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Warning, /pkg/atm_phys/old_doc/vert_diff.tech.ps is written in an unsupported language. File is not indexed.

view on githubraw file Latest commit 672ec4d5 on 2016-12-01 17:05:48 UTC
672ec4d5cf Jean*0001 %!PS-Adobe-2.0
                0002 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software
                0003 %%Title: vert_diff.dvi
                0004 %%Pages: 11
                0005 %%PageOrder: Ascend
                0006 %%BoundingBox: 0 0 596 842
                0007 %%EndComments
                0008 %DVIPSWebPage: (www.radicaleye.com)
                0009 %DVIPSCommandLine: dvips -o vert_diff.ps vert_diff.dvi
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                0013 %!
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                0016 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0
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                0980 %%EndProlog
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                0985 
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                0987 %%Page: 0 1
                0988 0 0 bop 936 2224 a Fk(The)52 b(v)l(ertical)i(di\013usion)g(mo)t(dule)
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                0990 %%Page: 1 2
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                1013 Fg(\030)k Fh(with)c(source)h(p)s(er)f(unit)f(mass)h Fg(S)6
                1014 b Fh(.)53 b(Only)328 2295 y(v)m(ertical)30 b(di\013usion)g(is)h
                1015 (considered)h(here.)44 b(W)-8 b(e)32 b(can)f(write)g(this)g(equation)g
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                1017 2523 155 4 v 1351 2614 a(@)g(t)1512 2546 y Fh(=)28 b
                1018 Ff(\000r)22 b(\001)g Fh(\()p Fg(\032v)t(\030)5 b Fh(\))22
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                1020 2217 2614 a(@)g(z)2370 2546 y Fh(+)22 b Fg(\032S)1612
                1021 2847 y(F)42 b Ff(\021)28 b(\000D)s Fh(\()p Fg(z)t Fh(\))p
                1022 Fg(\032)2165 2779 y(@)5 b(\030)p 2164 2823 107 4 v 2164
                1023 2915 a(@)g(z)328 3041 y Fh(or)32 b(adv)m(ectiv)m(e)i(form)1398
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                1025 3186 y Fh(=)28 b Ff(\000)p Fg(v)e Ff(\001)c(r)p Fg(\030)k
                1026 Ff(\000)2107 3119 y Fh(1)p 2106 3163 51 4 v 2106 3254
                1027 a Fg(\032)2176 3119 y(@)5 b(F)p 2176 3163 134 4 v 2189
                1028 3254 a(@)g(z)2342 3186 y Fh(+)22 b Fg(S)941 b Fh(\(1\))328
                1029 3400 y(Here)42 b Ff(D)i Fh(is)d(the)h(kinematic)e(di\013usivit)m(y)h
                1030 (and)h Fg(\032)f Fh(is)h(the)f(densit)m(y)i(of)e(the)h(air.)69
                1031 b Fg(\030)46 b Fh(is)41 b(a)328 3520 y(mixing)j(ratio:)69
                1032 b(the)46 b(amoun)m(t)g(of)f(a)h(substance)i(p)s(er)e(unit)g(mass)g(of)f
                1033 (air.)83 b Fg(F)60 b Fh(is)45 b(the)328 3641 y(up)m(w)m(ard)36
                1034 b(di\013usiv)m(e)e(\015ux)i(of)e(tracer,)h(with)f(v)-5
                1035 b(alue)34 b Fg(F)2221 3656 y Fe(s)2293 3641 y Fh(at)g(the)h(surface.)50
                1036 b(In)35 b(some)f(cases,)328 3761 y(i.e.)42 b(temp)s(erature,)31
                1037 b(there)g(can)f(also)g(b)s(e)g(a)g(distinction)f(b)s(et)m(w)m(een)j
                1038 (the)f(quan)m(tit)m(y)g(whose)328 3881 y(gradien)m(t)d(determines)g
                1039 (the)h(di\013usiv)m(e)f(\015ux)h(and)g(the)f(quan)m(tit)m(y)h(itself,)f
                1040 (a)g(p)s(oin)m(t)g(that)g(w)m(e)328 4002 y(return)33
                1041 b(to)f(b)s(elo)m(w.)474 4122 y(The)j(di\013usion)e(is)h(computed)g(in)f
                1042 (adv)m(ectiv)m(e)j(form.)47 b(This)34 b(need)h(presen)m(t)h(no)e
                1043 (di\016-)328 4242 y(cult)m(y)e(with)f(conserv)-5 b(ation)31
                1044 b(of)g(tracer)h(as)g(long)e(as)i(one)g(is)f(careful)g(that)g(the)h(adv)
                1045 m(ection)328 4363 y(is)43 b(treated)i(consisten)m(tly)g(with)e(the)i
                1046 (treatmen)m(t)f(of)f(conserv)-5 b(ation)44 b(of)g(mass.)78
                1047 b(Note,)328 4483 y(ho)m(w)m(ev)m(er,)36 b(that)d(if)f(one)h(sets)h
                1048 Fg(v)f Fh(=)28 b(0)33 b(in)f(the)i(adv)m(ectiv)m(e)g(form)e(of)h(the)h
                1049 (tracer)f(equation,)328 4604 y(while)40 b(still)e(allo)m(wing)g(the)j
                1050 (\015o)m(w)g(to)f(c)m(hange)h(the)g(mass)g(distribution,)f
                1051 Fg(\032)p Fh(,)j(then)e(total)328 4724 y(tracer)k(substance)h(will)c
                1052 (no)j(longer)f(b)s(e)g(conserv)m(ed,)50 b(whereas)c(conserv)-5
                1053 b(ation)45 b(w)m(ould)328 4844 y(still)28 b(hold)h(if)g(one)h(set)h
                1054 Fg(v)h Fh(=)27 b(0)j(in)f(the)i(\015ux)g(form)e(of)g(the)i(equation.)42
                1055 b(So)30 b(if)f(one)i(w)m(an)m(ts)g(to)328 4965 y(di\013use)i(a)f
                1056 (tracer)h(but)g(not)f(adv)m(ect)i(it,)e(this)g(m)m(ust)h(b)s(e)f(done)h
                1057 (with)g(care.)1922 5214 y(1)p eop
                1058 %%Page: 2 3
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                1064 1091 y Fg(h)28 b Fh(=)f Fg(c)557 1106 y Fe(p)597 1091
                1065 y Fg(T)14 b Fh(,)1543 1262 y Fg(c)1585 1277 y Fe(p)1635
                1066 1195 y Fg(D)s(T)p 1635 1239 155 4 v 1653 1331 a(D)s(t)1827
                1067 1262 y Fh(=)1941 1195 y(1)p 1940 1239 51 4 v 1940 1331
                1068 a Fg(\032)2011 1195 y(D)s(p)p 2011 1239 133 4 v 2018
                1069 1331 a(D)s(t)2175 1262 y Fh(+)22 b Fg(Q)1091 b Fh(\(2\))328
                1070 1481 y(where)50 b Fg(D)s(=D)s(t)e Fh(is)h(the)g(material)d(deriv)-5
                1071 b(ativ)m(e,)52 b Fg(c)2162 1496 y Fe(p)2251 1481 y Fh(the)d(heat)g
                1072 (capacit)m(y)g(at)g(constan)m(t)328 1601 y(pressure)26
                1073 b(p)s(er)e(unit)f(mass,)j(and)e Fg(Q)g Fh(is)g(the)g(heating)f(rate)h
                1074 (p)s(er)g(unit)g(mass.)40 b(Equiv)-5 b(alen)m(tly)d(,)1535
                1075 1782 y Fg(D)s(T)p 1535 1826 155 4 v 1552 1918 a(D)s(t)1727
                1076 1849 y Fh(=)1840 1782 y Fg(\024T)14 b(!)p 1840 1826 192
                1077 4 v 1911 1918 a(p)2063 1849 y Fh(+)22 b Fg(Q=c)2329 1864
                1078 y Fe(p)3441 1849 y Fh(\(3\))328 2105 y(where)34 b Fg(\024)28
                1079 b Fh(=)f Fg(R)q(=c)963 2120 y Fe(p)1003 2105 y Fh(,)32
                1080 b Fg(!)f Fh(=)d Fg(D)s(p=D)s(t)p Fh(,)k(and)g Fg(R)i
                1081 Fh(is)e(the)h(gas)g(constan)m(t.)474 2225 y(W)-8 b(e)29
                1082 b(add)f(to)g(the)g(RHS)g(of)g(\(26\))f(the)i(e\013ect)g(of)f(mixing)e
                1083 (the)i(the)h("dry)f(static)g(energy)328 2346 y(temp)s(erature")k
                1084 Fg(\033)1696 2484 y(\033)f Ff(\021)d Fg(T)36 b Fh(+)2088
                1085 2416 y Fg(g)t(z)p 2088 2460 101 4 v 2097 2552 a(c)2139
                1086 2567 y Fe(p)3441 2484 y Fh(\(4\))328 2710 y(where)g Fg(g)t(=c)754
                1087 2725 y Fe(p)828 2710 y Fh(is)f(the)g(dry-adiabatic)f(lapse)h(rate.)51
                1088 b(Th)m(us,)37 b(the)f(temp)s(erature)f(equation)328 2831
                1089 y(is)d(mo)s(di\014ed)f(to)i(read)1233 2993 y Fg(D)s(T)p
                1090 1233 3037 155 4 v 1251 3129 a(D)s(t)1425 3061 y Fh(=)1539
                1091 2993 y Fg(\024T)14 b(!)p 1539 3037 192 4 v 1610 3129
                1092 a(p)1762 3061 y Fh(+)1872 2993 y Fg(Q)p 1870 3037 82
                1093 4 v 1870 3129 a(c)1912 3144 y Fe(p)1984 3061 y Fh(+)2093
                1094 2993 y(1)p 2092 3037 51 4 v 2092 3129 a Fg(\032)2187
                1095 2993 y(@)p 2162 3037 107 4 v 2162 3129 a(@)5 b(z)2279
                1096 3061 y Ff(D)s Fh(\()p Fg(z)t Fh(\))p Fg(\032)2544 2993
                1097 y(@)g(\033)p 2544 3037 116 4 v 2549 3129 a(@)g(z)3441
                1098 3061 y Fh(\(5\))474 3319 y(P)m(oten)m(tial)38 b(temp)s(erature)h(is)f
                1099 (conserv)m(ed)j(if)d(the)h(\015o)m(w)h(is)e(adiabatic,)g(and)h(one)g
                1100 (ex-)328 3440 y(p)s(ects)30 b(the)g(fast)f(turbulence)h(b)s(eing)e(mo)s
                1101 (deled)g(here)i(to)f(mix)f(this)h(conserv)m(ed)i(quan)m(tit)m(y)-8
                1102 b(.)328 3560 y(Ho)m(w)m(ev)m(er,)32 b(one)e(w)m(ould)g(also)e(lik)m(e)h
                1103 (to)h(conserv)m(e)h(energy)-8 b(.)44 b(F)-8 b(or)28 b(a)i(h)m
                1104 (ydrostatic)g(ideal)e(gas,)328 3680 y(the)38 b(in)m(ternal)f(energy)i
                1105 Fg(c)1226 3695 y Fe(v)1267 3680 y Fg(T)51 b Fh(plus)38
                1106 b(the)g(p)s(oten)m(tial)f(energy)i Fg(g)t(z)t Fh(,)g(when)g(in)m
                1107 (tegrated)f(o)m(v)m(er)328 3801 y(the)h(en)m(tire)g(atmospheric)f
                1108 (column,)i(equals)f Fg(c)2057 3816 y Fe(p)2096 3801 y
                1109 Fg(T)53 b Fh(in)m(tegrated)39 b(o)m(v)m(er)g(the)h(column.)61
                1110 b(So)328 3921 y(conserv)-5 b(ation)33 b(of)h(in)m(ternal)e(plus)i(p)s
                1111 (oten)m(tial)e(energy)i(is)f(guaran)m(teed)i(if)d(one)i(conserv)m(es)
                1112 328 4041 y(the)f(mean)f(of)867 3970 y Fd(R)939 4041 y
                1113 Fg(\032T)14 b(dz)474 4162 y Fh(If)33 b(one)g(simply)e(di\013uses)i(p)s
                1114 (oten)m(tial)e(temp)s(erature,)h(\002,)1575 4344 y Fg(@)5
                1115 b Fh(\002)p 1575 4388 133 4 v 1595 4479 a Fg(@)g(t)1746
                1116 4411 y Fh(=)1860 4344 y(1)p 1859 4388 51 4 v 1859 4479
                1117 a Fg(\032)1954 4344 y(@)p 1929 4388 107 4 v 1929 4479
                1118 a(@)g(z)2046 4411 y(\032)p Ff(D)2186 4344 y Fg(@)g Fh(\002)p
                1119 2186 4388 133 4 v 2199 4479 a Fg(@)g(z)3441 4411 y Fh(\(6\))328
                1120 4661 y(one)33 b(will)d(not)i(conserv)m(e)j(energy)-8
                1121 b(.)45 b(One)33 b(alternativ)m(e)e(is)h(to)h(write)1578
                1122 4843 y Fg(@)5 b(T)p 1578 4887 128 4 v 1595 4979 a(@)g(t)1743
                1123 4910 y Fh(=)1857 4843 y(1)p 1856 4887 51 4 v 1856 4979
                1124 a Fg(\032)1952 4843 y(@)p 1927 4887 107 4 v 1927 4979
                1125 a(@)g(z)2043 4910 y(\032)p Ff(D)2183 4843 y Fg(@)g Fh(\002)p
                1126 2183 4887 133 4 v 2196 4979 a Fg(@)g(z)3441 4910 y Fh(\(7\))1922
                1127 5214 y(2)p eop
                1128 %%Page: 3 4
                1129 3 3 bop 328 631 a Fh(but)38 b(then)h(the)f(implied)d(dep)s(endence)41
                1130 b(on)c(the)i(reference)g(temp)s(erature)f(used)h(in)e(the)328
                1131 751 y(de\014nition)i(of)h(p)s(oten)m(tial)e(temp)s(erature)i(seems)h
                1132 (strange.)66 b(A)40 b(nice)g(alternativ)m(e)g(is)f(to)328
                1133 872 y(utilize)31 b(the)i(fact)f(that,)g(in)g(a)h(h)m(ydrostatic)g
                1134 (ideal)e(gas)1699 1046 y Fg(@)5 b(\033)p 1699 1090 116
                1135 4 v 1704 1182 a(@)g(z)1853 1113 y Fh(=)1969 1046 y Fg(T)p
                1136 1966 1090 76 4 v 1966 1182 a Fh(\002)2062 1046 y Fg(@)g
                1137 Fh(\002)p 2062 1090 133 4 v 2075 1182 a Fg(@)g(z)3441
                1138 1113 y Fh(\(8\))328 1337 y(so)33 b(that)f Fg(\033)37
                1139 b Fh(is)32 b(w)m(ell-mixed)e(when)k(\002)f(is)f(w)m(ell-mixed.)474
                1140 1457 y(In)j(a)e(future)i(release,)f(this)g(sc)m(heme)h(will)d(b)s(e)i
                1141 (mo)s(di\014ed)e(so)j(that)e(the)i(moist)d(static)328
                1142 1577 y(energy)k(and)f(and)g(total)f(\(v)-5 b(ap)s(or)34
                1143 b(plus)h(liquid\))e(w)m(ater)i(con)m(ten)m(t)i(are)e(homogenized)f(b)m
                1144 (y)328 1698 y(di\013usion,)d(rather)i(than)g(dry)g(static)f(energy)i
                1145 (and)e(sp)s(eci\014)h(h)m(umidit)m(y)-8 b(.)328 2027
                1146 y Fi(3)161 b(Time)54 b(di\013erencing)f({)h(t)l(w)l(o-time)d(lev)l(el)k
                1147 (mo)t(dels)328 2246 y Fh(A)45 b(fully)f(bac)m(kw)m(ards)j(time-step)d
                1148 (for)h(di\013usion)f(is)h(the)g(standard)h(c)m(hoice)f(in)g(atmo-)328
                1149 2367 y(spheric)c(GCMs)h(as)f(the)g(di\013usivities)f(are)h(in)m(v)-5
                1150 b(ariably)38 b(large)i(enough)i(that)e(explicit)328 2487
                1151 y(time)32 b(steps)k(are)e(to)s(o)f(restrictiv)m(e.)47
                1152 b(Cen)m(tered)36 b(implicit)30 b(steps,)36 b(while)d(more)g(accurate,)
                1153 328 2607 y(can)g(pro)s(duce)h(damp)s(ed)e(oscillations)e(when)k(the)g
                1154 (correct)f(solution)f(is)g(monotonically)328 2728 y(damp)s(ed.)43
                1155 b(F)-8 b(ully)31 b(bac)m(kw)m(ards)k(steps)f(are)e(the)h(safest.)474
                1156 2848 y(Consider)26 b(for)f(a)h(momen)m(t)f(the)h(generic)f(equation,)i
                1157 (with)e(bac)m(kw)m(ards)j(di\013erencing,)1453 3033 y
                1158 Fg(\030)1501 2997 y Fe(i)p Fc(+1)1641 3033 y Ff(\000)22
                1159 b Fg(\030)1788 2997 y Fe(i)p 1453 3077 364 4 v 1576 3169
                1160 a Fh(\001)p Fg(t)1853 3101 y Fh(=)28 b Fg(L)p Fh(\()p
                1161 Fg(\030)2109 3060 y Fe(i)p Fc(+1)2227 3101 y Fh(\))22
                1162 b(+)g Fg(S)328 3329 y Fh(The)31 b(sup)s(erscript)g(refers)g(to)f(the)g
                1163 (time)f(step,)i(and)g Fg(L)f Fh(is)g(a)f(linear)g(op)s(erator)g(\(the)i
                1164 (di\013u-)328 3450 y(sion)d(op)s(erator)g(in)g(our)h(case\).)43
                1165 b Fg(S)34 b Fh(is)29 b(the)g(rest)g(of)f(the)h(RHS)g(of)g(the)g
                1166 (equation,)g(assumed)328 3570 y(to)j(b)s(e)h(kno)m(wn)h(at)e(time)f
                1167 (step)j Fg(i)p Fh(.)43 b(Equiv)-5 b(alen)m(tly)d(,)32
                1168 b(w)m(e)i(can)f(write)1290 3768 y Fg(\030)1338 3727 y
                1169 Fe(i)p Fc(+1)1478 3768 y Ff(\000)22 b Fg(\030)1625 3727
                1170 y Fe(i)1680 3768 y Fh(=)28 b(\001)p Fg(tL)p Fh(\()p Fg(\030)2052
                1171 3727 y Fe(i)p Fc(+1)2193 3768 y Ff(\000)22 b Fg(\030)2340
                1172 3727 y Fe(i)2368 3768 y Fh(\))g(+)g Fg(E)328 3966 y Fh(where)34
                1173 b Fg(E)k Fh(includes)33 b(the)g(tendency)h(due)g(to)e
                1174 Fg(L)h Fh(as)g(computed)f(explicitly)-8 b(.)1544 4164
                1175 y Fg(E)34 b Fh(=)27 b(\001)p Fg(t)p Fh(\()p Fg(S)i Fh(+)22
                1176 b Fg(L)p Fh(\()p Fg(\030)2246 4122 y Fe(i)2274 4164 y
                1177 Fh(\)\))328 4361 y(T)-8 b(o)27 b(simplify)e(notation,)j(w)m(e)g(set)g
                1178 (\001)p Fg(\030)33 b Fh(equal)27 b(to)g(the)h(incremen)m(t)f(in)g
                1179 (tracer)h(mixing)d(ratio)328 4482 y(o)m(v)m(er)34 b(one)e(time)g(step.)
                1180 1635 4602 y(\001)p Fg(\030)g Ff(\021)c Fg(\030)1944 4561
                1181 y Fe(i)p Fc(+1)2084 4602 y Ff(\000)23 b Fg(\030)2232
                1182 4561 y Fe(i)328 4767 y Fh(Then)34 b(the)f(equation)f(to)g(b)s(e)h(solv)
                1183 m(ed)g(is)1524 4965 y(\001)p Fg(\030)f Fh(=)c(\001)p
                1184 Fg(tL)p Fh(\(\001)p Fg(\030)5 b Fh(\))23 b(+)f Fg(E)1922
                1185 5214 y Fh(3)p eop
                1186 %%Page: 4 5
                1187 4 4 bop 328 631 a Fh(This)33 b(equation)f(is)g(solv)m(ed)h(b)m(y)h(in)m
                1188 (v)m(erting)e(the)h(op)s(erator)f(\(1)22 b Ff(\000)g
                1189 Fg(L)p Fh(\001)p Fg(t)p Fh(\):)1516 844 y(\001)p Fg(\030)32
                1190 b Fh(=)c(\(1)22 b Ff(\000)g Fh(\001)p Fg(tL)p Fh(\))2204
                1191 803 y Fb(\000)p Fc(1)2300 844 y Fg(E)474 1057 y Fh(It)38
                1192 b(ma)m(y)f(at)h(times)e(b)s(e)i(useful)g(to)f(split)f(the)i(source)h
                1193 (term)e Fg(S)43 b Fh(in)m(to)37 b(a)g(part)h(that)f(is)328
                1194 1177 y(con)m(v)m(enien)m(tly)f(computed)f(b)s(efore)f(the)h
                1195 (di\013usion)f(is)g(p)s(erformed,)h(and)f(a)h(part)f(that)g(is)328
                1196 1297 y(computed)f(after)f(the)h(di\013usion)f Fg(S)h
                1197 Fh(=)28 b Fg(S)1827 1312 y Fe(bef)7 b(or)r(e)2058 1297
                1198 y Fh(+)22 b Fg(S)2216 1312 y Fe(af)7 b(ter)2391 1297
                1199 y Fh(,)32 b(resulting)g(in)1036 1495 y(\001)p Fg(\030)p
                1200 1036 1539 129 4 v 1042 1631 a Fh(\001)p Fg(t)1203 1562
                1201 y Fh(=)27 b Fg(S)1366 1577 y Fe(af)7 b(ter)1563 1562
                1202 y Fh(+)22 b(\(1)g Ff(\000)h Fh(\001)p Fg(tL)p Fh(\))2090
                1203 1521 y Fb(\000)p Fc(1)2185 1562 y Fh(\()p Fg(S)2283 1577
                1204 y Fe(bef)7 b(or)r(e)2514 1562 y Fh(+)22 b Fg(L)p Fh(\()p
                1205 Fg(\030)2764 1521 y Fe(i)2792 1562 y Fh(\)\))573 b(\(9\))328
                1206 1801 y(whic)m(h)36 b(di\013ers)f(from)f(the)i(result)f(that)h(w)m(ould)
                1207 f(b)s(e)h(obtained)e(if)h(one)g(computed)h(all)d(of)328
                1208 1921 y(the)g(explicit)e(tendencies)j(b)s(efore)f(the)g(implicit)28
                1209 b(di\013usion)k(step)1036 2110 y(\001)p Fg(\030)p 1036
                1210 2154 V 1042 2246 a Fh(\001)p Fg(t)1203 2178 y Fh(=)27
                1211 b(\(1)22 b Ff(\000)h Fh(\001)p Fg(tL)p Fh(\))1735 2137
                1212 y Fb(\000)p Fc(1)1830 2178 y Fh(\()p Fg(S)1928 2193 y
                1213 Fe(bef)7 b(or)r(e)2159 2178 y Fh(+)22 b Fg(L)p Fh(\()p
                1214 Fg(\030)2409 2137 y Fe(i)2437 2178 y Fh(\))g(+)g Fg(S)2655
                1215 2193 y Fe(af)7 b(ter)2830 2178 y Fh(\))328 2416 y(Splitting)24
                1216 b(ma)m(y)i(b)s(e)h(una)m(v)m(oidable)f(at)g(times,)h(but)g(it)f(is)g(p)
                1217 s(oten)m(tially)e(problematic.)40 b(The)328 2536 y(di\016cult)m(y)31
                1218 b(arises)g(when)h(there)g(is)e(close)h(comp)s(ensation)f(b)s(et)m(w)m
                1219 (een)j Fg(S)2876 2551 y Fe(bef)7 b(or)r(e)3116 2536 y
                1220 Fh(and)31 b Fg(S)3364 2551 y Fe(af)7 b(ter)3539 2536
                1221 y Fh(,)328 2657 y(and)35 b(the)h(implicit)c(treatmen)m(t)j(of)g(the)h
                1222 (equation)f(then)h(creates)h(tendencies)g(that)e(are)328
                1223 2777 y(to)s(o)d(large.)474 2897 y(The)c(di\013usion)d(mo)s(dule)g(up)s
                1224 (dates)i(the)g(tendency)i(b)m(y)e(replacing)e(the)i(old)f(tendency)328
                1225 3018 y Fg(S)388 3033 y Fe(bef)7 b(or)r(e)632 3018 y Fh(with)34
                1226 b(\(1)24 b Ff(\000)g Fh(\001)p Fg(tL)p Fh(\))1288 2982
                1227 y Fb(\000)p Fc(1)1383 3018 y Fh(\()p Fg(S)1481 3033 y
                1228 Fe(bef)7 b(or)r(e)1714 3018 y Fh(+)24 b Fg(L)p Fh(\()p
                1229 Fg(\030)1966 2982 y Fe(i)1993 3018 y Fh(\)\).)51 b(It)35
                1230 b(do)s(es)g(not)g(kno)m(w)h(an)m(ything)f(ab)s(out)328
                1231 3138 y Fg(S)388 3153 y Fe(af)7 b(ter)328 3470 y Fi(4)161
                1232 b(Leapfrog)328 3689 y Fh(Using)35 b(bac)m(kw)m(ards)k(di\013usion)34
                1233 b(in)i(the)g(con)m(text)h(of)f(the)g(leapfrog)f(step)h(used)i(in)d(man)
                1234 m(y)328 3809 y(GCMs,)e(w)m(e)h(ha)m(v)m(e)1408 3917 y
                1235 Fg(\030)1456 3881 y Fe(i)p Fc(+1)1596 3917 y Ff(\000)22
                1236 b Fg(\030)1743 3881 y Fe(i)p Fb(\000)p Fc(1)p 1408 3961
                1237 454 4 v 1552 4053 a Fh(2\001)p Fg(t)1899 3984 y Fh(=)27
                1238 b Fg(L)p Fh(\()p Fg(\030)2154 3943 y Fe(i)p Fc(+1)2272
                1239 3984 y Fh(\))22 b(+)g Fg(S)328 4181 y Fh(With)32 b(the)h(same)f
                1240 (manipulation)d(as)k(b)s(efore,)g(w)m(e)h(obtain)d(the)i(same)g
                1241 (equation)1492 4394 y(\001)p Fg(\030)f Fh(=)c(\(1)21
                1242 b Ff(\000)i Fh(2\001)p Fg(tL)p Fh(\))2229 4353 y Fb(\000)p
                1243 Fc(1)2324 4394 y Fg(E)328 4607 y Fh(where)34 b(no)m(w)1425
                1244 4819 y Fg(E)90 b Fh(=)83 b(\(2\001)p Fg(t)p Fh(\))p Fg(S)29
                1245 b Fh(+)22 b Fg(L)p Fh(\()p Fg(\030)2326 4778 y Fe(i)p
                1246 Fb(\000)p Fc(1)2444 4819 y Fh(\)\))1374 4965 y(\001)p
                1247 Fg(\030)88 b Ff(\021)83 b Fg(\030)1794 4924 y Fe(i)p
                1248 Fc(+1)1934 4965 y Ff(\000)23 b Fg(\030)2082 4924 y Fe(i)p
                1249 Fb(\000)p Fc(1)2200 4965 y Fg(:)1922 5214 y Fh(4)p eop
                1250 %%Page: 5 6
                1251 5 5 bop 328 631 a Fh(The)27 b(di\013usion)f(mo)s(dule)f(do)s(es)h(not)h
                1252 (need)g(to)f(kno)m(w)i(whether)f(or)f(not)h(a)f(leapfrog)f(step)i(is)
                1253 328 751 y(b)s(eing)k(used.)45 b(In)32 b(the)g(leapfrog)f(case,)i(the)g
                1254 (\014eld)e Fg(\030)2169 715 y Fe(i)p Fb(\000)p Fc(1)2319
                1255 751 y Fh(is)h(input)f(to)h(the)g(mo)s(dule)f(rather)328
                1256 872 y(than)38 b Fg(\030)609 836 y Fe(i)636 872 y Fh(,)i(and)e(the)g
                1257 (time)f(step)h(passed)i(as)e(input)f(is)h(2\001)p Fg(t)g
                1258 Fh(rather)g(than)g(\001)p Fg(t)p Fh(.)59 b(W)-8 b(e)39
                1259 b(use)328 992 y(the)f(notation)e Fg(\016)t(t)h Fh(to)h(stand)g(for)e
                1260 (the)i(time)e(step)j(\001)p Fg(t)f Fh(in)e(the)i(t)m(w)m(o)g(time)e
                1261 (lev)m(el)h(sc)m(heme)328 1112 y(and)30 b(for)g(2\001)p
                1262 Fg(t)h Fh(in)f(the)g(leapfrog)f(sc)m(heme.)45 b(Also,)30
                1263 b(the)h(sym)m(b)s(ol)e Fg(\016)t(\030)35 b Fh(refers)c(to)f
                1264 Fg(\030)3169 1076 y Fe(i)p Fc(+1)3305 1112 y Ff(\000)18
                1265 b Fg(\030)3448 1076 y Fe(i)p Fb(\000)p Fc(1)328 1233
                1266 y Fh(in)32 b(the)h(leapfrog)e(case)i(and)g Fg(\030)1425
                1267 1197 y Fe(i)p Fc(+1)1565 1233 y Ff(\000)23 b Fg(\030)1713
                1268 1197 y Fe(i)1773 1233 y Fh(for)32 b(the)h(t)m(w)m(o-time)e(lev)m(el)h
                1269 (v)m(ersion.)328 1566 y Fi(5)161 b(V)-13 b(ertical)54
                1270 b(di\013erencing)328 1785 y Fh(Using)32 b(the)h(h)m(ydrostatic)g
                1271 (equation,)g(w)m(e)g(can)g(write)f(the)h(di\013usion)f(as)1845
                1272 2049 y Fg(g)1906 1981 y(@)5 b(F)p 1906 2025 134 4 v 1920
                1273 2117 a(@)g(p)328 2313 y Fh(Dividing)30 b(the)j(atmosphere)g(in)m(to)f
                1274 Fg(N)43 b Fh(la)m(y)m(ers,)34 b(with)e Fg(k)f Fh(=)d(1)k(the)h(top)g
                1275 (la)m(y)m(er)g(and)g Fg(k)e Fh(=)c Fg(N)328 2434 y Fh(the)37
                1276 b(b)s(ottom)e(la)m(y)m(er,)j(the)e(di\013usiv)m(e)h(\015ux)g(is)f
                1277 (de\014ned)i(at)e(the)h(in)m(terfaces,)h(with)e Fg(F)3362
                1278 2449 y Fe(k)r Fc(+1)p Fe(=)p Fc(2)328 2554 y Fh(b)s(et)m(w)m(een)47
                1279 b(the)e Fg(k)i Fh(and)e Fg(k)33 b Fh(+)d(1)45 b(la)m(y)m(ers,)j(so)d
                1280 (that)f Fg(F)2215 2570 y Fc(1)p Fe(=)p Fc(2)2374 2554
                1281 y Fh(=)k(0)c(and)h Fg(F)2856 2570 y Fe(N)7 b Fc(+1)p
                1282 Fe(=)p Fc(2)3132 2554 y Fh(=)48 b Fg(F)3319 2569 y Fe(s)3356
                1283 2554 y Fh(,)f(the)328 2675 y(surface)d(\015ux.)77 b(W)-8
                1284 b(e)44 b(also)e(require)i(the)g(pressure)h(de\014ned)g(at)e(the)h(in)m
                1285 (terface)f Fg(p)3335 2690 y Fe(k)r Fc(+1)p Fe(=)p Fc(2)3539
                1286 2675 y Fh(.)328 2795 y(The)33 b(simplest)f(di\013erencing)g(then)h
                1287 (yields)1531 3015 y Fg(\026)1590 3030 y Fe(k)1632 3015
                1288 y Fh(\()p Fg(F)1733 3030 y Fe(k)r Fc(+1)p Fe(=)p Fc(2)1959
                1289 3015 y Ff(\000)22 b Fg(F)2121 3030 y Fe(k)r Fb(\000)p
                1290 Fc(1)p Fe(=)p Fc(2)2325 3015 y Fh(\))328 3235 y(where)1268
                1291 3355 y Fg(\026)1327 3370 y Fe(k)1397 3355 y Ff(\021)1801
                1292 3288 y Fg(g)p 1513 3332 627 4 v 1513 3424 a(p)1562 3439
                1293 y Fe(k)r Fc(+1)p Fe(=)p Fc(2)1787 3424 y Ff(\000)h Fg(p)1936
                1294 3439 y Fe(k)r Fb(\000)p Fc(1)p Fe(=)p Fc(2)2149 3355
                1295 y Fg(;)83 b(k)31 b Fh(=)c(1)p Fg(;)17 b(N)777 b Fh(\(10\))328
                1296 3587 y(Note)44 b(that)g(the)g(\015uxes)i(are)e(considered)h(p)s(ositiv)
                1297 m(e)e(when)i(directed)f(up)m(w)m(ards,)49 b(ev)m(en)328
                1298 3708 y(though)e(the)h(indexing)f(starts)h(from)e(the)h(top)h(\(to)e(b)s
                1299 (e)i(consisten)m(t)g(with)f(standard)328 3828 y(practice)32
                1300 b(in)g(GCM)h(co)s(des\).)474 3948 y(The)25 b(\015uxes,)i(in)c(turn,)j
                1301 (are)d(giv)m(en)h(in)f(terms)h(of)f(the)h(di\013usivit)m(y)f
                1302 Ff(D)2854 3964 y Fe(k)r Fc(+1)p Fe(=)p Fc(2)3081 3948
                1303 y Fh(and)h(densit)m(y)328 4069 y Fg(\032)378 4084 y Fe(k)r
                1304 Fc(+1)p Fe(=)p Fc(2)614 4069 y Fh(de\014ned)34 b(at)e(the)h(in)m
                1305 (terfaces)h(and)e(the)h(heigh)m(ts)g Fg(z)2408 4084 y
                1306 Fe(k)2484 4069 y Fh(of)f(the)h(full)d(lev)m(els,)1182
                1307 4345 y Fg(F)1245 4361 y Fe(k)r Fc(+1)p Fe(=)p Fc(2)1533
                1308 4345 y Fh(=)83 b Ff(\000D)1846 4361 y Fe(k)r Fc(+1)p
                1309 Fe(=)p Fc(2)2050 4345 y Fg(\032)2100 4361 y Fe(k)r Fc(+1)p
                1310 Fe(=)p Fc(2)2316 4278 y Fg(\030)2359 4293 y Fe(k)r Fc(+1)2514
                1311 4278 y Ff(\000)23 b Fg(\030)2657 4293 y Fe(k)p 2314 4322
                1312 388 4 v 2314 4414 a Fg(z)2359 4429 y Fe(k)r Fc(+1)2514
                1313 4414 y Ff(\000)g Fg(z)2659 4429 y Fe(k)1532 4543 y Ff(\021)83
                1314 b Fg(\027)1740 4559 y Fe(k)r Fc(+1)p Fe(=)p Fc(2)1944
                1315 4543 y Fh(\()p Fg(\030)2025 4558 y Fe(k)r Fc(+1)2179
                1316 4543 y Ff(\000)23 b Fg(\030)2322 4558 y Fe(k)2364 4543
                1317 y Fh(\))990 b(\(11\))328 4764 y(Note)33 b(that)f Fg(\027)39
                1318 b Fh(is)32 b(p)s(ositiv)m(e)g(\(as)h(is)f Fg(\026)p Fh(\))g(since)h
                1319 Fg(k)j Fh(increases)d(do)m(wn)m(w)m(ards.)1922 5214 y(5)p
                1320 eop
                1321 %%Page: 6 7
                1322 6 6 bop 328 631 a Fi(6)161 b(T)-13 b(ridiagonal)42 b(solv)l(er)e(with)g
                1323 (explicit)i(surface)d(\015ux)328 850 y Fh(W)-8 b(e)31
                1324 b(de\014ne)g Fg(E)37 b Fh(to)30 b(b)s(e)h(the)g(incremen)m(t)f(in)g
                1325 Fg(\030)k Fh(due)e(to)e(all)e(terms)i(treated)h(explicitly)e(plus)328
                1326 970 y(the)41 b(explicit)e(con)m(tribution)g(from)g(di\013usion)h(in)f
                1327 (the)i(in)m(terior.)66 b(W)-8 b(e)41 b(do)f(not)g(include)328
                1328 1091 y(the)d(e\013ect)g(of)f(the)h(surface)h(\015ux)f
                1329 Fg(F)1640 1106 y Fe(s)1713 1091 y Fh(itself)f(on)g(the)h(incremen)m(t)g
                1330 (in)e(the)i(lo)m(w)m(est)g(mo)s(del)328 1211 y(la)m(y)m(er,)c(ev)m(en)h
                1331 (when)g(explicit,)d(in)h(the)h(expression)h(for)e Fg(E)6
                1332 b Fh(.)474 1332 y(With)1731 1452 y Fg(\026)1790 1411
                1333 y Fb(0)1841 1452 y Ff(\021)28 b Fh(\()p Fg(\016)t(t)p
                1334 Fh(\))p Fg(\026)328 1618 y Fh(w)m(e)34 b(ha)m(v)m(e)400
                1335 1820 y Fg(\016)t(\030)490 1835 y Fe(k)615 1820 y Fh(=)83
                1336 b Fg(E)846 1835 y Fe(k)911 1820 y Fh(+)22 b Fg(\026)1068
                1337 1779 y Fb(0)1068 1844 y Fe(k)1110 1820 y Fh(\()p Fg(\027)1196
                1338 1835 y Fe(k)r Fc(+1)p Fe(=)p Fc(2)1400 1820 y Fh(\()p
                1339 Fg(\016)t(\030)1528 1835 y Fe(k)r Fc(+1)1682 1820 y Ff(\000)h
                1340 Fg(\016)t(\030)1872 1835 y Fe(k)1914 1820 y Fh(\))f Ff(\000)h
                1341 Fg(\027)2122 1835 y Fe(k)r Fb(\000)p Fc(1)p Fe(=)p Fc(2)2325
                1342 1820 y Fh(\()p Fg(\016)t(\030)2453 1835 y Fe(k)2518 1820
                1343 y Ff(\000)f Fg(\016)t(\030)2707 1835 y Fe(k)r Fb(\000)p
                1344 Fc(1)2840 1820 y Fh(\)\))66 b Fg(k)31 b Fh(=)c(2)p Fg(;)17
                1345 b(N)32 b Ff(\000)23 b Fh(1)403 1965 y Fg(\016)t(\030)493
                1346 1980 y Fc(1)615 1965 y Fh(=)83 b Fg(E)846 1980 y Fc(1)908
                1347 1965 y Fh(+)22 b Fg(\026)1065 1924 y Fb(0)1065 1990 y
                1348 Fc(1)1104 1965 y Fg(\027)1152 1980 y Fc(3)p Fe(=)p Fc(2)1262
                1349 1965 y Fh(\()p Fg(\016)t(\030)1390 1980 y Fc(2)1451 1965
                1350 y Ff(\000)h Fg(\016)t(\030)1641 1980 y Fc(1)1680 1965
                1351 y Fh(\))375 2110 y Fg(\016)t(\030)465 2125 y Fe(N)615
                1352 2110 y Fh(=)83 b Fg(E)846 2125 y Fe(N)935 2110 y Fh(+)22
                1353 b Fg(\026)1092 2069 y Fb(0)1092 2135 y Fe(N)1159 2110
                1354 y Fg(F)1222 2125 y Fe(s)1281 2110 y Ff(\000)h Fg(\026)1440
                1355 2069 y Fb(0)1440 2135 y Fe(N)1507 2110 y Fg(\027)1555
                1356 2126 y Fe(N)7 b Fb(\000)p Fc(1)p Fe(=)p Fc(2)1783 2110
                1357 y Fh(\()p Fg(\016)t(\030)1911 2125 y Fe(N)2000 2110 y
                1358 Ff(\000)23 b Fg(\016)t(\030)2190 2125 y Fe(N)7 b Fb(\000)p
                1359 Fc(1)2347 2110 y Fh(\))1007 b(\(12\))328 2312 y(So)32
                1360 b(then)1004 2432 y Fg(A)1077 2447 y Fe(k)1120 2432 y
                1361 Fg(\016)t(\030)1210 2447 y Fe(k)r Fc(+1)1365 2432 y Fh(+)22
                1362 b Fg(B)1537 2447 y Fe(k)1580 2432 y Fg(\016)t(\030)1670
                1363 2447 y Fe(k)1734 2432 y Fh(+)g Fg(C)1902 2447 y Fe(k)1944
                1364 2432 y Fg(\016)t(\030)2034 2447 y Fe(k)r Fb(\000)p Fc(1)2194
                1365 2432 y Fh(=)28 b Fg(E)2370 2447 y Fe(k)2413 2432 y Fg(;)83
                1366 b(k)31 b Fh(=)c(1)p Fg(;)17 b(N)328 2598 y Fh(where)973
                1367 2800 y Fg(A)1046 2815 y Fe(k)1171 2800 y Fh(=)83 b Ff(\000)p
                1368 Fg(\026)1466 2759 y Fb(0)1466 2824 y Fe(k)1509 2800 y
                1369 Fg(\027)1557 2815 y Fe(k)r Fc(+1)p Fe(=)p Fc(2)1761 2800
                1370 y Fg(;)g(k)31 b Fh(=)c(1)p Fg(;)17 b(N)32 b Ff(\000)23
                1371 b Fh(1)948 2945 y Fg(A)1021 2960 y Fe(N)1171 2945 y Fh(=)83
                1372 b(0)976 3090 y Fg(C)1046 3105 y Fe(k)1171 3090 y Fh(=)g
                1373 Ff(\000)p Fg(\026)1466 3049 y Fb(0)1466 3115 y Fe(k)1509
                1374 3090 y Fg(\027)1557 3106 y Fe(k)r Fb(\000)p Fc(1)p Fe(=)p
                1375 Fc(2)1761 3090 y Fg(;)g(k)31 b Fh(=)c(2)p Fg(;)17 b(N)979
                1376 3236 y(C)1049 3251 y Fc(1)1171 3236 y Fh(=)83 b(0)972
                1377 3381 y Fg(B)1046 3396 y Fe(k)1171 3381 y Fh(=)g(1)22
                1378 b(+)g Fg(\026)1558 3340 y Fb(0)1558 3406 y Fe(k)1601
                1379 3381 y Fh(\()p Fg(\027)1687 3396 y Fe(k)r Fc(+1)p Fe(=)p
                1380 Fc(1)1912 3381 y Fh(+)g Fg(\027)2058 3396 y Fe(k)r Fb(\000)p
                1381 Fc(1)p Fe(=)p Fc(2)2262 3381 y Fh(\))p Fg(;)83 b(k)31
                1382 b Fh(=)c(2)p Fg(;)17 b(N)32 b Ff(\000)23 b Fh(1)975 3526
                1383 y Fg(B)1049 3541 y Fc(1)1171 3526 y Fh(=)83 b(1)22 b(+)g
                1384 Fg(\026)1558 3485 y Fb(0)1558 3551 y Fc(1)1597 3526 y
                1385 Fg(\027)1645 3542 y Fc(3)p Fe(=)p Fc(2)947 3672 y Fg(B)1021
                1386 3687 y Fe(N)1171 3672 y Fh(=)83 b(1)22 b(+)g Fg(\026)1558
                1387 3630 y Fb(0)1558 3696 y Fe(N)1625 3672 y Fg(\027)1673
                1388 3687 y Fe(N)7 b Fb(\000)p Fc(1)p Fe(=)p Fc(2)328 3873
                1389 y Fh(T)-8 b(o)32 b(solv)m(e)g(these)g(equations)g(with)g(the)g
                1390 (standard)g(tridiagnonal)c(substitution,)j(w)m(e)i(set)1316
                1391 4074 y Fg(\016)t(\030)1406 4089 y Fe(k)1476 4074 y Fh(=)28
                1392 b Fg(e)1625 4089 y Fe(k)1668 4074 y Fg(\016)t(\030)1758
                1393 4089 y Fe(k)r Fc(+1)1912 4074 y Fh(+)22 b Fg(f)2058 4089
                1394 y Fe(k)2101 4074 y Fg(;)83 b(k)31 b Fh(=)c(1)p Fg(;)17
                1395 b(N)825 b Fh(\(13\))328 4276 y(Substituting,)32 b(w)m(e)h(ha)m(v)m(e)
                1396 939 4521 y Fg(A)1012 4536 y Fe(k)1065 4454 y Fg(\016)t(\030)1155
                1397 4469 y Fe(k)1219 4454 y Ff(\000)23 b Fg(f)1367 4469 y
                1398 Fe(k)p 1065 4498 345 4 v 1193 4589 a Fg(e)1238 4604 y
                1399 Fe(k)1442 4521 y Fh(+)f Fg(B)1614 4536 y Fe(k)1656 4521
                1400 y Fg(\016)t(\030)1746 4536 y Fe(k)1811 4521 y Fh(+)g
                1401 Fg(C)1979 4536 y Fe(k)2021 4521 y Fh(\()p Fg(e)2104 4536
                1402 y Fe(k)r Fb(\000)p Fc(1)2237 4521 y Fg(\016)t(\030)2327
                1403 4536 y Fe(k)2392 4521 y Fh(+)g Fg(f)2538 4536 y Fe(k)r
                1404 Fb(\000)p Fc(1)2671 4521 y Fh(\))27 b(=)h Fg(E)2912 4536
                1405 y Fe(k)328 4763 y Fh(so)33 b(that)1144 4908 y Fg(e)1189
                1406 4923 y Fe(k)1260 4908 y Fh(=)1541 4840 y Ff(\000)p Fg(A)1691
                1407 4855 y Fe(k)p 1374 4884 528 4 v 1374 4976 a Fg(B)1448
                1408 4991 y Fe(k)1512 4976 y Fh(+)22 b Fg(C)1680 4991 y Fe(k)1723
                1409 4976 y Fg(e)1768 4991 y Fe(k)r Fb(\000)p Fc(1)1977 4908
                1410 y Fg(f)2025 4923 y Fe(k)2096 4908 y Fh(=)2209 4840 y
                1411 Fg(E)2281 4855 y Fe(k)2346 4840 y Ff(\000)h Fg(C)2516
                1412 4855 y Fe(k)2558 4840 y Fg(f)2606 4855 y Fe(k)r Fb(\000)p
                1413 Fc(1)p 2209 4884 531 4 v 2210 4976 a Fg(B)2284 4991 y
                1414 Fe(k)2349 4976 y Fh(+)f Fg(C)2517 4991 y Fe(k)2560 4976
                1415 y Fg(e)2605 4991 y Fe(k)r Fb(\000)p Fc(1)1922 5214 y
                1416 Fh(6)p eop
                1417 %%Page: 7 8
                1418 7 7 bop 328 631 a Fh(Starting)31 b(at)i(the)g(top)f(of)g(the)h
                1419 (atmosphere,)g(w)m(e)h(ha)m(v)m(e)1502 894 y Fg(e)1547
                1420 909 y Fc(1)1615 894 y Fh(=)27 b Ff(\000)1805 826 y Fg(A)1878
                1421 841 y Fc(1)p 1805 870 114 4 v 1805 962 a Fg(B)1879 977
                1422 y Fc(1)1929 894 y Fg(;)83 b(f)2087 909 y Fc(1)2154 894
                1423 y Fh(=)2269 826 y Fg(E)2341 841 y Fc(1)p 2268 870 V 2268
                1424 962 a Fg(B)2342 977 y Fc(1)328 1154 y Fh(One)34 b(can)g(then)g
                1425 (progress)g(do)m(wn)m(w)m(ards,)i(computing)c Fg(e)p
                1426 Fh('s)j(and)e Fg(f)11 b Fh('s)34 b(up)g(to)f Fg(e)3132
                1427 1169 y Fe(N)3233 1154 y Fh(and)g Fg(f)3471 1169 y Fe(N)3539
                1428 1154 y Fh(.)328 1275 y(The)g(last)f(step)i(in)e(this)g(sw)m(eep)i(is)
                1429 1104 1495 y Fg(e)1149 1510 y Fe(N)1299 1495 y Fh(=)83
                1430 b(0)1101 1683 y Fg(f)1149 1698 y Fe(N)1299 1683 y Fh(=)g
                1431 Fg(\016)t(\030)1548 1698 y Fe(N)1642 1683 y Fh(=)28 b
                1432 Ff(\000)1833 1616 y Fg(E)1905 1631 y Fe(N)1995 1616 y
                1433 Fh(+)22 b Fg(\026)2152 1580 y Fb(0)2152 1640 y Fe(N)2219
                1434 1616 y Fg(F)2282 1631 y Fe(s)2341 1616 y Ff(\000)h Fg(C)2511
                1435 1631 y Fe(N)2578 1616 y Fg(f)2626 1631 y Fe(N)7 b Fb(\000)p
                1436 Fc(1)p 1833 1660 950 4 v 2008 1751 a Fg(B)2082 1766 y
                1437 Fe(N)2171 1751 y Fh(+)22 b Fg(C)2339 1766 y Fe(N)2406
                1438 1751 y Fg(e)2451 1766 y Fe(N)7 b Fb(\000)p Fc(1)3392
                1439 1683 y Fh(\(14\))328 1956 y(The)40 b(up)m(w)m(ard)f(part)g(of)f(the)h
                1440 (tridiagonal)c(reduction)k(pro)s(ceeds)h(b)m(y)g(returning)e(to)g
                1441 (\(6\))328 2076 y(and)33 b(sw)m(eeping)g(recursiv)m(ely)h(up)f(the)g
                1442 (atmosphere)g(to)f Fg(k)f Fh(=)c(1.)474 2197 y(Substituting)32
                1443 b(the)h(v)-5 b(alues)32 b(of)h Fg(A)1676 2212 y Fe(N)1775
                1444 2197 y Fh(and)g Fg(B)2039 2212 y Fe(N)2139 2197 y Fh(in)m(to)f(\(7\))
                1445 1492 2417 y Fg(\016)t(\030)1582 2432 y Fe(N)1676 2417
                1446 y Fh(=)c(\000\()p Fg(E)1957 2376 y Fb(\003)1951 2441
                1447 y Fe(N)2040 2417 y Fh(+)22 b Fg(\026)2197 2376 y Fb(0)2197
                1448 2441 y Fe(N)2264 2417 y Fg(F)2327 2432 y Fe(s)2364 2417
                1449 y Fh(\))990 b(\(15\))328 2637 y(where)1100 2857 y Fg(E)1178
                1450 2816 y Fb(\003)1172 2881 y Fe(N)1323 2857 y Ff(\021)83
                1451 b Fg(E)1555 2872 y Fe(N)1645 2857 y Fh(+)22 b Fg(\026)1802
                1452 2816 y Fb(0)1802 2881 y Fe(N)1869 2857 y Fg(\027)1917
                1453 2872 y Fe(N)7 b Fb(\000)p Fc(1)p Fe(=)p Fc(2)2145 2857
                1454 y Fg(f)2193 2872 y Fe(N)g Fb(\000)p Fc(1)3392 2857 y
                1455 Fh(\(16\))1179 3054 y(\000)83 b Ff(\021)1686 2986 y Fh(1)p
                1456 1493 3030 436 4 v 1493 3122 a(1)22 b Ff(\000)h Fg(\026)1723
                1457 3088 y Fb(0)1723 3146 y Fe(N)1790 3122 y Fg(D)1874 3093
                1458 y Fe(T)3392 3054 y Fh(\(17\))1101 3316 y Fg(D)1185 3275
                1459 y Fe(T)1323 3316 y Ff(\021)83 b(\000)1570 3249 y Fg(@)5
                1460 b(F)1689 3264 y Fe(N)i Fb(\000)p Fc(1)p 1570 3293 278
                1461 4 v 1626 3385 a Fg(@)e(\030)1725 3400 y Fe(N)1886 3316
                1462 y Ff(\021)28 b(\000)p Fg(\027)2116 3332 y Fe(N)7 b Fb(\000)p
                1463 Fc(1)p Fe(=)p Fc(2)2345 3316 y Fh(\(1)21 b Ff(\000)i
                1464 Fg(e)2598 3331 y Fe(N)7 b Fb(\000)p Fc(1)2756 3316 y
                1465 Fh(\))598 b(\(18\))3392 3512 y(\(19\))328 3732 y Fg(D)412
                1466 3696 y Fe(T)511 3732 y Fh(is)44 b(the)g(sensitivit)m(y)h(of)e(the)i
                1467 Fa(downwar)-5 b(d)53 b Fh(\015ux)45 b(through)g(the)f(top)h(of)e(the)i
                1468 (lo)m(w)m(est)328 3852 y(mo)s(del)33 b(la)m(y)m(er)i(to)f(the)h(v)-5
                1469 b(alue)34 b(of)g Fg(\030)k Fh(in)c(the)h(lo)m(w)m(est)g(mo)s(del)e(la)m
                1470 (y)m(er,)i(taking)f(in)m(to)g(accoun)m(t)328 3973 y(the)f(implicit)c
                1471 (corrections)j(to)h(all)d(of)i(the)h(other)g(la)m(y)m(ers.)328
                1472 4305 y Fi(7)161 b(Implicit)47 b(dep)t(endence)c(of)j(surface)f(\015ux)g
                1473 (on)g(lo)l(w-)570 4488 y(est)53 b(atmospheric)f(la)l(y)l(er)328
                1474 4707 y Fh(Using)43 b(a)g(linear)e(T)-8 b(a)m(ylor's)43
                1475 b(expansion)h(w)m(e)g(assume)g(that)f(the)g(surface)h(\015ux)g(can)g(b)
                1476 s(e)328 4827 y(written)32 b(in)g(the)h(form)1544 4948
                1477 y Fg(F)1607 4963 y Fe(s)1672 4948 y Fh(=)27 b Fg(F)1852
                1478 4907 y Fe(ex)1838 4972 y(s)1951 4948 y Fh(+)22 b Fg(D)2133
                1479 4907 y Fe(B)2193 4948 y Fg(\016)t(\030)2283 4963 y Fe(N)3392
                1480 4948 y Fh(\(20\))1922 5214 y(7)p eop
                1481 %%Page: 8 9
                1482 8 8 bop 328 631 a Fh(where)40 b Fg(F)693 595 y Fe(ex)679
                1483 656 y(s)809 631 y Fh(and)f Fg(D)1089 595 y Fe(B)1189
                1484 631 y Fh(are)g(kno)m(wn)i(explicitly)-8 b(.)61 b(F)-8
                1485 b(ollo)m(wing)36 b(through)k(the)f(deriv)-5 b(ation)328
                1486 751 y(in)33 b(the)h(preceeding)g(section,)g(the)g(only)f(c)m(hange)h
                1487 (in)f(the)h(resulting)f(algorithm)d(is)j(that)328 872
                1488 y(in)e(the)i(\014nal)f(step)h(in)e(the)i(do)m(wn)m(w)m(ard)h(sw)m(eep)g
                1489 (w)m(e)f(need)h(to)e(replace)g Fg(F)2938 887 y Fe(s)3007
                1490 872 y Fh(b)m(y)h Fg(F)3219 836 y Fe(ex)3205 896 y(s)3328
                1491 872 y Fh(in)e(\(8\))328 992 y(and)i(w)m(e)g(need)h(to)e(c)m(hange)i
                1492 Fg(D)1412 1007 y Fe(T)1499 992 y Fh(to)e Fg(D)1699 1007
                1493 y Fe(T)1776 992 y Fh(+)22 b Fg(D)1955 1007 y Fe(B)2049
                1494 992 y Fh(in)31 b(the)i(de\014nition)f(of)g(\000.)1349
                1495 1173 y Fg(\016)t(\030)1439 1188 y Fe(N)1590 1173 y Fh(=)83
                1496 b(\000\()p Fg(E)1926 1132 y Fb(\003)1920 1197 y Fe(N)2010
                1497 1173 y Fh(+)22 b Fg(\026)2167 1132 y Fb(0)2167 1197 y
                1498 Fe(N)2234 1173 y Fg(F)2311 1132 y Fe(ex)2297 1197 y(s)2387
                1499 1173 y Fh(\))967 b(\(21\))1445 1362 y(\000)83 b Ff(\021)2123
                1500 1295 y Fh(1)p 1759 1339 776 4 v 1759 1431 a(1)22 b Ff(\000)h
                1501 Fg(\026)1989 1396 y Fb(0)1989 1455 y Fe(N)2056 1431 y
                1502 Fh(\()p Fg(D)2178 1402 y Fe(T)2255 1431 y Fh(+)f Fg(D)2437
                1503 1402 y Fe(B)2497 1431 y Fh(\))3392 1362 y(\(22\))328
                1504 1708 y Fi(8)161 b(Implicit)50 b(dep)t(endence)c(of)j(surface)f(\015ux)h
                1505 (on)f(sur-)570 1890 y(face)54 b(parameters)328 2110 y
                1506 Fh(No)m(w)36 b(supp)s(ose)h(that)f(the)g(surface)g(\015ux)h(also)d(dep)
                1507 s(ends)k(on)d(some)h(surface)g(parameter)328 2230 y Fg(\020)8
                1508 b Fh(.)42 b(\(This)29 b(could)g(also)g(b)s(e)h(a)f(set)h(of)f
                1509 (parameters,)i(but)e(it)g(is)g(easier)h(to)f(think)g(ab)s(out)g(the)328
                1510 2350 y(case)k(in)f(whic)m(h)h(there)h(is)e(only)g(1\).)43
                1511 b(A)33 b(linear)e(T)-8 b(a)m(ylor)32 b(expansion)h(no)m(w)g(reads)1368
                1512 2531 y Fg(F)1431 2546 y Fe(s)1496 2531 y Fh(=)27 b Fg(F)1676
                1513 2490 y Fe(ex)1662 2555 y(s)1775 2531 y Fh(+)22 b Fg(D)1957
                1514 2490 y Fe(B)2017 2531 y Fg(\016)t(\030)2107 2546 y Fe(N)2196
                1515 2531 y Fh(+)g Fg(D)2378 2490 y Fe(S)2429 2531 y Fg(\016)t(\020)873
                1516 b Fh(\(23\))328 2711 y(Revising)32 b(\(14\),)g(w)m(e)h(ha)m(v)m(e)1251
                1517 2892 y Fg(\016)t(\030)1341 2907 y Fe(N)1436 2892 y Fh(=)27
                1518 b(\000\()p Fg(E)1716 2851 y Fb(\003)1710 2917 y Fe(N)1800
                1519 2892 y Fh(+)22 b Fg(\026)1957 2851 y Fb(0)1957 2917 y
                1520 Fe(N)2024 2892 y Fh(\()p Fg(F)2139 2851 y Fe(ex)2125
                1521 2917 y(s)2237 2892 y Fh(+)g Fg(D)2419 2851 y Fe(S)2470
                1522 2892 y Fg(\016)t(\020)8 b Fh(\)\))748 b(\(24\))328 3072
                1523 y(or)1580 3193 y Fg(\016)t(\030)1670 3208 y Fe(N)1765
                1524 3193 y Fh(=)27 b Fg(e)1913 3208 y Fe(N)1981 3193 y Fg(\016)t(\020)i
                1525 Fh(+)22 b Fg(f)2246 3208 y Fe(N)3392 3193 y Fh(\(25\))328
                1526 3350 y(where)1471 3531 y Fg(e)1516 3546 y Fe(N)1667 3531
                1527 y Fh(=)83 b(\000)p Fg(\026)1946 3490 y Fb(0)1946 3555
                1528 y Fe(N)2012 3531 y Fg(D)2096 3490 y Fe(s)3392 3531 y
                1529 Fh(\(26\))1468 3676 y Fg(f)1516 3691 y Fe(N)1667 3676
                1530 y Fh(=)g(\000)p Fg(E)1965 3635 y Fb(\003)1959 3701 y
                1531 Fe(N)2048 3676 y Fh(+)22 b Fg(\026)2205 3635 y Fb(0)2205
                1532 3701 y Fe(N)2272 3676 y Fg(F)2349 3635 y Fe(ex)2335 3701
                1533 y(s)3392 3676 y Fh(\(27\))474 3857 y(Substituting)32
                1534 b(bac)m(k)i(in)m(to)d(the)i(expression)h(for)e(the)h(\015ux,)1661
                1535 4037 y Fg(F)1724 4052 y Fe(s)1789 4037 y Fh(=)27 b Fg(\013)c
                1536 Fh(+)f Fg(\014)6 b(\016)t(\020)1166 b Fh(\(28\))328 4218
                1537 y(where)1527 4398 y Fg(\013)84 b Ff(\021)f Fg(F)1910
                1538 4357 y Fe(ex)1896 4423 y(s)2009 4398 y Fh(+)22 b Fg(D)2191
                1539 4357 y Fe(B)2251 4398 y Fg(f)2299 4413 y Fe(N)3392 4398
                1540 y Fh(\(29\))1529 4543 y Fg(\014)89 b Ff(\021)83 b Fg(D)1917
                1541 4502 y Fe(S)1990 4543 y Fh(+)22 b Fg(D)2172 4502 y Fe(B)2232
                1542 4543 y Fg(e)2277 4558 y Fe(N)3392 4543 y Fh(\(30\))474
                1543 4724 y(So)h(w)m(e)h(can)g(ask)g(the)f(surface)h(mo)s(dules)e(to)h
                1544 (compute)g Fg(\016)t(\020)30 b Fh(giv)m(en)23 b(a)g(b)s(oundary)h
                1545 (comdi-)328 4844 y(tion)h(of)h(the)h(form)e(\(21\).)40
                1546 b(One)27 b(can)g(then)g(compute)f Fg(\016)t(\030)2322
                1547 4859 y Fe(N)2415 4844 y Fh(from)f(\(18\))h(and)g(pro)s(ceed)h(with)328
                1548 4965 y(the)33 b(up)m(w)m(ard)h(sw)m(eep.)1922 5214 y(8)p
                1549 eop
                1550 %%Page: 9 10
                1551 9 9 bop 328 631 a Fi(9)161 b(Implemen)l(tation)328 850
                1552 y Fh(If)27 b(there)g(is)g(no)f(implicit)d(dep)s(endence)29
                1553 b(of)e(surface)g(\015uxes)i(on)e(surface)g(parameters,)h(the)328
                1554 970 y(v)m(ertical)35 b(di\013usion)g(can)h(up)s(date)g(the)h(mo)s(del)d
                1555 (tendencies)j(to)f(tak)m(e)g(in)m(to)g(accoun)m(t)g(the)328
                1556 1091 y(v)m(ertical)f(di\013usion)f(b)m(y)i(itself.)51
                1557 b(This)36 b(is)e(done)i(with)f(a)h(single)e(call)g(to)h(the)g
                1558 (subroutine)328 1211 y Fg(v)t(er)s(t)p 512 1211 30 4
                1559 v 35 w(dif)11 b(f)44 b Fh(for)34 b(a)f(single)g(\014eld,)i(or)e
                1560 Fg(g)t(cm)p 1827 1211 V 35 w(v)t(er)s(t)p 2040 1211 V
                1561 35 w(dif)11 b(f)44 b Fh(whic)m(h)35 b(handles)f Fg(u)p
                1562 Fh(,)g Fg(v)t Fh(,)g Fg(T)14 b Fh(,)34 b Fg(q)t Fh(,)h(and)328
                1563 1332 y(tracers)e(sim)m(ultaneously)-8 b(.)474 1452 y(If)42
                1564 b(there)g(is)f(implicit)d(dep)s(endence)43 b(of)e(surface)i(\015uxes)g
                1565 (on)e(surface)i(parameters,)328 1572 y(then)30 b(the)g(job)g(is)f
                1566 (split)f(b)s(et)m(w)m(een)k(the)e(v)m(ertical)f(di\013usion)g(mo)s
                1567 (dule,)f(the)j(\015ux)f(exc)m(hange)328 1693 y(mo)s(dule,)h(and)i(the)g
                1568 (surface)g(mo)s(dule\(s\))474 1813 y(The)49 b("GCM")e(calls)g(assume)h
                1569 (the)g(sp)s(eci\014c)g(h)m(umidit)m(y)f(and)h(heat)g(are)f(di\013used)
                1570 328 1933 y(with)c(the)h(same)f(di\013usivit)m(y)-8 b(,)45
                1571 b(but)e(momen)m(tum)f(can)i(p)s(oten)m(tially)d(ha)m(v)m(e)k(a)e
                1572 (di\013eren)m(t)328 2054 y(di\013usivit)m(y)-8 b(.)43
                1573 b(Other)32 b(tracers)i(share)f(the)g(di\013usivit)m(y)f(for)g(heat)g
                1574 (and)h(moisture.)474 2174 y(The)42 b(GCM)e(calls)f(also)h(assume)g
                1575 (that)h(the)f(surface)i(\015uxes)g(of)d(momen)m(tem)g(ha)m(v)m(e)328
                1576 2295 y(no)29 b(implicit)d(dep)s(endence)32 b(on)d(surface)h
                1577 (parameters,)g(but)g(that)f(the)h(surface)g(heat)g(and)328
                1578 2415 y(moisture)i(\015uxes)i(p)s(oten)m(tially)c(do.)44
                1579 b(\(All)31 b(other)h(tracers)i(are)e(lik)m(e)g(momen)m(tum)f(in)h(this)
                1580 328 2535 y(regard.\))43 b(The)34 b(mo)s(dule)d(is)h(not)g(a)m(w)m(are)i
                1581 (of)e(the)h(details)e(of)i(this)f(dep)s(endency)-8 b(.)474
                1582 2656 y(One)32 b(\014rst)f(calls)f Fg(g)t(cm)p 1278 2656
                1583 V 35 w(v)t(er)s(t)p 1491 2656 V 35 w(dif)11 b(f)p 1727
                1584 2656 V 45 w(dow)s(n)30 b Fh(to)h(do)g(the)g(do)m(wn)m(w)m(ard)i(sw)m
                1585 (eep.)45 b(The)32 b(v)-5 b(al-)328 2776 y(ues)33 b(of)g
                1586 Fg(e)653 2791 y Fe(k)728 2776 y Fh(and)g Fg(f)966 2791
                1587 y Fe(k)1041 2776 y Fh(for)f Fg(k)f Fh(=)c(1)p Fg(;)17
                1588 b(N)32 b Ff(\000)23 b Fh(1)32 b(are)h(sa)m(v)m(ed)h(as)f(mo)s(dule)e(v)
                1589 -5 b(ariables)31 b(for)h(later)g(use)328 2896 y(b)m(y)h(the)g(up)m(w)m
                1590 (ard)h(sw)m(eep.)46 b(The)33 b(v)-5 b(ariable)31 b Fg(e)1907
                1591 2911 y Fe(k)1982 2896 y Fh(is)h(the)h(same)f(for)g(moisture)g(and)g
                1592 (temp)s(er-)328 3017 y(ature,)40 b(but)e(there)h(is)f(are)g(separate)h
                1593 Fg(f)1767 3032 y Fe(k)1848 3017 y Fh(\014elds)g(for)e(moisture)h(and)g
                1594 (for)g(temp)s(erature.)328 3137 y(\(If)d(temp)s(erature)g(and)h
                1595 (moisture)e(are)h(giv)m(en)h(di\013eren)m(t)f(di\013usivities,)g(then)h
                1596 (the)f(co)s(de)328 3258 y(will)27 b(ha)m(v)m(e)j(to)f(b)s(e)h(mo)s
                1597 (di\014ed)e(to)g(sa)m(v)m(e)j(di\013eren)m(t)f Fg(e)2119
                1598 3273 y Fe(k)2191 3258 y Fh(\014elds)f(for)g(the)h(di\013eren)m(t)f(v)-5
                1599 b(ariables.\))328 3378 y(Additionally)d(,)29 b(the)k(v)-5
                1600 b(alues)33 b(of)f(the)h(follo)m(wing)d(\014elds)j(are)f(output)1335
                1601 3598 y Fg(\026)1394 3557 y Fb(0)1394 3623 y Fe(N)1544
                1602 3598 y Ff(\021)84 b Fg(\016)t(t\026)1846 3613 y Fe(N)3392
                1603 3598 y Fh(\(31\))1322 3743 y Fg(E)1400 3702 y Fb(\003)1394
                1604 3768 y Fe(N)1544 3743 y Ff(\021)g Fg(E)1777 3758 y Fe(N)1866
                1605 3743 y Fh(+)22 b Fg(\026)2023 3758 y Fe(N)2090 3743 y
                1606 Fg(\027)2138 3759 y Fe(N)7 b Fb(\000)p Fc(1)p Fe(=)p
                1607 Fc(2)2366 3743 y Fg(f)2414 3758 y Fe(N)g Fb(\000)p Fc(1)1322
                1608 3889 y Fg(D)1406 3847 y Fe(T)1544 3889 y Ff(\021)84 b(\000)p
                1609 Fg(\027)1830 3904 y Fe(N)7 b Fb(\000)p Fc(1)p Fe(=)p
                1610 Fc(2)2058 3889 y Fh(\(1)22 b Ff(\000)h Fg(e)2312 3904
                1611 y Fe(N)7 b Fb(\000)p Fc(1)2470 3889 y Fh(\))328 4109
                1612 y(The)41 b(latter)d(t)m(w)m(o)j(are)f(computed)g(separately)g(for)f
                1613 (temp)s(erature)h(and)f(for)h(moisture,)328 4229 y(resulting)32
                1614 b(in)f(\014v)m(e)j(\014elds)f(that)f(are)h(placed)f(in)g(a)h
                1615 Fg(ty)t(pe)p Fh(\()p Fg(sur)s(f)p 2577 4229 V 44 w(dif)11
                1616 b(f)p 2812 4229 V 46 w(ty)t(pe)p Fh(\).)474 4349 y Fg(\026)533
                1617 4313 y Fb(0)533 4374 y Fe(N)625 4349 y Fh(is)25 b(the)h(atmospheric)f
                1618 (time)f(step)i(\(2)f(\001)p Fg(t)g Fh(for)g(leapfrog\))f(divided)h(b)m
                1619 (y)h(the)g(mass)f(of)328 4470 y(the)i(lo)m(w)m(est)f(atmospheric)g(la)m
                1620 (y)m(er.)41 b Fg(E)1667 4434 y Fb(\003)1661 4494 y Fe(N)1755
                1621 4470 y Fh(is)26 b(the)g(incremen)m(t)g(\(of)g(temp)s(erature)g(or)g(sp)
                1622 s(eci\014c)328 4590 y(h)m(umidit)m(y\))i(in)g(the)h(lo)m(w)m(est)h
                1623 (atmospheric)e(la)m(y)m(er,)i(due)f(to)g(all)e(terms)i(computed)g
                1624 (explic-)328 4711 y(itly)-8 b(,)37 b(including)e(the)j(di\013usiv)m(e)f
                1625 (\015ux)h(at)f(the)h(top)f(of)g(this)f(la)m(y)m(er)i(\()f(but)h(not)f
                1626 (including)328 4831 y(the)f(surface)g(\015ux\),)h(corrected)f(for)f
                1627 (the)h(implicit)31 b(treatmen)m(t)k(of)g(the)h(di\013usiv)m(e)f
                1628 (\015uxes)328 4951 y(throughout)f(the)h(atmosphere)g(\(b)m(y)h(adding)e
                1629 (the)h(term)f Fg(\026)2494 4966 y Fe(N)2561 4951 y Fg(\027)2609
                1630 4967 y Fe(N)7 b Fb(\000)p Fc(1)p Fe(=)p Fc(2)2837 4951
                1631 y Fg(f)2885 4966 y Fe(N)g Fb(\000)p Fc(1)3042 4951 y
                1632 Fh(\).)50 b Fg(D)3241 4915 y Fe(T)3330 4951 y Fh(is)34
                1633 b(the)1922 5214 y(9)p eop
                1634 %%Page: 10 11
                1635 10 10 bop 328 631 a Fh(sensitivit)m(y)38 b(of)f(the)h
                1636 Fa(downwar)-5 b(d)47 b Fh(\015ux)39 b(through)f(the)g(top)g(of)f(the)i
                1637 (lo)m(w)m(est)f(mo)s(del)f(la)m(y)m(er)328 751 y(to)k(the)g(v)-5
                1638 b(alue)41 b(of)f(temp)s(eature)h(or)g(sp)s(eci\014c)h(h)m(umidit)m(y)e
                1639 (in)g(the)h(lo)m(w)m(est)h(mo)s(del)d(la)m(y)m(er,)328
                1640 872 y(taking)32 b(in)m(to)g(accoun)m(t)h(the)g(implicit)c(corrections)j
                1641 (to)h(all)d(of)i(the)h(other)g(la)m(y)m(ers.)474 992
                1642 y(The)g(calling)d(program)h(requests)j(that)e(a)g(v)-5
                1643 b(ariable)30 b(of)i Fg(ty)t(pe)p Fh(\()p Fg(sur)s(f)p
                1644 2950 992 30 4 v 44 w(dif)11 b(f)p 3185 992 V 46 w(ty)t(pe)p
                1645 Fh(\))32 b(b)s(e)328 1112 y(allo)s(cated)42 b(when)i(initializing)39
                1646 b Fg(v)t(er)s(t)p 1697 1112 V 35 w(dif)11 b(f)p 1933
                1647 1112 V 45 w(mod)44 b Fh(.)76 b(Let's)45 b(call)d(this)h(v)-5
                1648 b(ariable)42 b Fg(S)6 b(ur)s(f)11 b Fh(.)328 1233 y(Then)34
                1649 b(the)f(\014v)m(e)g(\014elds)g(output)g(from)e Fg(g)t(cm)p
                1650 1913 1233 V 35 w(v)t(er)s(t)p 2126 1233 V 35 w(dif)11
                1651 b(f)p 2362 1233 V 46 w(dow)s(n)31 b Fh(are)1150 1453
                1652 y Fg(S)6 b(ur)s(f)11 b Fh(\045)p Fg(dtmass)82 b Fh(=)h
                1653 Fg(\026)2073 1468 y Fe(N)1142 1598 y Fg(S)6 b(ur)s(f)11
                1654 b Fh(\045)p Fg(dl)r(f)g(ux)p 1709 1598 V 34 w(t)83 b
                1655 Fh(=)g Fg(D)2098 1557 y Fe(T)2153 1598 y Ff(j)2181 1613
                1656 y Fe(temper)r(atur)r(e)1131 1743 y Fg(S)6 b(ur)s(f)11
                1657 b Fh(\045)p Fg(dl)r(f)g(ux)p 1698 1743 V 33 w(q)87 b
                1658 Fh(=)c Fg(D)2098 1702 y Fe(T)2153 1743 y Ff(j)2181 1758
                1659 y Fe(specif)7 b(ic)p 2440 1758 V 33 w(humidity)1180 1889
                1660 y Fg(S)f(ur)s(f)11 b Fh(\045)p Fg(del)r(ta)p 1708 1889
                1661 V 35 w(t)83 b Fh(=)g Fg(E)2092 1848 y Fb(\003)2086 1913
                1662 y Fe(N)2154 1889 y Ff(j)2182 1904 y Fe(temper)r(atur)r(e)1169
                1663 2034 y Fg(S)6 b(ur)s(f)11 b Fh(\045)p Fg(del)r(ta)p 1697
                1664 2034 V 34 w(q)87 b Fh(=)c Fg(E)2092 1993 y Fb(\003)2086
                1665 2059 y Fe(N)2154 2034 y Ff(j)2182 2049 y Fe(specif)7
                1666 b(ic)p 2441 2049 V 33 w(humidity)474 2254 y Fh(All)25
                1667 b(surface)i(\015uxes)h(and)e(deriv)-5 b(ativ)m(es)26
                1668 b(are)h(computed)f(in)g(the)g(\015ux)h(exc)m(hange)h(mo)s(d-)328
                1669 2374 y(ule.)42 b(It)29 b(computes)h Fg(e)1100 2389 y
                1670 Fe(N)1197 2374 y Fh(and)f Fg(f)1431 2389 y Fe(N)1528
                1671 2374 y Fh(and)h(then)g Fg(\013)g Fh(and)f Fg(\014)6 b
                1672 Fh(.)42 b(The)30 b(surface)g(mo)s(dules)f(then)h(do)328
                1673 2495 y(their)c(job,)i(after)e(whic)m(h)i(the)f(\015ux)g(exc)m(hange)i
                1674 (mo)s(dule)c(computes)i(the)g(\014nal)f(\015uxes)j(and)328
                1675 2615 y(the)i(\014nal)e(incremen)m(ts)h(in)g(temp)s(erature)g(and)g
                1676 (moisture)f(for)h(the)h(lo)m(w)m(est)f(atmospheric)328
                1677 2736 y(la)m(y)m(er.)52 b(These)38 b(incremen)m(ts)d(are)h(passed)h(bac)
                1678 m(k)f(to)f(the)h(v)m(ertical)f(di\013usion)f(mo)s(dule)h(in)328
                1679 2856 y Fg(S)6 b(ur)s(f)11 b Fh(\045)p Fg(d)-16 b(f)11
                1680 b(l)r(ux)p 879 2856 V 33 w(t)37 b Fh(and)g Fg(S)6 b(ur)s(f)11
                1681 b Fh(\045)p Fg(d)-16 b(f)11 b(l)r(ux)p 1723 2856 V 34
                1682 w(q)t Fh(.)56 b(The)38 b(subroutine)f Fg(g)t(cm)p 2757
                1683 2856 V 34 w(v)t(er)s(t)p 2969 2856 V 35 w(dif)11 b(f)p
                1684 3205 2856 V 46 w(up)36 b Fh(then)328 2976 y(computes)d(the)g(incremen)m
                1685 (ts)g(for)f(all)f(of)h(the)h(other)f(atmospheric)g(la)m(y)m(ers.)1898
                1686 5214 y(10)p eop
                1687 %%Trailer
                1688 end
                1689 userdict /end-hook known{end-hook}if
                1690 %%EOF
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