1)46 b Fb(\(i\))35 b(When)g(the)g Fk(\014)5 b Fb(-e\013e)-5 b(ct)34 b(is)h(at)g(the)g(or)-5 b(der)37 b(of)e(the)g(R)-5 b(ossby)36 b(numb)-5 b(er)35 b Fk(\017)p Fb(,)448 4371 y(al)5 b(l)32 b(meridional)5 b(ly)34 b(incr)-5 b(e)g(asing)32 b(ste)-5 b(ady)33 b(fr)-5 b(onts)33 b(ar)-5 b(e)33 b(formal)5 b(ly)33 b(stable,)f(and)h(henc)-5 b(e)32 b(it)f(is)448 4509 y(line)-5 b(arly)34 b(stable)g(in)e(the)h(sense)g(of)g(Lyapunov)g (in)g(the)g(norm)h Ff(jj)p Fk(h)p Ff(jj)26 b Fp(=)2831 4403 y Fd(q)p 2914 4403 361 4 v 106 x Ff(\000)p Fk(\016)3028 4483 y Fg(2)3092 4486 y Fp(~)3068 4509 y Fk(H)7 b Fp(\()p Fk(h)p Fp(\))q Fb(.)589 4622 y(\(ii\))39 b(When)f(the)h Fk(\017)p Fb(-e\013e)-5 b(ct)38 b(is)g(zer)-5 b(o,)41 b(al)5 b(l)39 b(ste)-5 b(ady)40 b(fr)-5 b(onts)40 b(ar)-5 b(e)39 b(formal)5 b(ly)40 b(and)g(henc)-5 b(e)448 4735 y(line)g(arly)34 b(stable)g(in)e(the)h(sense)g(of)g(Lyapunov.)1920 5162 y Fp(5)p eop %%Page: 6 6 6 5 bop 589 623 a Fp(W)-8 b(e)24 b(remark)d(that)i(the)f(second)g (statemen)m(t)i(ab)s(o)m(v)m(e)f(w)m(as)g(pro)m(v)m(ed)f(in)f(Sw)m (aters)h([1993])448 736 y(and)40 b(can)h(b)s(e)f(easily)g(seen)g(from)h (\(3.6\).)73 b(The)40 b(statemen)m(t)i(\(i\))f(in)e(Theorem)h(1)h(also) 448 848 y(implies)31 b(the)j(result)e(b)m(y)h(Benilo)m(v)g([1994])j (who)d(sho)m(w)m(ed)h(that)g(when)e(the)i Fk(\014)5 b Fp(-)34 b(e\013ect)h(is)448 961 y(presen)m(t,)40 b(all)d(meridionally)d (increasing)i(steady)i(fron)m(ts)g(are)g(sp)s(ectrally)e(stable)i(\(no) 448 1074 y(p)s(ositiv)m(e)30 b(eigen)m(v)-5 b(alue\).)41 b(Our)29 b(result)g(is)h(generally)f(stronger)i(than)f(Benilo)m(v's)g (result.)589 1187 y(F)-8 b(or)28 b(the)e(equation)g(\(1.3\),)j(w)m(e)e (also)f(consider)g(a)g(constrained)g(Hamiltonian)f(func-)448 1300 y(tion)42 b(as)g(in)f(\(3.1\))j(but)d(with)g Fk(H)7 b Fp(\()p Fk(h)p Fp(\))43 b(de\014ned)e(b)m(y)h(\(2.12\).)78 b(W)-8 b(e)43 b(c)m(ho)s(ose)g Fk(C)7 b Fp(\()p Fk(h)p Fp(\))42 b(suc)m(h)448 1413 y(that)1068 1622 y(0)26 b(=)f Fk(\016)1303 1599 y Fp(~)1278 1622 y Fk(H)8 b Fp(\()p Fk(h)1449 1636 y Fg(0)1489 1622 y Fp(\))26 b(=)1646 1507 y Fd(Z)52 b(Z)1827 1622 y Fp([)p Fk(y)s(h)1952 1636 y Fg(0)2012 1622 y Fp(+)20 b Fk(g)2149 1584 y Fc(0)2173 1622 y Fp(\()p Fk(h)2260 1636 y Fg(0)2300 1622 y Fp(\)])p Fk(\016)s(hdxdy)604 b Fp(\(3.7\))448 1820 y(or)1678 2048 y Fk(g)s Fp(\()p Fk(h)p Fp(\))26 b(=)f Ff(\000)2054 1934 y Fd(Z)2137 1960 y Fe(h)2197 2048 y Fk(f)10 b Fp(\()p Fk(\030)t Fp(\))p Fk(d\030)t(;)768 b Fp(\(3.8\))1221 2286 y Fk(C)7 b Fp(\()p Fk(h)p Fp(\))26 b(=)f Ff(\000)1623 2171 y Fd(Z)52 b(Z)1804 2286 y Fp(\()1839 2171 y Fd(Z)1923 2197 y Fe(h)1983 2286 y Fk(f)10 b Fp(\()p Fk(\030)t Fp(\))p Fk(d\030)t Fp(\))p Fk(dxdy)s(:)753 b Fp(\(3.9\))448 2512 y(W)-8 b(e)32 b(calculate)1123 2721 y Fk(\016)1166 2683 y Fg(2)1230 2698 y Fp(~)1206 2721 y Fk(H)7 b Fp(\()p Fk(h)1376 2735 y Fg(0)1416 2721 y Fp(\))26 b(=)1573 2606 y Fd(Z)52 b(Z)1754 2721 y Fp([)p Fk(y)23 b Ff(\000)d Fk(g)1984 2683 y Fc(0)2008 2721 y Fp(\()p Fk(h)2095 2735 y Fg(0)2135 2721 y Fp(\)]\()p Fk(\016)s(h)p Fp(\))2360 2683 y Fg(2)2402 2721 y Fk(dxdy)1643 2948 y Fp(=)25 b Ff(\000)1825 2833 y Fd(Z)52 b(Z)2050 2887 y Fk(h)2102 2901 y Fg(0)p 2032 2927 130 4 v 2032 3011 a Fk(h)2084 3025 y Fg(0)p Fe(y)2171 2948 y Fp(\()p Fk(\016)s(h)p Fp(\))2336 2911 y Fg(2)2377 2948 y Fk(dxdy)s(;)609 b Fp(\(3.10\))448 3199 y(whic)m(h)21 b(is)g(negativ)m(e-de\014nite)h(for) g(meridionally)d(increasing)h(steady)j(fron)m(ts)e(\()p Fk(h)3132 3213 y Fg(0)p Fe(y)3235 3199 y Fk(>)k Fp(0\),)448 3312 y(and)30 b(p)s(ositiv)m(e-de\014nite)f(for)h(meridionally)d (decreasing)j(steady)h(fron)m(ts)f(\()p Fk(h)3018 3326 y Fg(0)p Fe(y)3121 3312 y Fk(<)25 b Fp(0\).)589 3425 y(Hence)32 b(w)m(e)e(ha)m(v)m(e)i(the)f(follo)m(wing)d(result:)448 3632 y Fl(Theorem)34 b(2)46 b Fb(When)35 b(the)g Fk(\014)5 b Fb(-)35 b(e\013e)-5 b(ct)35 b(is)g(much)g(lar)-5 b(ger)36 b(than)g(the)g(R)-5 b(ossby)36 b(numb)-5 b(er)35 b Fk(\017)p Fb(,)448 3744 y(b)-5 b(oth)27 b(meridional)5 b(ly)26 b(incr)-5 b(e)g(asing)26 b(and)g(de)-5 b(cr)g(e)g(asing)27 b(ste)-5 b(ady)27 b(fr)-5 b(onts)26 b(ar)-5 b(e)26 b(formal)5 b(ly)27 b(stable,)448 3857 y(and)46 b(henc)-5 b(e)46 b(line)-5 b(arly)46 b(stable)g(in)f(the)h(sense)f(of)g(Lyapunov,)k(in)c (the)h(norm)g Ff(jj)p Fk(h)p Ff(jj)j Fp(=)448 3895 y Fd(q)p 531 3895 361 4 v 106 x Ff(\000)p Fk(\016)645 3974 y Fg(2)709 3978 y Fp(~)685 4001 y Fk(H)7 b Fp(\()p Fk(h)p Fp(\))34 b Fb(or)f Ff(jj)p Fk(h)p Ff(jj)27 b Fp(=)1316 3895 y Fd(q)p 1399 3895 290 4 v 106 x Fk(\016)1442 3974 y Fg(2)1506 3978 y Fp(~)1482 4001 y Fk(H)7 b Fp(\()p Fk(h)p Fp(\))q Fb(,)33 b(r)-5 b(esp)g(e)g(ctively.)448 4286 y Fh(4)135 b(Nonlinear)46 b(Stabilit)l(y)448 4489 y Fp(T)-8 b(o)25 b(sho)m(w)f(nonlinear)e(stabilit)m(y)g(of)i(steady)h (fron)m(ts,)g(w)m(e)g(need)f(to)g(deriv)m(e)g(global)f(b)s(ounds)448 4602 y(\(in)28 b(some)i(norm\))e(of)h(the)g(disturbances)e(in)h(terms)h (of)g(kno)m(wn)f(data.)41 b(T)-8 b(o)30 b(this)d(end)i(w)m(e)448 4715 y(consider)h(the)g(conserv)m(ed)h(function)e(as)i(in)e(Sw)m(aters) h([1993])876 4913 y Ff(L)p Fp(\()p Fk(h)p Fp(\))c(=)f Fk(H)7 b Fp(\()p Fk(h)1353 4927 y Fg(0)1414 4913 y Fp(+)20 b Fk(h)p Fp(\))h Ff(\000)e Fk(H)7 b Fp(\()p Fk(h)1873 4927 y Fg(0)1914 4913 y Fp(\))20 b(+)g Fk(C)7 b Fp(\()p Fk(h)2219 4927 y Fg(0)2279 4913 y Fp(+)20 b Fk(h)p Fp(\))h Ff(\000)f Fk(C)7 b Fp(\()p Fk(h)2728 4927 y Fg(0)2767 4913 y Fp(\))p Fk(;)423 b Fp(\(4.1\))1920 5162 y(6)p eop %%Page: 7 7 7 6 bop 448 623 a Fp(whic)m(h)29 b(b)s(ecomes,)i(after)g(using)e (\(2.2\))j(and)e(\(3.4\))601 843 y Ff(L)p Fp(\()p Fk(h)p Fp(\))c(=)918 781 y(1)p 918 822 46 4 v 918 905 a(2)988 728 y Fd(Z)52 b(Z)1169 843 y Fp([)p Ff(4)p Fk(h)1327 857 y Fg(0)1367 843 y Fk(h)1419 805 y Fg(2)1479 843 y Ff(\000)20 b Fp(\()p Fk(h)1657 857 y Fg(0)1718 843 y Fp(+)g Fk(h)p Fp(\))p Ff(r)p Fk(h)h Ff(\001)f(r)p Fk(h)g Fp(+)g Fk(\013y)s Fp(\(2)p Fk(h)2567 857 y Fg(0)2608 843 y Fk(h)h Fp(+)f Fk(h)2824 805 y Fg(2)2864 843 y Fp(\)])p Fk(dxdy)1705 1080 y Fp(+)1791 965 y Fd(Z)52 b(Z)1972 1080 y Fp([)1997 965 y Fd(Z)2080 992 y Fe(h)2121 1001 y Fa(0)2043 1154 y Fe(h)2084 1163 y Fa(0)2118 1154 y Fg(+)p Fe(h)2233 1080 y Fk(f)10 b Fp(\()p Fk(\030)t Fp(\))p Fk(d\030)25 b Fp(+)19 b Fk(f)10 b Fp(\()p Fk(h)2746 1094 y Fg(0)2786 1080 y Fp(\))p Fk(h)p Fp(])p Fk(dxdy)s(:)133 b Fp(\(4.2\))448 1303 y(W)-8 b(e)32 b(assume)e(that)1577 1485 y Fk(a)1625 1499 y Fg(1)1665 1485 y Fk(\030)f(<)c(f)10 b Fp(\()p Fk(\030)t Fp(\))25 b Fk(<)g(b)2159 1499 y Fg(1)2199 1485 y Fk(\030)t(;)982 b Fp(\(4.3\))1436 1623 y(sup)1475 1700 y Fc(R)1573 1623 y Fp(\()p Ff(4)p Fk(h)1741 1637 y Fg(0)1801 1623 y Fp(+)20 b Fk(\013y)s Fp(\))26 b Fk(<)f(a)2203 1637 y Fg(1)2243 1623 y Fk(;)982 b Fp(\(4.4\))448 1849 y(where)30 b Fk(a)759 1863 y Fg(1)799 1849 y Fk(;)15 b(b)878 1863 y Fg(1)948 1849 y Fp(are)31 b(\014nite)e(real)h(n)m(um)m (b)s(ers.)39 b(Hence)31 b(w)m(e)g(ha)m(v)m(e)825 2064 y Ff(L)p Fp(\()p Fk(h)p Fp(\))26 b Fk(<)1142 2002 y Fp(1)p 1142 2043 V 1142 2126 a(2)1212 1949 y Fd(Z)52 b(Z)1393 2064 y Fp(\()p Ff(4)p Fk(h)1561 2078 y Fg(0)1622 2064 y Fp(+)20 b Fk(\013y)j Ff(\000)d Fk(b)1969 2078 y Fg(1)2008 2064 y Fp(\))p Fk(h)2095 2026 y Fg(2)2156 2064 y Ff(\000)g Fp(\()p Fk(h)2334 2078 y Fg(0)2394 2064 y Fp(+)g Fk(h)p Fp(\))p Ff(r)p Fk(h)h Ff(\001)g(r)p Fk(h)1717 2281 y Fp(+)1803 2166 y Fd(Z)52 b(Z)1984 2281 y Fp([)p Fk(\013y)s(h)2167 2295 y Fg(0)2228 2281 y Ff(\000)20 b Fk(b)2358 2295 y Fg(1)2397 2281 y Fk(h)2449 2295 y Fg(0)2509 2281 y Fp(+)g Fk(f)10 b Fp(\()p Fk(h)2742 2295 y Fg(0)2782 2281 y Fp(\)])p Fk(h)1177 2504 y(<)1283 2443 y Fp(1)p 1283 2483 V 1283 2567 a(2)1353 2389 y Fd(Z)53 b(Z)1535 2504 y Fp(\()p Ff(4)p Fk(h)1703 2518 y Fg(0)1763 2504 y Fp(+)20 b Fk(\013y)j Ff(\000)d Fk(b)2110 2518 y Fg(1)2150 2504 y Fp(\))p Fk(h)2237 2467 y Fg(2)2297 2504 y Fp(+)g Fk(A)p Fp(\()2491 2389 y Fd(Z)53 b(Z)2688 2504 y Fk(h)2740 2467 y Fg(2)2780 2504 y Fp(\))2825 2440 y Fa(1)p 2825 2452 31 4 v 2825 2493 a(2)2869 2504 y Fk(;)356 b Fp(\(4.5\))448 2746 y(where)25 b Fk(A)g Fp(=)g Ff(f)940 2677 y Fd(R)32 b(R)1067 2746 y Fp([)p Fk(\013y)s(h)1250 2760 y Fg(0)1300 2746 y Ff(\000)10 b Fk(b)1420 2760 y Fg(1)1458 2746 y Fk(h)1510 2760 y Fg(0)1560 2746 y Fp(+)g Fk(f)g Fp(\()p Fk(h)1783 2760 y Fg(0)1822 2746 y Fp(\)])1882 2713 y Fg(2)1922 2746 y Ff(g)1977 2683 y Fa(1)p 1977 2695 V 1977 2736 a(2)2022 2746 y Fp(,)26 b(and)f(in)e(the)j(last)f(step)g(ab)s(o)m(v)m(e,)i(w)m (e)f(ha)m(v)m(e)448 2859 y(used)k(the)g(H\177)-45 b(older)30 b(inequalit)m(y)f(in)g Fk(L)1698 2826 y Fg(2)1737 2859 y Fp(\(\012\).)41 b(Therefore)715 3074 y Ff(L)p Fp(\()p Fk(h)p Fp(\))21 b Ff(\000)f Fk(A)p Fp(\()1115 2959 y Fd(Z)53 b(Z)1312 3074 y Fk(h)1364 3036 y Fg(2)1404 3074 y Fp(\))1449 3009 y Fa(1)p 1449 3021 V 1449 3062 a(2)1519 3074 y Fk(<)25 b Fp(\000)1687 2959 y Fd(Z)52 b(Z)1868 3074 y Fp([)p Fk(h)1945 3036 y Fg(2)2005 3074 y Fp(+)20 b(\()p Fk(h)2183 3088 y Fg(0)2244 3074 y Fp(+)f Fk(h)p Fp(\))p Ff(r)p Fk(h)j Ff(\001)e(r)p Fk(h)p Fp(])p Fk(dxdy)s(;)262 b Fp(\(4.6\))448 3299 y(where)41 b(\000)i(=)f(max)p Ff(f)1159 3263 y Fg(1)p 1159 3278 36 4 v 1159 3331 a(2)1220 3299 y Fp(sup)1357 3321 y Fg(\012)1412 3299 y Fp(\()p Ff(4)p Fk(h)1580 3313 y Fg(0)1647 3299 y Fp(+)27 b Fk(\013y)k Ff(\000)c Fk(b)2016 3313 y Fg(1)2055 3299 y Fp(\))p Fk(;)15 b Ff(\000)p Fp(1)p Ff(g)45 b Fk(<)d Fp(0.)73 b(Moreo)m(v)m(er,)46 b(w)m(e)41 b(denote)449 3402 y(^)448 3426 y Fk(h)q Fp(\()p Fk(x;)15 b(y)s Fp(\))26 b(=)f Fk(h)p Fp(\()p Fk(x;)15 b(y)s(;)g Fp(0\),)33 b(and)c(w)m(e)i(get)820 3646 y Ff(L)p Fp(\()919 3622 y(^)918 3646 y Fk(h)q Fp(\))25 b Fk(>)1137 3585 y Fp(1)p 1137 3625 46 4 v 1137 3708 a(2)1208 3531 y Fd(Z)52 b(Z)1389 3646 y Fp(\()p Ff(4)p Fk(h)1557 3660 y Fg(0)1617 3646 y Fp(+)20 b Fk(\013y)k Ff(\000)19 b Fk(a)1973 3660 y Fg(1)2013 3646 y Fp(\))2049 3622 y(^)2048 3646 y Fk(h)2100 3608 y Fg(2)2160 3646 y Ff(\000)h Fp(\()p Fk(h)2338 3660 y Fg(0)2399 3646 y Fp(+)2490 3622 y(^)2490 3646 y Fk(h)p Fp(\))p Ff(r)2654 3622 y Fp(^)2653 3646 y Fk(h)h Ff(\001)f(r)2848 3622 y Fp(^)2847 3646 y Fk(h)1518 3864 y Fp(+)1604 3749 y Fd(Z)52 b(Z)1785 3864 y Fp([)p Fk(\013y)s(h)1968 3878 y Fg(0)2029 3864 y Ff(\000)20 b Fk(a)2168 3878 y Fg(1)2207 3864 y Fk(h)2259 3878 y Fg(0)2319 3864 y Fp(+)g Fk(f)10 b Fp(\()p Fk(h)2552 3878 y Fg(0)2592 3864 y Fp(\)])2653 3840 y(^)2652 3864 y Fk(h)q(dxdy)1036 4081 y(>)1137 4058 y Fp(^)1131 4081 y(\000)1203 3966 y Fd(Z)53 b(Z)1385 4081 y Fp([)1411 4057 y(^)1410 4081 y Fk(h)1462 4044 y Fg(2)1522 4081 y Fp(+)20 b(\()p Fk(h)1700 4095 y Fg(0)1760 4081 y Fp(+)1852 4057 y(^)1851 4081 y Fk(h)p Fp(\))p Ff(r)2015 4057 y Fp(^)2014 4081 y Fk(h)h Ff(\001)g(r)2210 4057 y Fp(^)2209 4081 y Fk(h)f Ff(\000)g Fk(B)5 b Fp(\()2481 3966 y Fd(Z)52 b(Z)2678 4057 y Fp(^)2677 4081 y Fk(h)2729 4044 y Fg(2)2769 4081 y Fp(\))2814 4017 y Fa(1)p 2814 4029 31 4 v 2814 4070 a(2)2859 4081 y Fk(;)366 b Fp(\(4.7\))448 4323 y(where)30 b Fk(B)g Fp(=)25 b Ff(f)951 4254 y Fd(R)32 b(R)1077 4323 y Fp([)p Fk(\013y)s(h)1260 4337 y Fg(0)1320 4323 y Ff(\000)19 b Fk(a)1458 4337 y Fg(1)1497 4323 y Fk(h)1549 4337 y Fg(0)1609 4323 y Fp(+)g Fk(f)10 b Fp(\()p Fk(h)1841 4337 y Fg(0)1880 4323 y Fp(\)])1940 4290 y Fg(2)1980 4323 y Ff(g)2035 4260 y Fa(1)p 2036 4272 V 2036 4313 a(2)2080 4323 y Fp(,)31 b(and)2318 4300 y(^)2312 4323 y(\000)25 b(=)g(min)n Ff(f)2696 4287 y Fg(1)p 2696 4302 36 4 v 2696 4355 a(2)2757 4323 y Fp(inf)2868 4337 y Fg(\012)2923 4323 y Fp(\()p Ff(4)p Fk(h)3091 4337 y Fg(0)3150 4323 y Fp(+)20 b Fk(\013y)i Ff(\000)448 4436 y Fk(a)496 4450 y Fg(1)536 4436 y Fp(\))p Fk(;)15 b Ff(\000)p Fp(1)p Ff(g)26 b Fk(<)f Fp(0.)589 4549 y(Note)50 b(that)1041 4480 y Fd(R)31 b(R)1182 4549 y Fk(h)1234 4516 y Fg(2)1274 4549 y Fk(dxdy)51 b Fp(is)c(also)g(a)i(constan)m(t)g(of)e(motion)h (\(Cushman-Roisin)448 4662 y([1986]\).)c(So)1008 4877 y Ff(L)p Fp(\()p Fk(h)p Fp(\))21 b Ff(\000)f Fk(A)p Fp(\()1408 4762 y Fd(Z)53 b(Z)1605 4877 y Fk(h)1657 4839 y Fg(2)1697 4877 y Fp(\))1742 4812 y Fa(1)p 1742 4824 31 4 v 1742 4865 a(2)1812 4877 y Fp(=)25 b Ff(L)p Fp(\()2007 4853 y(^)2006 4877 y Fk(h)p Fp(\))c Ff(\000)f Fk(A)p Fp(\()2308 4762 y Fd(Z)52 b(Z)2505 4853 y Fp(^)2504 4877 y Fk(h)2556 4839 y Fg(2)2596 4877 y Fp(\))2641 4812 y Fa(1)p 2642 4824 V 2642 4865 a(2)2686 4877 y Fk(:)539 b Fp(\(4.8\))1920 5162 y(7)p eop %%Page: 8 8 8 7 bop 448 623 a Fp(Th)m(us)30 b(from)g(\(4.6\),)i(\(4.7\))g(and)e (\(4.8\),)i(w)m(e)f(get)1616 710 y Fd(Z)52 b(Z)1812 825 y Fk(h)1864 788 y Fg(2)1930 825 y Ff(\024)2026 710 y Fd(Z)g(Z)2222 825 y Fk(h)2274 788 y Fg(2)2334 825 y Fp(+)20 b(\()p Fk(h)2512 839 y Fg(0)2572 825 y Fp(+)g Fk(h)p Fp(\))p Ff(r)p Fk(h)h Ff(\001)g(r)p Fk(h)1997 1043 y(<)k Fp(\000)2150 1005 y Fc(\000)p Fg(1)2244 1043 y Fp([)p Ff(L)p Fp(\()p Fk(h)p Fp(\))c Ff(\000)f Fk(A)p Fp(\()2669 928 y Fd(Z)53 b(Z)2866 1043 y Fk(h)2918 1005 y Fg(2)2958 1043 y Fp(\))3003 978 y Fa(1)p 3003 990 31 4 v 3003 1031 a(2)3047 1043 y Fp(])1997 1260 y(=)25 b(\000)2150 1223 y Fc(\000)p Fg(1)2244 1260 y Fp([)p Ff(L)p Fp(\()2368 1236 y(^)2367 1260 y Fk(h)p Fp(\))c Ff(\000)f Fk(A)p Fp(\()2669 1146 y Fd(Z)53 b(Z)2867 1236 y Fp(^)2866 1260 y Fk(h)2918 1223 y Fg(2)2958 1260 y Fp(\))3003 1196 y Fa(1)p 3003 1208 V 3003 1249 a(2)3047 1260 y Fp(])647 1478 y Fk(<)25 b Fp(\000)800 1441 y Fc(\000)p Fg(1)899 1455 y Fp(^)894 1478 y(\000)966 1363 y Fd(Z)52 b(Z)1147 1478 y Fp([)1173 1454 y(^)1172 1478 y Fk(h)1224 1441 y Fg(2)1284 1478 y Fp(+)20 b(\()p Fk(h)1462 1492 y Fg(0)1522 1478 y Fp(+)1614 1454 y(^)1613 1478 y Fk(h)q Fp(\))p Ff(r)1778 1454 y Fp(^)1777 1478 y Fk(h)g Ff(\001)h(r)1972 1454 y Fp(^)1971 1478 y Fk(h)p Fp(])g Ff(\000)e Fp(\000)2216 1441 y Fc(\000)p Fg(1)2311 1478 y Fp(\()p Fk(A)h Fp(+)g Fk(B)5 b Fp(\)\()2669 1363 y Fd(Z)53 b(Z)2867 1454 y Fp(^)2866 1478 y Fk(h)2918 1441 y Fg(2)2958 1478 y Fp(\))3003 1413 y Fa(1)p 3003 1425 V 3003 1467 a(2)3047 1478 y Fk(:)178 b Fp(\(4.9\))448 1676 y(Note)39 b(that)e(the)g(righ)m(t)f(hand)g(side)g (is)g(nonnegativ)m(e)i(and)e(dep)s(ends)f(only)h(on)h(kno)m(wn)448 1788 y(data.)i(This)20 b(sho)m(ws)h(the)h(nonlinear)d(stabilit)m(y)i (of)g Fk(h)2169 1802 y Fg(0)2209 1788 y Fp(\()p Fk(x;)15 b(y)s Fp(\))23 b(and)e(w)m(e)h(ha)m(v)m(e)h(the)f(follo)m(wing)448 1901 y(result.)448 2079 y Fl(Theorem)34 b(3)46 b Fb(L)-5 b(et)42 b Fk(h)1208 2093 y Fg(0)1248 2079 y Fp(\()p Fk(x;)15 b(y)s Fp(\))43 b Fb(b)-5 b(e)42 b(a)h(ste)-5 b(ady)44 b(fr)-5 b(ont)43 b(for)g(the)g(e)-5 b(quation)43 b(\(1.1\),)i(i.e.,)f (it)448 2192 y(satis\014es)1057 2394 y Fk(h)1109 2408 y Fg(0)1169 2394 y Ff(4)20 b Fk(h)1322 2408 y Fg(0)1382 2394 y Fp(+)1483 2333 y(1)p 1483 2373 46 4 v 1483 2457 a(2)1559 2394 y Ff(5)g Fk(h)1712 2408 y Fg(0)1772 2394 y Ff(\001)g(5)p Fk(h)1950 2408 y Fg(0)2010 2394 y Fp(+)g Fk(\013y)s(h)2259 2408 y Fg(0)2324 2394 y Fp(=)25 b Fk(f)10 b Fp(\()p Fk(h)2562 2408 y Fg(0)2602 2394 y Fp(\))p Fk(;)543 b Fp(\(4.10\))448 2590 y Fb(for)33 b(an)g(arbitr)-5 b(ary)34 b(\(smo)-5 b(oth\))35 b(function)d Fk(f)10 b Fp(\()p Fk(h)p Fp(\))p Fb(.)42 b(If)32 b Fk(f)10 b Fp(\()p Fk(h)p Fp(\))32 b Fb(satis\014es)h(the)g(fol)5 b(lowing)33 b(c)-5 b(on-)448 2702 y(dition)1577 2872 y Fk(a)1625 2886 y Fg(1)1665 2872 y Fk(\030)29 b(<)c(f)10 b Fp(\()p Fk(\030)t Fp(\))25 b Fk(<)g(b)2159 2886 y Fg(1)2199 2872 y Fk(\030)t(;)937 b Fp(\(4.11\))1436 3010 y(sup)1475 3087 y Fc(R)1573 3010 y Fp(\()p Ff(4)p Fk(h)1741 3024 y Fg(0)1801 3010 y Fp(+)20 b Fk(\013y)s Fp(\))26 b Fk(<)f(a)2203 3024 y Fg(1)2243 3010 y Fk(;)937 b Fp(\(4.12\))448 3228 y Fb(for)43 b(some)g(\014nite)f (r)-5 b(e)g(al)43 b(numb)-5 b(ers)43 b Fk(a)1690 3242 y Fg(1)1729 3228 y Fk(;)15 b(b)1808 3242 y Fg(1)1848 3228 y Fb(,)44 b(then,)h Fk(h)2214 3242 y Fg(0)2254 3228 y Fp(\()p Fk(x;)15 b(y)s Fp(\))42 b Fb(is)g(nonline)-5 b(arly)45 b(stable)d(in)448 3341 y Fk(L)510 3308 y Fg(2)550 3341 y Fp(\(\012\))33 b Fb(\(in)f(the)h(usual)g(norm\).)589 3518 y Fp(W)-8 b(e)40 b(remark)d(that,)k(if)c(the)i Fk(\014)5 b Fp(-e\013ect)40 b(is)d(not)h(zero,)j(i.e.,)f Fk(\013)f Fp(=)2786 3478 y Fe(\014)p 2786 3497 44 4 v 2793 3550 a(\017)2877 3518 y Ff(6)p Fp(=)e(0,)k(then)d(the)448 3631 y(nonlinear)29 b(stabilit)m(y)g(condition)h(directly)f(in)m(v)m (olv)m(es)i(the)g(meridional)d(stretc)m(h)j(or)g(size)448 3744 y(of)25 b(the)f(fron)m(tal)g(region;)j(see)d(ab)s(o)m(v)m(e)i (condition)d(\(4.12\))j(in)d(the)i(statemen)m(t)h(of)e(Theorem)448 3857 y(3.)40 b(When)27 b Fk(\014)j Fp(=)25 b(0,)j(the)g(stabilit)m(y)d (condition)g(in)m(v)m(olv)m(es)i Fk(h)2384 3871 y Fg(0)2451 3857 y Fp(itself.)38 b(This)25 b(also)i(indicates)448 3970 y(that)k(the)g(set)g(of)f(stable)g(fron)m(ts)h(is)e(generally)h (smaller)f(when)g Fk(\014)i Ff(6)p Fp(==)24 b(0.)589 4083 y(No)m(w)36 b(w)m(e)f(consider)e(the)i(equation)f(\(1.3\).)55 b(W)-8 b(e)36 b(follo)m(w)e(the)h(sc)m(heme)g(in)e(Holm)h(et)448 4196 y(al.)41 b([1985].)i(F)-8 b(rom)31 b(\(2.12\))h(and)e(\(3.9\),)i (w)m(e)f(calculate)554 4403 y Fk(H)7 b Fp(\()p Fk(h)724 4417 y Fg(0)784 4403 y Fp(+)20 b Fk(h)p Fp(\))h Ff(\000)f Fk(H)7 b Fp(\()p Fk(h)1244 4417 y Fg(0)1284 4403 y Fp(\))21 b Ff(\000)f Fk(D)s(H)7 b Fp(\()p Fk(h)1679 4417 y Fg(0)1719 4403 y Fp(\))20 b Ff(\001)h Fk(h)k Fp(=)2003 4342 y(1)p 2003 4382 46 4 v 2003 4466 a(2)2074 4288 y Fd(Z)52 b(Z)2255 4403 y Fp([)p Fk(y)s Fp(\(2)p Fk(h)2460 4417 y Fg(0)2501 4403 y Fk(h)20 b Fp(+)g Fk(h)2716 4366 y Fg(2)2756 4403 y Fp(\))h Ff(\000)f Fp(2)p Fk(y)s(h)3048 4417 y Fg(0)3088 4403 y Fk(h)p Fp(])2257 4626 y(=)2363 4565 y(1)p 2363 4605 V 2363 4688 a(2)2433 4511 y Fd(Z)52 b(Z)2630 4626 y Fk(y)s(h)2730 4588 y Fg(2)2794 4626 y Ff(\021)25 b Fk(Q)2962 4640 y Fg(1)3002 4626 y Fp(\()p Fk(h)p Fp(\))p Fk(;)56 b Fp(\(4.13\))627 4864 y Fk(C)7 b Fp(\()p Fk(h)786 4878 y Fg(0)846 4864 y Fp(+)20 b Fk(h)p Fp(\))g Ff(\000)g Fk(C)7 b Fp(\()p Fk(h)1294 4878 y Fg(0)1334 4864 y Fp(\))21 b Ff(\000)e Fk(D)s(C)7 b Fp(\()p Fk(h)1717 4878 y Fg(0)1757 4864 y Fp(\))20 b Ff(\001)h Fk(h)26 b Fp(=)2032 4749 y Fd(Z)52 b(Z)2213 4864 y Fp([)2238 4749 y Fd(Z)2321 4775 y Fe(h)2362 4784 y Fa(0)2284 4937 y Fe(h)2325 4946 y Fa(0)2359 4937 y Fg(+)p Fe(h)2474 4864 y Fk(f)10 b Fp(\()p Fk(\030)t Fp(\))p Fk(d\030)25 b Fp(+)20 b Fk(f)10 b Fp(\()p Fk(h)2988 4878 y Fg(0)3027 4864 y Fp(\))p Fk(h)p Fp(])p Fk(:)41 b Fp(\(4.14\))1920 5162 y(8)p eop %%Page: 9 9 9 8 bop 589 623 a Fp(W)-8 b(e)32 b(assume)e(that)1426 819 y Fk(a)1474 833 y Fg(2)1539 819 y Fk(<)25 b(f)1690 782 y Fc(0)1712 819 y Fp(\()p Fk(\030)t Fp(\))h Fk(<)f(b)1987 833 y Fg(2)2027 819 y Fk(;)40 b(on)25 b Ff(R)912 b Fp(\(4.15\))448 1016 y(for)30 b(some)h(real)f(\014nite)g(n)m(um)m(b)s(ers)f Fk(a)1640 1030 y Fg(2)1709 1016 y Fp(and)h Fk(b)1925 1030 y Fg(2)1965 1016 y Fp(.)40 b(Then)30 b(from)f(\(4.14\))k(w)m(e)e (ha)m(v)m(e)708 1251 y Fk(C)7 b Fp(\()p Fk(h)867 1265 y Fg(0)927 1251 y Fp(+)20 b Fk(h)p Fp(\))h Ff(\000)f Fk(C)7 b Fp(\()p Fk(h)1376 1265 y Fg(0)1415 1251 y Fp(\))21 b Ff(\000)f Fk(D)s(C)7 b Fp(\()p Fk(h)1799 1265 y Fg(0)1838 1251 y Fp(\))21 b Ff(\001)f Fk(h)26 b(>)f Ff(\000)2194 1189 y Fp(1)p 2194 1230 46 4 v 2194 1313 a(2)2249 1251 y Fk(b)2288 1265 y Fg(2)2342 1136 y Fd(Z)53 b(Z)2539 1251 y Fk(h)2591 1213 y Fg(2)2656 1251 y Ff(\021)2773 1228 y Fp(^)2752 1251 y Fk(Q)2824 1265 y Fg(2)2863 1251 y Fp(\()p Fk(h)p Fp(\);)195 b(\(4.16\))448 1447 y(or)703 1659 y Fk(C)7 b Fp(\()p Fk(h)862 1673 y Fg(0)922 1659 y Fp(+)20 b Fk(h)p Fp(\))h Ff(\000)f Fk(C)7 b Fp(\()p Fk(h)1371 1673 y Fg(0)1411 1659 y Fp(\))20 b Ff(\000)g Fk(D)s(C)7 b Fp(\()p Fk(h)1794 1673 y Fg(0)1834 1659 y Fp(\))20 b Ff(\001)h Fk(h)k(<)g Ff(\000)2189 1598 y Fp(1)p 2189 1638 V 2189 1722 a(2)2244 1659 y Fk(a)2292 1673 y Fg(2)2347 1545 y Fd(Z)52 b(Z)2543 1659 y Fk(h)2595 1622 y Fg(2)2660 1659 y Ff(\021)2777 1637 y Fp(\024)2756 1659 y Fk(Q)2828 1673 y Fg(2)2868 1659 y Fp(\()p Fk(h)p Fp(\))p Fk(:)190 b Fp(\(4.17\))448 1889 y(F)-8 b(rom)39 b(Holm)e(et)i(al.)63 b([1985],)42 b(the)c(nonlinear)e(stabilit)m(y)g (of)i Fk(h)2612 1903 y Fg(0)2652 1889 y Fp(\()p Fk(x;)15 b(y)s Fp(\))39 b(is)e(ensured)f(b)m(y)448 2002 y(either)1123 2209 y Fk(Q)1195 2223 y Fg(1)1234 2209 y Fp(\()p Fk(h)p Fp(\))21 b(+)1489 2186 y(^)1468 2209 y Fk(Q)1540 2223 y Fg(2)1579 2209 y Fp(\()p Fk(h)p Fp(\))27 b(=)1824 2094 y Fd(Z)52 b(Z)2005 2209 y Fp(\()p Fk(y)23 b Ff(\000)d Fk(b)2238 2223 y Fg(2)2278 2209 y Fp(\))p Fk(h)2365 2172 y Fg(2)2430 2209 y Fk(>)25 b Fp(0)p Fk(;)609 b Fp(\(4.18\))448 2431 y(i.e.,)1625 2628 y(inf)1650 2687 y Fc(R)1736 2628 y Fp(\()p Fk(y)s Fp(\))25 b Fk(>)g(b)2014 2642 y Fg(2)2054 2628 y Fk(;)1126 b Fp(\(4.19\))448 2825 y(or)1118 3032 y Fk(Q)1190 3046 y Fg(1)1229 3032 y Fp(\()p Fk(h)p Fp(\))22 b(+)1484 3009 y(\024)1464 3032 y Fk(Q)1536 3046 y Fg(2)1575 3032 y Fp(\()p Fk(h)p Fp(\))k(=)1819 2917 y Fd(Z)52 b(Z)2000 3032 y Fp(\()p Fk(y)24 b Ff(\000)c Fk(a)2243 3046 y Fg(2)2282 3032 y Fp(\))p Fk(h)2369 2994 y Fg(2)2435 3032 y Fk(<)25 b Fp(0)p Fk(;)604 b Fp(\(4.20\))448 3254 y(i.e.,)1615 3451 y(sup)1653 3527 y Fc(R)1752 3451 y Fp(\()p Fk(y)s Fp(\))26 b Fk(<)f(a)2040 3465 y Fg(2)2079 3451 y Fk(:)1101 b Fp(\(4.21\))589 3691 y(Th)m(us)30 b(w)m(e)h(ha)m(v)m(e)g(pro)m(v)m (ed)g(the)f(follo)m(wing)f(conclusion.)448 3896 y Fl(Theorem)34 b(4)46 b Fb(L)-5 b(et)33 b Fk(h)1199 3910 y Fg(0)1239 3896 y Fp(\()p Fk(x;)15 b(y)s Fp(\))33 b Fb(b)-5 b(e)33 b(a)g(ste)-5 b(ady)34 b(solution)g(of)e(the)h(e)-5 b(quation)34 b(\(1.3\),)f(i.e.,)1621 4092 y Fk(y)s(h)1721 4106 y Fg(0)1786 4092 y Fp(=)25 b Fk(f)10 b Fp(\()p Fk(h)2024 4106 y Fg(0)2063 4092 y Fp(\))1107 b(\(4.22\))448 4289 y Fb(for)34 b(an)f(arbitr)-5 b(ary)34 b(\(smo)-5 b(oth\))36 b(function)d Fk(f)10 b Fp(\()p Fk(h)p Fp(\))p Fb(.)42 b(If)32 b Fk(f)10 b Fp(\()p Fk(h)p Fp(\))33 b Fb(satis\014es)1364 4486 y Fk(a)1412 4500 y Fg(2)1477 4486 y Fk(<)25 b(f)1628 4448 y Fc(0)1650 4486 y Fp(\()p Fk(\030)t Fp(\))h Fk(<)f(b)1925 4500 y Fg(2)1965 4486 y Fk(;)40 b(on)25 b(on)g Ff(R)850 b Fp(\(4.23\))448 4682 y Fb(for)34 b(some)f(r)-5 b(e)g(al)34 b(\014nite)f(numb)-5 b(ers)33 b Fk(a)1643 4696 y Fg(2)1715 4682 y Fb(and)g Fk(b)1930 4696 y Fg(2)1970 4682 y Fb(,)f(and)i(if)e(either)1633 4879 y Fp(inf)1658 4938 y Fc(R)1743 4879 y Fp(\()p Fk(y)s Fp(\))26 b Fk(>)f(b)2022 4893 y Fg(2)2061 4879 y Fk(;)1119 b Fp(\(4.24\))1920 5162 y(9)p eop %%Page: 10 10 10 9 bop 448 623 a Fb(or)1628 817 y Fp(sup)1666 894 y Fc(R)1765 817 y Fp(\()p Fk(y)s Fp(\))25 b Fk(<)g(a)2052 831 y Fg(2)3205 817 y Fp(\(4.25\))448 1105 y Fb(then,)34 b Fk(h)731 1119 y Fg(0)771 1105 y Fp(\()p Fk(x;)15 b(y)s Fp(\))34 b Fb(is)g(nonline)-5 b(arly)35 b(stable)f(under)g(the)g(norm)h Ff(jj)p Fk(h)p Ff(jj)28 b Fp(=)2774 998 y Fd(q)p 2857 998 580 4 v 107 x Fk(Q)2929 1119 y Fg(1)2969 1105 y Fp(\()p Fk(h)p Fp(\))21 b(+)3224 1082 y(^)3203 1105 y Fk(Q)3275 1119 y Fg(2)3314 1105 y Fp(\()p Fk(h)p Fp(\))448 1263 y Fb(or)34 b Ff(jj)p Fk(h)p Ff(jj)26 b Fp(=)840 1159 y Fd(q)p 923 1159 651 4 v 104 x Ff(\000)p Fk(Q)1066 1277 y Fg(1)1105 1263 y Fp(\()p Fk(h)p Fp(\))c Ff(\000)1360 1240 y Fp(\024)1339 1263 y Fk(Q)1411 1277 y Fg(2)1451 1263 y Fp(\()p Fk(h)p Fp(\))q Fb(,)32 b(r)-5 b(esp)g(e)g(ctively.)589 1465 y Fp(W)d(e)32 b(remark)f(that,)h(when)d(the)i Fk(\014)5 b Fp(-)32 b(e\013ect)g(is)e(large)h(comparing)f(with)f(the)i(Rossb)m(y) 448 1578 y(n)m(um)m(b)s(er)g Fk(\017)p Fp(,)i(the)g(nonlinear)d (stabilit)m(y)h(condition)g(also)h(dep)s(ends)f(on)h(the)h(meridional) 448 1691 y(stretc)m(h)f(or)e(size)g(of)h(the)f(fron)m(tal)g(region)g (\(Theorem)h(4\).)448 1976 y Fh(5)135 b(Discussions)448 2179 y Fp(In)39 b(this)g(pap)s(er,)i(w)m(e)f(ha)m(v)m(e)h(studied)c (the)j(formal)f(and)g(nonlinear)e(stabilit)m(y)i(of)g(one-)448 2292 y(la)m(y)m(er)e(geostrophic)f(fron)m(ts.)58 b(W)-8 b(e)38 b(ha)m(v)m(e)f(seen)f(that)h(when)e(the)h Fk(\014)5 b Fp(-)37 b(e\013ect)h(is)d(presen)m(t,)448 2405 y(the)30 b(class)f(of)h(formally)e(and)h(linearly)e(stable)i(fron)m(ts)g(is)g (more)h(restricted)f(than)g(when)448 2518 y(the)j Fk(\014)5 b Fp(-)31 b(e\013ect)i(is)d(absen)m(t.)44 b(When)31 b(the)g Fk(\014)5 b Fp(-)32 b(e\013ect)g(is)f(presn)m(t,)g(the)g(nonlinear)e (stabilit)m(y)448 2631 y(condition)g(in)m(v)m(olv)m(es)i(with)e(the)h (meridional)e(stretc)m(h)j(or)g(size)f(of)g(the)h(fron)m(tal)f(region.) 589 2744 y(W)-8 b(e)25 b(should)c(remark)i(that)h(the)f(mo)s(dels)f (used)g(here)h(do)g(not)g(include)e(strati\014cation)448 2857 y(or)38 b(baro)s(clinicit)m(y)-8 b(.)61 b(The)38 b(result)f(ma)m(y)h(not)g(directly)f(b)s(e)g(applied)f(to)i(real)g (geoph)m(ysi-)448 2969 y(cal)d(fron)m(ts)f(\(suc)m(h)g(as)g(Gulf)g (Stream)g(rings\))f(without)g(the)i(analysis)d(of)j(the)f(e\013ect)i (of)448 3082 y(baro)s(clinicit)m(y)-8 b(.)589 3195 y(The)44 b(fron)m(ts)f(studied)f(here)h(are)h(densit)m(y)f(fron)m(ts.)81 b(Constan)m(tin)42 b(et)j(al.)80 b([1994])448 3308 y(recen)m(tly)31 b(studied)e(temp)s(erature)h(fron)m(ts.)448 3593 y Fh(References)539 3796 y Fp([1])47 b(E.)42 b(Benilo)m(v,)k(A)d(note)g(on)f(the)h (stabilit)m(y)e(of)i(one-la)m(y)m(er)h(geostrophic)f(fron)m(ts,)681 3909 y Fb(Ge)-5 b(ophys.)34 b(Astr)-5 b(ophys.)35 b(Fluid)e(Dynamics)e Fl(66)g Fp(\(1992\),)i(81-86.)539 4093 y([2])47 b(J.)35 b(G.)h(Charney)e(and)h(G.)g(R.)h(Flierl,)f(Oceanic)g(analogues)g(of)h (large-scale)g(at-)681 4206 y(mospheric)25 b(motions,)i(In)e Fb(Evolution)30 b(of)f(Physic)-5 b(al)30 b(Oc)-5 b(e)g(ano)g(gr)g(aphy) 30 b Fp(\(eds.)d(B.)g(A.)681 4319 y(W)-8 b(arren)30 b(and)g(C.)g(W)-8 b(unsc)m(h\),)31 b(514-548,)j(Massac)m(h)m(usetts:)43 b(MIT)30 b(Press,)g(1981.)539 4503 y([3])47 b(P)-8 b(.)35 b(Constan)m(tin,)h(A.)g(Ma)5 b(jda)35 b(and)g(E.)g(G.)h(T)-8 b(abak,)37 b(Singular)32 b(fron)m(t)j(formation)681 4616 y(in)29 b(a)h(mo)s(del)g(for)g(quasigeostrophic)f(\015o)m(w,)i Fb(Phys.)i(Fluids)e Fl(6)g Fp(\(1994\),)i(9-11.)539 4800 y([4])47 b(B.)28 b(Cushman-Roisin,)e(F)-8 b(ron)m(tal)29 b(geostrophic)g(dynamics,)e Fb(J.)j(Phys.)h(Oc)-5 b(e)g(ano)g(g-)681 4913 y(r)g(aphy)32 b Fl(16)f Fp(\(1986\),)i(132-143.)1897 5162 y(10)p eop %%Page: 11 11 11 10 bop 539 623 a Fp([5])47 b(B.)36 b(Cushman-Roisin)c(and)j(B.)h(Y) -8 b(ang,)38 b(Geostrophic)e(turbulence)e(and)h(emer-)681 736 y(gence)g(of)e(eddies)g(b)s(ey)m(ond)g(the)h(radius)e(of)i (deformation,)g Fb(J.)h(Phys.)h(Oc)-5 b(e)g(ano)g(g-)681 848 y(r)g(aphy)32 b Fl(20)f Fp(\(1990\),)i(97-113.)539 1036 y([6])47 b(D.)40 b(D.)h(Holm,)h(J.)e(E.)g(Marsden,)i(T.)e(Ratiu)f (and)g(A.)i(W)-8 b(einstein,)41 b(Nonlinear)681 1149 y(stabilit)m(y)27 b(of)j(\015uid)d(and)h(plasma)g(equilibria,)e Fb(Phys.)32 b(R)-5 b(ept.)30 b Fl(123)g Fp(\(1985\),1-116.)539 1337 y([7])47 b(H.)42 b(Hukuda)g(and)g(T.)g(Y)-8 b(amagata,)49 b(A)42 b(uni\014ed)e(geostrophic)j(equation)f(with)681 1450 y(application)28 b(to)j(a)g(cold)f(core)h(ring,)f Fb(T)-7 b(el)5 b(lus)30 b Fl(40A)h Fp(\(1988\),)i(407-418.)539 1637 y([8])47 b(P)-8 b(.)49 b(J.)g(Olv)m(er,)k Fb(Applic)-5 b(ations)51 b(of)f(Lie)g(Gr)-5 b(oups)51 b(to)f(Di\013er)-5 b(ential)50 b(Equations)p Fp(,)681 1750 y(Springer-V)-8 b(erlag,)29 b(New)i(Y)-8 b(ork,)31 b(1986.)539 1938 y([9])47 b(J.)40 b(P)m(edlosky)-8 b(,)43 b Fb(Ge)-5 b(ophysic)g(al)45 b(Fluid)e(Dynamics)p Fp(,)h(Springer-V)-8 b(erlag,)42 b(2nd)e(edi-)681 2051 y(tion,)30 b(1987.)494 2238 y([10])47 b(T.)21 b(W.)i(Sp)s(ence)e(and)g(R.)i(Legec)m(kis,)h(Satellite)e(and)f (h)m(ydrographic)f(observ)-5 b(ations)681 2351 y(of)22 b(lo)m(w-frequency)h(w)m(a)m(v)m(e)h(motions)e(asso)s(ciated)i(with)d (a)i(cold)f(core)i(Gulf)d(Stream)681 2464 y(ring,)29 b Fb(J.)j(Ge)-5 b(ophys.)35 b(R)-5 b(es.)31 b Fl(86)f Fp(\(1981\),)k(No.)d(C3,)f(1945-1953.)494 2652 y([11])47 b(G.)34 b(E.)h(Sw)m(aters,)g(On)f(the)g(baro)s(clinic)e(dynamics,)i (hamiltonian)e(form)m(ulation)681 2765 y(and)d(general)g(stabilit)m(y)f (c)m(haracteristics)i(of)g(densit)m(y-driv)m(en)e(surface)h(curren)m (ts)681 2878 y(and)f(fron)m(ts)h(o)m(v)m(er)h(a)f(sloping)e(con)m (tinen)m(tal)j(shelf,)e Fb(Phil.)j(T)-7 b(r)i(ans.)33 b(R.)e(So)-5 b(c.)32 b(L)-5 b(ond.)681 2990 y(A)30 b Fl(345)p Fp(\(1993\),)j(295-325.)494 3178 y([12])47 b(G.)40 b(P)-8 b(.)41 b(Williams)c(and)j(T.)g(Y)-8 b(amagata,)46 b(Geostrophic)40 b(eddies,)h(in)m(termediate)681 3291 y(solitary)28 b(v)m(ortices)i(and)e(Jo)m(vian)h(eddies,)g Fb(J.)i(A)n(tmos.)g(Sci.)e Fl(41)p Fp(\(1984\),)k(453-478.)494 3479 y([13])47 b(T.)h(Y)-8 b(amagata,)57 b(On)47 b(nonlinear)g (planetary)h(w)m(a)m(v)m(es:)79 b(a)49 b(class)g(of)g(solutions)681 3592 y(missed)23 b(b)m(y)h(the)h(quasi-geostrophic)f(appro)m(ximation,) 50 b Fb(J.)27 b(Oc)-5 b(e)g(ano)g(gr)g(aphy)30 b(So)-5 b(c.)681 3704 y(Jap)g(an)32 b Fl(38)e Fp(\(1982\),)j(236-244.)1897 5162 y(11)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF