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Fj(1)1362 1161 y Fr(\))e(for)h(scalar)f(functions)h(f.)41 b(The)28 b(function)e Fq( )31 b Fr(is)c(the)g(stream)324 1281 y(function.)470 1401 y(There)37 b(are)e(man)m(y)g(ph)m(ysical)g(examples)g(where)i(the) e(solutions)f(satisfy)i(the)f(equa-)324 1522 y(tions)h(ab)s(o)m(v)m(e,) j(with)e(an)g(extra)g(equation)g(or)g(op)s(erator)f(that)h(relates)g Fq(\022)j Fr(with)d(the)g(v)m(e-)324 1642 y(lo)s(cit)m(y)32 b(\014eld.)44 b(Examples)33 b(include;)g(P)m(assiv)m(e)i(scalars,)e (Unsteady)i(Prandtl)d(equations,)324 1763 y(2D)j(incompressible)h (Euler)g(equations,)i(Boussinesq,)h(2D)d(Ideal)g(Magnetoh)m(ydro)s(dy-) 324 1883 y(namics)31 b(and)i(the)g(Quasi-geostrophic)e(equation.)470 2003 y(In)49 b(the)g(literature)e(on)i(n)m(umerical)e(sim)m(ulations)f (for)i(the)h(2D)f(Ideal)g(Magneto-)324 2124 y(h)m(ydro)s(dynamics)38 b(\(MHD\))g(a)g(standard)g(candidate)g(for)f(a)h(curren)m(t)h(sheet)g (formation)324 2244 y(\(see)e(Fig.)54 b(1\))36 b(is)g(when)h(the)g(lev) m(el)f(sets)h(of)f(the)h(magnetic)e(stream)h(function)g(\()g(repre-)324 2365 y(sen)m(ted)j(in)e(\(1\))g(b)m(y)h Fq(\022)s Fr(\))g(con)m(tain)f (a)g(h)m(yp)s(erb)s(olic)f(saddle)i(\(an)f(X-p)s(oin)m(t)f (con\014guration\).)324 2485 y(The)46 b(fron)m(t)g(is)f(formed)g(when)h (the)g(h)m(yp)s(erb)s(olic)f(saddle)h(closes,)j(and)d(b)s(ecomes)g(t)m (w)m(o)324 2605 y(Y-p)s(oin)m(ts)39 b(con\014guration)h(joined)g(b)m(y) h(a)f(curren)m(t)i(sheet.)68 b(\(See)41 b(P)m(ark)m(er)h([12)o(],)h (Priest-)324 2726 y(Tito)m(v-Ric)m(k)-5 b(ard)31 b([13],)i(F)-8 b(riedel-Grauer-Marliani)27 b([10])32 b(and)h(Cordoba-Marliani)c ([8].\))1887 3444 y Fi(\000)1970 3361 y(\000)2041 3291 y(\000)1887 3055 y(@)1970 3138 y(@)2041 3208 y(@)708 b(\000)2915 3125 y(\000)2986 3055 y(\000)2832 3291 y(@)2915 3374 y(@)2986 3444 y(@)p 2124 3212 709 9 v 1391 3590 9 9 v 1387 3587 V 1383 3584 V 1379 3582 V 1375 3579 V 1371 3576 V 1366 3574 V 1362 3571 V 1358 3568 V 1354 3566 V 1350 3563 V 1346 3561 V 1342 3558 V 1337 3556 V 1333 3553 V 1329 3551 V 1325 3548 V 1321 3546 V 1317 3543 V 1312 3541 V 1308 3538 V 1304 3536 V 1300 3534 V 1296 3531 V 1292 3529 V 1288 3527 V 1283 3525 V 1279 3522 V 1275 3520 V 1271 3518 V 1267 3516 V 1263 3514 V 1258 3512 V 1254 3509 V 1250 3507 V 1246 3505 V 1242 3503 V 1238 3501 V 1234 3499 V 1229 3497 V 1225 3495 V 1221 3493 V 1217 3491 V 1213 3490 V 1209 3488 V 1204 3486 V 1200 3484 V 1196 3482 V 1192 3480 V 1188 3478 V 1184 3477 V 1180 3475 V 1175 3473 V 1171 3472 V 1167 3470 V 1163 3468 V 1159 3467 V 1155 3465 V 1150 3463 V 1146 3462 V 1142 3460 V 1138 3459 V 1134 3457 V 1130 3456 V 1126 3454 V 1121 3453 V 1117 3451 V 1113 3450 V 1109 3448 V 1105 3447 V 1101 3446 V 1096 3444 V 1092 3443 V 1088 3442 V 1084 3440 V 1080 3439 V 1076 3438 V 1072 3437 V 1067 3436 V 1063 3434 V 1059 3433 V 1055 3432 V 1051 3431 V 1047 3430 V 1042 3429 V 1038 3428 V 1034 3427 V 1030 3426 V 1026 3425 V 1022 3424 V 1018 3423 V 1013 3422 V 1009 3421 V 1005 3420 V 1001 3419 V 997 3418 V 993 3417 V 988 3417 V 984 3416 V 980 3415 V 976 3414 V 972 3414 V 968 3413 V 964 3412 V 959 3412 V 955 3411 V 951 3410 V 947 3410 V 943 3409 V 939 3409 V 935 3408 V 930 3407 V 926 3407 V 922 3406 V 918 3406 V 914 3406 V 910 3405 V 905 3405 V 901 3404 V 897 3404 V 893 3404 V 889 3403 V 885 3403 V 881 3403 V 876 3402 V 872 3402 V 868 3402 V 864 3402 V 860 3402 V 856 3401 V 851 3401 V 847 3401 V 843 3401 V 839 3401 V 835 3401 V 831 3401 V 827 3401 V 822 3401 V 818 3401 V 814 3401 V 810 3401 V 806 3401 V 802 3401 V 797 3401 V 793 3401 V 789 3402 V 785 3402 V 781 3402 V 777 3402 V 773 3402 V 768 3403 V 764 3403 V 760 3403 V 756 3404 V 752 3404 V 748 3404 V 743 3405 V 739 3405 V 735 3406 V 731 3406 V 727 3406 V 723 3407 V 719 3407 V 714 3408 V 710 3409 V 706 3409 V 702 3410 V 698 3410 V 694 3411 V 689 3412 V 685 3412 V 681 3413 V 677 3414 V 673 3414 V 669 3415 V 665 3416 V 660 3417 V 656 3417 V 652 3418 V 648 3419 V 644 3420 V 640 3421 V 635 3422 V 631 3423 V 627 3424 V 623 3425 V 619 3426 V 615 3427 V 611 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V 1371 2847 V 1366 2850 V 1362 2853 V 1358 2855 V 1354 2858 V 1350 2861 V 1346 2863 V 1342 2866 V 1337 2868 V 1333 2871 V 1329 2873 V 1325 2876 V 1321 2878 V 1317 2880 V 1312 2883 V 1308 2885 V 1304 2888 V 1300 2890 V 1296 2892 V 1292 2895 V 1288 2897 V 1283 2899 V 1279 2901 V 1275 2903 V 1271 2906 V 1267 2908 V 1263 2910 V 1258 2912 V 1254 2914 V 1250 2916 V 1246 2918 V 1242 2920 V 1238 2922 V 1234 2924 V 1229 2926 V 1225 2928 V 1221 2930 V 1217 2932 V 1213 2934 V 1209 2936 V 1204 2938 V 1200 2940 V 1196 2942 V 1192 2943 V 1188 2945 V 1184 2947 V 1180 2949 V 1175 2950 V 1171 2952 V 1167 2954 V 1163 2956 V 1159 2957 V 1155 2959 V 1150 2960 V 1146 2962 V 1142 2964 V 1138 2965 V 1134 2967 V 1130 2968 V 1126 2970 V 1121 2971 V 1117 2972 V 1113 2974 V 1109 2975 V 1105 2977 V 1101 2978 V 1096 2979 V 1092 2981 V 1088 2982 V 1084 2983 V 1080 2985 V 1076 2986 V 1072 2987 V 1067 2988 V 1063 2989 V 1059 2991 V 1055 2992 V 1051 2993 V 1047 2994 V 1042 2995 V 1038 2996 V 1034 2997 V 1030 2998 V 1026 2999 V 1022 3000 V 1018 3001 V 1013 3002 V 1009 3003 V 1005 3004 V 1001 3005 V 997 3005 V 993 3006 V 988 3007 V 984 3008 V 980 3009 V 976 3009 V 972 3010 V 968 3011 V 964 3011 V 959 3012 V 955 3013 V 951 3013 V 947 3014 V 943 3015 V 939 3015 V 935 3016 V 930 3016 V 926 3017 V 922 3017 V 918 3018 V 914 3018 V 910 3019 V 905 3019 V 901 3019 V 897 3020 V 893 3020 V 889 3020 V 885 3021 V 881 3021 V 876 3021 V 872 3022 V 868 3022 V 864 3022 V 860 3022 V 856 3022 V 851 3022 V 847 3023 V 843 3023 V 839 3023 V 835 3023 V 831 3023 V 827 3023 V 822 3023 V 818 3023 V 814 3023 V 810 3023 V 806 3023 V 802 3023 V 797 3022 V 793 3022 V 789 3022 V 785 3022 V 781 3022 V 777 3022 V 773 3021 V 768 3021 V 764 3021 V 760 3020 V 756 3020 V 752 3020 V 748 3019 V 743 3019 V 739 3019 V 735 3018 V 731 3018 V 727 3017 V 723 3017 V 719 3016 V 714 3016 V 710 3015 V 706 3015 V 702 3014 V 698 3013 V 694 3013 V 689 3012 V 685 3011 V 681 3011 V 677 3010 V 673 3009 V 669 3009 V 665 3008 V 660 3007 V 656 3006 V 652 3005 V 648 3005 V 644 3004 V 640 3003 V 635 3002 V 631 3001 V 627 3000 V 623 2999 V 619 2998 V 615 2997 V 611 2996 V 606 2995 V 602 2994 V 598 2993 V 594 2992 V 590 2991 V 586 2989 V 581 2988 V 577 2987 V 573 2986 V 569 2985 V 565 2983 V 561 2982 V 557 2981 V 552 2979 V 548 2978 V 544 2977 V 540 2975 V 536 2974 V 532 2972 V 527 2971 V 523 2970 V 519 2968 V 515 2967 V 511 2965 V 507 2964 V 503 2962 V 498 2960 V 494 2959 V 490 2957 V 486 2956 V 482 2954 V 478 2952 V 473 2950 V 469 2949 V 465 2947 V 461 2945 V 457 2943 V 453 2942 V 449 2940 V 444 2938 V 440 2936 V 436 2934 V 432 2932 V 428 2930 V 424 2928 V 420 2926 V 415 2924 V 411 2922 V 407 2920 V 403 2918 V 399 2916 V 395 2914 V 390 2912 V 386 2910 V 382 2908 V 378 2906 V 374 2903 V 370 2901 V 366 2899 V 361 2897 V 357 2895 V 353 2892 V 349 2890 V 345 2888 V 341 2885 V 336 2883 V 332 2880 V 328 2878 V 324 2876 V 320 2873 V 316 2871 V 312 2868 V 307 2866 V 303 2863 V 299 2861 V 295 2858 V 291 2855 V 287 2853 V 282 2850 V 278 2848 V 274 2845 V 270 2842 V 266 2839 V 262 2837 V 258 2834 V 1116 3776 a Fr(Fig.)42 b(1.)h(Lev)m(el)33 b(curv)m(es)i(of)d Fq(\022)s Fr(.)706 3713 y Fq(t)c(<)f(T)2359 3714 y(t)h Fr(=)g Fq(T)p 1415 3540 4 4 v 1413 3539 V 1410 3537 V 1408 3536 V 1406 3535 V 1403 3533 V 1401 3532 V 1399 3531 V 1396 3529 V 1394 3528 V 1392 3527 V 1390 3526 V 1387 3524 V 1385 3523 V 1383 3522 V 1381 3520 V 1378 3519 V 1376 3518 V 1374 3516 V 1372 3515 V 1370 3514 V 1367 3512 V 1365 3511 V 1363 3510 V 1361 3508 V 1359 3507 V 1357 3506 V 1355 3504 V 1353 3503 V 1350 3502 V 1348 3500 V 1346 3499 V 1344 3498 V 1342 3496 V 1340 3495 V 1338 3494 V 1336 3492 V 1334 3491 V 1332 3490 V 1330 3488 V 1328 3487 V 1326 3486 V 1324 3485 V 1322 3483 V 1320 3482 V 1318 3481 V 1316 3479 V 1314 3478 V 1312 3477 V 1311 3475 V 1309 3474 V 1307 3473 V 1305 3471 V 1303 3470 V 1301 3469 V 1299 3467 V 1298 3466 V 1296 3465 V 1294 3463 V 1292 3462 V 1290 3461 V 1288 3459 V 1287 3458 V 1285 3457 V 1283 3455 V 1281 3454 V 1280 3453 V 1278 3451 V 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3340 V 1165 3339 V 1164 3338 V 1163 3336 V 1162 3335 V 1161 3334 V 1161 3332 V 1160 3331 V 1159 3330 V 1158 3328 V 1157 3327 V 1156 3326 V 1155 3324 V 1155 3323 V 1154 3322 V 1153 3320 V 1152 3319 V 1151 3318 V 1151 3317 V 1150 3315 V 1149 3314 V 1148 3313 V 1148 3311 V 1147 3310 V 1146 3309 V 1146 3307 V 1145 3306 V 1144 3305 V 1144 3303 V 1143 3302 V 1142 3301 V 1142 3299 V 1141 3298 V 1140 3297 V 1140 3295 V 1139 3294 V 1138 3293 V 1138 3291 V 1137 3290 V 1137 3289 V 1136 3287 V 1136 3286 V 1135 3285 V 1135 3283 V 1134 3282 V 1133 3281 V 1133 3279 V 1133 3278 V 1132 3277 V 1132 3276 V 1131 3274 V 1131 3273 V 1130 3272 V 1130 3270 V 1129 3269 V 1129 3268 V 1128 3266 V 1128 3265 V 1128 3264 V 1127 3262 V 1127 3261 V 1127 3260 V 1126 3258 V 1126 3257 V 1126 3256 V 1125 3254 V 1125 3253 V 1125 3252 V 1124 3250 V 1124 3249 V 1124 3248 V 1123 3246 V 1123 3245 V 1123 3244 V 1123 3242 V 1122 3241 V 1122 3240 V 1122 3238 V 1122 3237 V 1122 3236 V 1121 3235 V 1121 3233 V 1121 3232 V 1121 3231 V 1121 3229 V 1121 3228 V 1121 3227 V 1120 3225 V 1120 3224 V 1120 3223 V 1120 3221 V 1120 3220 V 1120 3219 V 1120 3217 V 1120 3216 V 1120 3215 V 1120 3213 V 1120 3212 V 1120 3211 V 1120 3209 V 1120 3208 V 1120 3207 V 1120 3205 V 1120 3204 V 1120 3203 V 1120 3201 V 1120 3200 V 1120 3199 V 1120 3197 V 1120 3196 V 1120 3195 V 1120 3193 V 1121 3192 V 1121 3191 V 1121 3190 V 1121 3188 V 1121 3187 V 1121 3186 V 1121 3184 V 1122 3183 V 1122 3182 V 1122 3180 V 1122 3179 V 1122 3178 V 1123 3176 V 1123 3175 V 1123 3174 V 1123 3172 V 1124 3171 V 1124 3170 V 1124 3168 V 1125 3167 V 1125 3166 V 1125 3164 V 1126 3163 V 1126 3162 V 1126 3160 V 1127 3159 V 1127 3158 V 1127 3156 V 1128 3155 V 1128 3154 V 1128 3152 V 1129 3151 V 1129 3150 V 1130 3149 V 1130 3147 V 1131 3146 V 1131 3145 V 1132 3143 V 1132 3142 V 1132 3141 V 1133 3139 V 1134 3138 V 1134 3137 V 1135 3135 V 1135 3134 V 1136 3133 V 1136 3131 V 1137 3130 V 1137 3129 V 1138 3127 V 1138 3126 V 1139 3125 V 1140 3123 V 1140 3122 V 1141 3121 V 1142 3119 V 1142 3118 V 1143 3117 V 1144 3115 V 1144 3114 V 1145 3113 V 1146 3111 V 1146 3110 V 1147 3109 V 1148 3108 V 1148 3106 V 1149 3105 V 1150 3104 V 1151 3102 V 1151 3101 V 1152 3100 V 1153 3098 V 1154 3097 V 1155 3096 V 1155 3094 V 1156 3093 V 1157 3092 V 1158 3090 V 1159 3089 V 1160 3088 V 1161 3086 V 1161 3085 V 1162 3084 V 1163 3082 V 1164 3081 V 1165 3080 V 1166 3078 V 1167 3077 V 1168 3076 V 1169 3074 V 1170 3073 V 1171 3072 V 1172 3070 V 1173 3069 V 1174 3068 V 1175 3067 V 1176 3065 V 1177 3064 V 1178 3063 V 1179 3061 V 1180 3060 V 1181 3059 V 1182 3057 V 1183 3056 V 1184 3055 V 1185 3053 V 1187 3052 V 1188 3051 V 1189 3049 V 1190 3048 V 1191 3047 V 1192 3045 V 1194 3044 V 1195 3043 V 1196 3041 V 1197 3040 V 1198 3039 V 1200 3037 V 1201 3036 V 1202 3035 V 1203 3033 V 1205 3032 V 1206 3031 V 1207 3029 V 1208 3028 V 1210 3027 V 1211 3026 V 1212 3024 V 1214 3023 V 1215 3022 V 1216 3020 V 1218 3019 V 1219 3018 V 1220 3016 V 1222 3015 V 1223 3014 V 1225 3012 V 1226 3011 V 1227 3010 V 1229 3008 V 1230 3007 V 1232 3006 V 1233 3004 V 1235 3003 V 1236 3002 V 1238 3000 V 1239 2999 V 1241 2998 V 1242 2996 V 1244 2995 V 1245 2994 V 1247 2992 V 1248 2991 V 1250 2990 V 1251 2988 V 1253 2987 V 1255 2986 V 1256 2984 V 1258 2983 V 1259 2982 V 1261 2981 V 1263 2979 V 1264 2978 V 1266 2977 V 1268 2975 V 1269 2974 V 1271 2973 V 1273 2971 V 1275 2970 V 1276 2969 V 1278 2967 V 1280 2966 V 1281 2965 V 1283 2963 V 1285 2962 V 1287 2961 V 1288 2959 V 1290 2958 V 1292 2957 V 1294 2955 V 1296 2954 V 1297 2953 V 1299 2951 V 1301 2950 V 1303 2949 V 1305 2947 V 1307 2946 V 1309 2945 V 1311 2943 V 1312 2942 V 1314 2941 V 1316 2940 V 1318 2938 V 1320 2937 V 1322 2936 V 1324 2934 V 1326 2933 V 1328 2932 V 1330 2930 V 1332 2929 V 1334 2928 V 1336 2926 V 1338 2925 V 1340 2924 V 1342 2922 V 1344 2921 V 1346 2920 V 1348 2918 V 1350 2917 V 1353 2916 V 1355 2914 V 1357 2913 V 1359 2912 V 1361 2910 V 1363 2909 V 1365 2908 V 1367 2906 V 1370 2905 V 1372 2904 V 1374 2902 V 1376 2901 V 1378 2900 V 1381 2899 V 1383 2897 V 1385 2896 V 1387 2895 V 1390 2893 V 1392 2892 V 1394 2891 V 1396 2889 V 1399 2888 V 1401 2887 V 1403 2885 V 1406 2884 V 1408 2883 V 1410 2881 V 1413 2880 V 1415 2879 V 234 3540 V 236 3539 V 239 3537 V 241 3536 V 243 3535 V 246 3533 V 248 3532 V 250 3531 V 252 3529 V 255 3528 V 257 3527 V 259 3526 V 262 3524 V 264 3523 V 266 3522 V 268 3520 V 270 3519 V 273 3518 V 275 3516 V 277 3515 V 279 3514 V 281 3512 V 284 3511 V 286 3510 V 288 3508 V 290 3507 V 292 3506 V 294 3504 V 296 3503 V 298 3502 V 301 3500 V 303 3499 V 305 3498 V 307 3496 V 309 3495 V 311 3494 V 313 3492 V 315 3491 V 317 3490 V 319 3488 V 321 3487 V 323 3486 V 325 3485 V 327 3483 V 329 3482 V 331 3481 V 333 3479 V 334 3478 V 336 3477 V 338 3475 V 340 3474 V 342 3473 V 344 3471 V 346 3470 V 348 3469 V 350 3467 V 351 3466 V 353 3465 V 355 3463 V 357 3462 V 359 3461 V 360 3459 V 362 3458 V 364 3457 V 366 3455 V 367 3454 V 369 3453 V 371 3451 V 373 3450 V 374 3449 V 376 3447 V 378 3446 V 379 3445 V 381 3444 V 383 3442 V 384 3441 V 386 3440 V 388 3438 V 389 3437 V 391 3436 V 393 3434 V 394 3433 V 396 3432 V 397 3430 V 399 3429 V 401 3428 V 402 3426 V 404 3425 V 405 3424 V 407 3422 V 408 3421 V 410 3420 V 411 3418 V 413 3417 V 414 3416 V 416 3414 V 417 3413 V 419 3412 V 420 3410 V 421 3409 V 423 3408 V 424 3406 V 426 3405 V 427 3404 V 428 3403 V 430 3401 V 431 3400 V 433 3399 V 434 3397 V 435 3396 V 437 3395 V 438 3393 V 439 3392 V 440 3391 V 442 3389 V 443 3388 V 444 3387 V 446 3385 V 447 3384 V 448 3383 V 449 3381 V 451 3380 V 452 3379 V 453 3377 V 454 3376 V 455 3375 V 457 3373 V 458 3372 V 459 3371 V 460 3369 V 461 3368 V 462 3367 V 463 3365 V 464 3364 V 466 3363 V 467 3361 V 468 3360 V 469 3359 V 470 3358 V 471 3356 V 472 3355 V 473 3354 V 474 3352 V 475 3351 V 476 3350 V 477 3348 V 478 3347 V 479 3346 V 480 3344 V 481 3343 V 482 3342 V 483 3340 V 484 3339 V 485 3338 V 486 3336 V 487 3335 V 487 3334 V 488 3332 V 489 3331 V 490 3330 V 491 3328 V 492 3327 V 493 3326 V 493 3324 V 494 3323 V 495 3322 V 496 3320 V 497 3319 V 497 3318 V 498 3317 V 499 3315 V 500 3314 V 500 3313 V 501 3311 V 502 3310 V 503 3309 V 503 3307 V 504 3306 V 505 3305 V 505 3303 V 506 3302 V 507 3301 V 507 3299 V 508 3298 V 509 3297 V 509 3295 V 510 3294 V 510 3293 V 511 3291 V 512 3290 V 512 3289 V 513 3287 V 513 3286 V 514 3285 V 514 3283 V 515 3282 V 515 3281 V 516 3279 V 516 3278 V 517 3277 V 517 3276 V 518 3274 V 518 3273 V 519 3272 V 519 3270 V 520 3269 V 520 3268 V 520 3266 V 521 3265 V 521 3264 V 522 3262 V 522 3261 V 522 3260 V 523 3258 V 523 3257 V 523 3256 V 524 3254 V 524 3253 V 524 3252 V 525 3250 V 525 3249 V 525 3248 V 525 3246 V 526 3245 V 526 3244 V 526 3242 V 526 3241 V 527 3240 V 527 3238 V 527 3237 V 527 3236 V 527 3235 V 528 3233 V 528 3232 V 528 3231 V 528 3229 V 528 3228 V 528 3227 V 528 3225 V 529 3224 V 529 3223 V 529 3221 V 529 3220 V 529 3219 V 529 3217 V 529 3216 V 529 3215 V 529 3213 V 529 3212 V 529 3211 V 529 3209 V 529 3208 V 529 3207 V 529 3205 V 529 3204 V 529 3203 V 529 3201 V 529 3200 V 529 3199 V 529 3197 V 529 3196 V 529 3195 V 528 3193 V 528 3192 V 528 3191 V 528 3190 V 528 3188 V 528 3187 V 528 3186 V 527 3184 V 527 3183 V 527 3182 V 527 3180 V 527 3179 V 526 3178 V 526 3176 V 526 3175 V 526 3174 V 525 3172 V 525 3171 V 525 3170 V 525 3168 V 524 3167 V 524 3166 V 524 3164 V 523 3163 V 523 3162 V 523 3160 V 522 3159 V 522 3158 V 522 3156 V 521 3155 V 521 3154 V 520 3152 V 520 3151 V 520 3150 V 519 3149 V 519 3147 V 518 3146 V 518 3145 V 517 3143 V 517 3142 V 516 3141 V 516 3139 V 515 3138 V 515 3137 V 514 3135 V 514 3134 V 513 3133 V 513 3131 V 512 3130 V 512 3129 V 511 3127 V 510 3126 V 510 3125 V 509 3123 V 509 3122 V 508 3121 V 507 3119 V 507 3118 V 506 3117 V 505 3115 V 505 3114 V 504 3113 V 503 3111 V 503 3110 V 502 3109 V 501 3108 V 500 3106 V 500 3105 V 499 3104 V 498 3102 V 497 3101 V 497 3100 V 496 3098 V 495 3097 V 494 3096 V 493 3094 V 493 3093 V 492 3092 V 491 3090 V 490 3089 V 489 3088 V 488 3086 V 487 3085 V 487 3084 V 486 3082 V 485 3081 V 484 3080 V 483 3078 V 482 3077 V 481 3076 V 480 3074 V 479 3073 V 478 3072 V 477 3070 V 476 3069 V 475 3068 V 474 3067 V 473 3065 V 472 3064 V 471 3063 V 470 3061 V 469 3060 V 468 3059 V 467 3057 V 466 3056 V 464 3055 V 463 3053 V 462 3052 V 461 3051 V 460 3049 V 459 3048 V 458 3047 V 457 3045 V 455 3044 V 454 3043 V 453 3041 V 452 3040 V 451 3039 V 449 3037 V 448 3036 V 447 3035 V 446 3033 V 444 3032 V 443 3031 V 442 3029 V 440 3028 V 439 3027 V 438 3026 V 437 3024 V 435 3023 V 434 3022 V 433 3020 V 431 3019 V 430 3018 V 428 3016 V 427 3015 V 426 3014 V 424 3012 V 423 3011 V 421 3010 V 420 3008 V 419 3007 V 417 3006 V 416 3004 V 414 3003 V 413 3002 V 411 3000 V 410 2999 V 408 2998 V 407 2996 V 405 2995 V 404 2994 V 402 2992 V 401 2991 V 399 2990 V 397 2988 V 396 2987 V 394 2986 V 393 2984 V 391 2983 V 389 2982 V 388 2981 V 386 2979 V 384 2978 V 383 2977 V 381 2975 V 379 2974 V 378 2973 V 376 2971 V 374 2970 V 373 2969 V 371 2967 V 369 2966 V 367 2965 V 366 2963 V 364 2962 V 362 2961 V 360 2959 V 359 2958 V 357 2957 V 355 2955 V 353 2954 V 351 2953 V 350 2951 V 348 2950 V 346 2949 V 344 2947 V 342 2946 V 340 2945 V 338 2943 V 336 2942 V 334 2941 V 333 2940 V 331 2938 V 329 2937 V 327 2936 V 325 2934 V 323 2933 V 321 2932 V 319 2930 V 317 2929 V 315 2928 V 313 2926 V 311 2925 V 309 2924 V 307 2922 V 305 2921 V 303 2920 V 301 2918 V 298 2917 V 296 2916 V 294 2914 V 292 2913 V 290 2912 V 288 2910 V 286 2909 V 284 2908 V 281 2906 V 279 2905 V 277 2904 V 275 2902 V 273 2901 V 270 2900 V 268 2899 V 266 2897 V 264 2896 V 262 2895 V 259 2893 V 257 2892 V 255 2891 V 253 2889 V 250 2888 V 248 2887 V 246 2885 V 243 2884 V 241 2883 V 239 2881 V 236 2880 V 234 2879 V 234 3562 a Fi(")317 3512 y(")400 3462 y(")483 3413 y(")566 3363 y(")649 3313 y(")732 3263 y(")815 3213 y(")898 3164 y(")981 3114 y(")1064 3064 y(")1147 3014 y(")1230 2964 y(")1313 2914 y(")1332 2903 y(")-1181 b(b)317 2953 y(b)400 3003 y(b)483 3053 y(b)566 3102 y(b)649 3152 y(b)732 3202 y(b)815 3252 y(b)898 3302 y(b)981 3352 y(b)1064 3401 y(b)1147 3451 y(b)1230 3501 y(b)1313 3551 y(b)1332 3562 y(b)824 3208 y Fh(r)741 2973 y Fr(\000)802 2988 y Fp(+)741 3493 y Fr(\000)802 3508 y Fk(\000)2447 3373 y Fr(\000)470 3919 y(The)46 b(same)e (con\014guration)g(w)m(as)h(observ)m(ed)i(in)d(n)m(umerical)f(sim)m (ulations)f(for)i(the)324 4039 y(Quasi-geostrophic)21 b(equation)h(\(QG\).)f(In)i(this)f(case)h(the)g(geometry)f(of)g(the)h (lev)m(el)f(sets)h(of)324 4159 y(the)f(temp)s(erature)g(has)g(a)g(h)m (yp)s(erb)s(olic)f(structure)i(\(See)g(Constan)m(tin-Ma)5 b(jda-T)-8 b(abak)22 b([4)o(],)324 4280 y(Okhitani-Y)-8 b(amada)18 b([11],)24 b(Cordoba)e([6])g(and)f(Constan)m(tin-Nie-Sc)m (horghofer)h([5)o(]\).)40 b(The)324 4400 y(QG)e(literature)g(discusses) j(X-p)s(oin)m(ts,)f(but)f(not)g(Y-p)s(oin)m(ts.)62 b(In)39 b(the)h(case)g(of)e(Boussi-)324 4521 y(nesq)32 b(there)f(is)f(no)g(men) m(tion,)g(on)g(an)m(y)h(n)m(umerical)e(sim)m(ulation)f(study)-8 b(,)32 b(that)e(a)g(p)s(ossible)324 4641 y(singularit)m(y)25 b(is)h(due)h(to)f(the)h(closing)f(of)g(a)g(h)m(yp)s(erb)s(olic)g (saddle.)42 b(In)27 b(the)g(w)m(ork)g(of)f(Pumir-)324 4761 y(Siggia)21 b([14])j(there)h(has)f(b)s(een)h(observ)m(ed)h (evidence)f(for)e(a)h(formation)e(of)h(a)h(fron)m(t)g(in)f(\014nite)324 4882 y(time,)28 b(across)i(whic)m(h)g Fq(\022)i Fr(v)-5 b(aries)29 b(dramatically)-8 b(,)27 b(on)i(a)g(cap)g(of)g(a)g (symmetric)f(rising)g(bub-)324 5002 y(ble.)64 b(E-Sh)m(u)40 b([9])f(p)s(erformed)g(n)m(umerical)g(sim)m(ulations)e(with)i(the)h (same)f(initial)d(data)1918 5251 y(2)p eop %%Page: 3 3 3 2 bop 324 548 a Fr(as)43 b(in)85 b([14],)45 b(whic)m(h)f(suggest)f (that)g(the)g(thic)m(kness)i(of)e(the)g(bubble)g(decreases)i(only)324 668 y(exp)s(onen)m(tially)-8 b(.)470 789 y(The)34 b(equations)e(for)g (MHD,)h(QG)f(and)h(Boussinesq)h(are)e(as)h(follo)m(ws)470 909 y Fg(MHD)p Fr(:)1287 1129 y(\()p Fq(@)1376 1144 y Fl(t)1428 1129 y Fr(+)22 b Fq(u)g Fo(\001)f(r)p Fr(\))p Fq(\022)86 b Fr(=)d(0)1271 1274 y(\()p Fq(@)1360 1289 y Fl(t)1412 1274 y Fr(+)22 b Fq(u)f Fo(\001)h(r)p Fr(\))p Fq(!)86 b Fr(=)d Fo(r)2147 1233 y Fk(?)2206 1274 y Fq(\022)26 b Fo(\001)c(r)p Fr(\(\001)p Fq(\022)s Fr(\))1767 1420 y Fq(u)82 b Fr(=)h Fo(r)2147 1379 y Fk(?)2206 1420 y Fq( )324 1640 y Fr(and)32 b(initial)c(conditions)j Fq(\022)s Fr(\()p Fq(x;)17 b Fr(0\))28 b(=)g Fq(\022)1709 1655 y Fp(0)1780 1640 y Fr(and)33 b Fq(u)p Fr(\()p Fq(x;)17 b Fr(0\))27 b(=)g Fq(u)2436 1655 y Fp(0)2475 1640 y Fr(.)43 b(The)33 b Fo(r)2828 1604 y Fk(?)2887 1640 y Fq(\022)j Fr(represen)m(ts)e(the)324 1760 y(magnetic)24 b(\014eld,)j(\001)p Fq(\022)h Fr(represen)m(ts)g(the)e(curren)m(t)h(densit)m(y)f(and)g Fq(!)31 b Fr(=)c Fo(\000)p Fr(\001)p Fq( )j Fr(the)c(v)m(orticit)m(y)-8 b(.)470 1880 y Fg(QG)p Fr(:)1502 2100 y(\()p Fq(@)1591 2115 y Fl(t)1643 2100 y Fr(+)22 b Fq(u)g Fo(\001)f(r)p Fr(\))p Fq(\022)31 b Fr(=)d(0)1204 2334 y Fq(u)f Fr(=)h Fo(r)1474 2293 y Fk(?)1533 2334 y Fq( )102 b(w)s(her)s(e)97 b(\022)31 b Fr(=)c Fo(\000)p Fr(\()p Fo(\0004)p Fr(\))2569 2265 y Fj(1)p 2570 2277 31 3 v 2570 2319 a(2)2614 2334 y Fq( )324 2508 y Fr(and)32 b(initial)d(condition)i Fq(\022)s Fr(\()p Fq(x;)17 b Fr(0\))28 b(=)g Fq(\022)1672 2523 y Fp(0)1711 2508 y Fr(.)44 b(The)33 b(temp)s(erature)g(is)f(represen)m (ted)j(b)m(y)f Fq(\022)s Fr(.)470 2628 y Fg(Boussinesq)p Fr(:)1427 2848 y(\()p Fq(@)1516 2863 y Fl(t)1568 2848 y Fr(+)22 b Fq(u)g Fo(\001)g(r)1877 2863 y Fl(x)1921 2848 y Fr(\))16 b Fq(\022)86 b Fr(=)d(0)1411 2994 y(\()p Fq(@)1500 3009 y Fl(t)1552 2994 y Fr(+)22 b Fq(u)g Fo(\001)g(r)1861 3009 y Fl(x)1905 2994 y Fr(\))16 b Fq(!)86 b Fr(=)d Fo(\000)p Fq(\022)2387 3009 y Fl(x)2427 3018 y Fj(1)1968 3139 y Fq(u)f Fr(=)h Fo(r)2348 3098 y Fk(?)2407 3139 y Fq( )324 3359 y Fr(Again,)31 b Fq(\022)36 b Fr(and)d(u)f(are)h(sp)s(eci\014ed)g (at)g(time)e(t=0.)324 3692 y Fs(3)161 b(Criterion)324 3911 y Fr(A)25 b(singularit)m(y)f(can)h(b)s(e)h(formed)e(b)m(y)i (collision)c(of)j(t)m(w)m(o)h(particle)e(tra)5 b(jectories.)41 b(A)26 b(tra)5 b(jec-)324 4031 y(tory)32 b(X\(q,t\))h(is)g(obtain)e(b)m (y)j(solving)d(the)i(follo)m(wing)d(ordinary)i(di\013eren)m(tial)e (equation)1401 4239 y Fq(dX)8 b Fr(\()p Fq(q)t(;)17 b(t)p Fr(\))p 1401 4283 342 4 v 1528 4374 a Fq(dt)1835 4306 y Fr(=)83 b Fq(u)p Fr(\()p Fq(X)8 b Fr(\()p Fq(q)t(;)17 b(t)p Fr(\))p Fq(;)g(t)p Fr(\))1448 4482 y Fq(X)8 b Fr(\()p Fq(q)t(;)17 b Fr(0\))82 b(=)h Fq(q)324 4702 y Fr(Therefore,)936 4922 y(\()p Fq(X)8 b Fr(\()p Fq(q)t(;)17 b(t)p Fr(\))k Fo(\000)i Fq(X)8 b Fr(\()p Fq(p;)17 b(t)p Fr(\)\))1717 4937 y Fl(t)1774 4922 y Fo(\024)28 b(j)p Fq(X)8 b Fr(\()p Fq(q)t(;)17 b(t)p Fr(\))k Fo(\000)i Fq(X)8 b Fr(\()p Fq(p;)17 b(t)p Fr(\))p Fo(jjr)p Fq(u)p Fo(j)2835 4937 y Fl(L)2883 4918 y Ff(1)1918 5251 y Fr(3)p eop %%Page: 4 4 4 3 bop 842 553 a Fo(j)p Fq(X)8 b Fr(\()p Fq(q)t(;)17 b(t)p Fr(\))k Fo(\000)i Fq(X)8 b Fr(\()p Fq(p;)17 b(t)p Fr(\))p Fo(j)26 b(\025)j(j)p Fq(X)8 b Fr(\()p Fq(q)t(;)17 b Fr(0\))j Fo(\000)j Fq(X)8 b Fr(\()p Fq(p;)17 b Fr(0\))p Fo(j)p Fq(e)2568 512 y Fk(\000)2634 458 y Fe(R)2681 479 y Fd(t)2667 536 y Fj(0)2720 512 y Fk(jr)p Fl(u)p Fk(j)2860 525 y Fd(L)2902 511 y Ff(1)2971 512 y Fl(ds)324 725 y Fr(By)24 b(this)f(trivial)e(argumen)m(t;)26 b(in)d(order)h(to)f(ha)m(v) m(e)i(a)e(collision)d(the)k(quan)m(tit)m(y)3041 645 y Fc(R)3108 671 y Fl(t)3088 760 y Fp(0)3154 725 y Fo(jr)p Fq(u)p Fo(j)3349 740 y Fl(L)3397 721 y Ff(1)3465 725 y Fq(ds)324 845 y Fr(has)33 b(to)f(div)m(erge.)470 966 y(A)k(classic)f(criterion)f(for)h(formation)e(of)i(singularities)d(in)j (\015uid)g(\015o)m(ws)i(is)e(the)h(the-)324 1086 y(orem)h(of)g (Beale-Kato-Ma)5 b(jda)36 b(\(BKM\);)i(\(see)76 b([1]\),)39 b(whic)m(h)f(impro)m(v)m(es)g(the)g(estimate)324 1206 y(describ)s(ed)k(ab)s(o)m(v)m(e,)k(and)41 b(deals)h(with)g(arbitrary)e (singularities,)i(not)g(just)g(collisions.)324 1327 y(Analogues)e(of)g (the)h(BKM)g(theorem)g(for)f(the)h(ab)s(o)m(v)m(e)g(2-dimensional)d (equations)j(in-)324 1447 y(clude)32 b(the)i(follo)m(wing)29 b(results)470 1568 y(F)-8 b(or)27 b Fg(MHD)p Fr(,a)h(singularit)m(y)f (cannot)h(dev)m(elop)h(at)f(a)g(\014nite)g(time)e(T,)j(unless)g(w)m(e)g (ha)m(v)m(e)960 1708 y Fc(Z)1059 1734 y Fl(T)1015 1934 y Fp(0)1131 1844 y Fq(sup)1282 1859 y Fl(x)1325 1844 y Fo(j)p Fq(!)t Fr(\()p Fq(x;)17 b(t)p Fr(\))p Fo(j)k Fr(+)h Fq(sup)1926 1859 y Fl(x)1970 1844 y Fo(j4)2087 1859 y Fl(x)2130 1844 y Fq(\022)s Fr(\()p Fq(x;)17 b(t)p Fr(\))p Fo(j)p Fq(dt)27 b Fr(=)h Fo(1)p Fq(;)324 2111 y Fr(where)34 b Fq(!)h Fr(denotes)f(the)f(v)m(orticit)m(y)-8 b(.\(See)33 b(Ca\015isc)m(h-Klapp)s(er-Steele)f([2].\))470 2231 y(F)-8 b(or)32 b Fg(QG)p Fr(,)h(a)f(singularit)m(y)f(cannot)i(dev) m(elop)g(at)f(a)g(\014nite)h(time)e(T,)i(unless)g(w)m(e)h(ha)m(v)m(e) 1285 2371 y Fc(Z)1384 2398 y Fl(T)1340 2597 y Fp(0)1456 2507 y Fq(sup)1607 2522 y Fl(x)1650 2507 y Fo(jr)1761 2522 y Fl(x)1805 2507 y Fq(\022)s Fr(\()p Fq(x;)17 b(t)p Fr(\))p Fo(j)p Fq(dt)27 b Fr(=)h Fo(1)p Fq(;)324 2774 y Fr(\(See)33 b(Constan)m(tin-Ma)5 b(jda-T)-8 b(abak)33 b([4]\).)470 2894 y(F)-8 b(or)32 b Fg(Boussinesq)p Fr(,)h(if)e(a)i (singularit)m(y)d(dev)m(elops)k(at)e(a)h(\014nite)f(time)f(T)i(then)499 3035 y Fc(Z)598 3061 y Fl(T)554 3260 y Fp(0)670 3170 y Fq(sup)821 3185 y Fl(x)864 3170 y Fo(j)p Fq(!)t Fr(\()p Fq(x;)17 b(t)p Fr(\))p Fo(j)p Fq(dt)27 b Fr(=)g Fo(1)65 b Fq(and)1818 3035 y Fc(Z)1917 3061 y Fl(T)1873 3260 y Fp(0)1989 3035 y Fc(Z)2088 3061 y Fl(t)2044 3260 y Fp(0)2135 3170 y Fq(sup)2286 3185 y Fl(x)2329 3170 y Fo(jr)2440 3185 y Fl(x)2484 3170 y Fq(\022)s Fr(\()p Fq(x;)17 b(s)p Fr(\))p Fo(j)p Fq(dsdt)27 b Fr(=)g Fo(1)p Fq(:)324 3437 y Fr(\(See)33 b(E-Sh)m(u)g([9].\))470 3557 y(See)23 b(also)d(Constan)m(tin-Ma)5 b(jda-T)-8 b(abak)22 b([4])g(and)g(Constan)m(tin-F)-8 b(e\013erman-Ma)5 b(jda)21 b([3])324 3678 y(for)30 b(other)g(conditions)g(in)m(v)m(olving)f (direction)g(\014elds,)i(that)f(rule)g(out)h(formation)c(of)j(sin-)324 3798 y(gularities)g(in)i(\015uids.)470 3919 y(In)43 b(the)h(case)g(of)e (2D)g(Euler,)k(a)c(singularit)m(y)f(cannot)i(dev)m(elop)h(at)f(a)f (\014nite)h(time.)324 4039 y(F)-8 b(rom)37 b(the)j(BKM)f(viewp)s(oin)m (t)f(this)h(follo)m(ws)f(from)f(the)j(fact)e(that)h Fq(!)j Fr(is)d(adv)m(ected)i(b)m(y)324 4159 y(the)33 b(\015uid,)f(and)h (therefore)g Fq(sup)1491 4174 y Fl(x)1534 4159 y Fo(j)p Fq(!)t Fr(\()p Fq(x;)17 b(t)p Fr(\))p Fo(j)32 b Fr(is)g(indep)s(enden)m (t)i(of)e(t.)43 b(\(See)33 b(BKM)66 b([1].\))470 4280 y(Instead)28 b(of)e(lo)s(oking)e(at)i(particle)f(tra)5 b(jectories)27 b(w)m(e)g(lo)s(ok)e(at)h(lev)m(el)h(curv)m(es.)43 b(Because)324 4400 y(the)g(scalar)g(function)f Fq(\022)47 b Fr(is)42 b(con)m(v)m(ected)k(b)m(y)e(the)f(\015o)m(w,)k(that)c (implies)e(that)h(the)i(lev)m(el)324 4521 y(curv)m(es)f(are)e(transp)s (orted)g(b)m(y)h(the)g(\015o)m(w.)69 b(A)41 b(p)s(ossible)f(singular)g (scenario)h(is)f(due)i(to)324 4641 y(lev)m(el)27 b(curv)m(es)j(approac) m(hing)d(eac)m(h)i(other)f(v)m(ery)i(fast)d(whic)m(h)i(will)c(lead)i (to)h(a)g(fast)f(gro)m(wth)324 4761 y(of)34 b(the)i(gradien)m(t)f(of)f (the)i(scalar)f(function.)50 b(In)36 b(this)f(pap)s(er)g(w)m(e)h (presen)m(t)h(a)e(v)-5 b(arian)m(t)34 b(of)324 4882 y(the)26 b(BKM)g(criterion)f(for)g(sharp)i(fron)m(t)e(formation.)39 b(W)-8 b(e)26 b(pro)m(vide)h(a)e(test)i(for)e(n)m(umerical)324 5002 y(sim)m(ulations)31 b(that)i(app)s(ear)h(to)f(dev)m(elop)h(sharp)g (fron)m(ts.)47 b(The)34 b(BKM)g(Theorem)g(sho)m(ws)1918 5251 y(4)p eop %%Page: 5 5 5 4 bop 324 548 a Fr(that)31 b(the)h(v)m(orticit)m(y)f(gro)m(ws)i (large)d(if)h(an)m(y)h(singularit)m(y)e(forms;)h(our)h(Theorem)g(1)f (sho)m(ws)324 668 y(that)h(the)h(v)m(elo)s(cit)m(y)g(gro)m(ws)g(large)e (if)h(a)g(sharp)h(fron)m(t)f(forms.)470 789 y(The)i(theorem)e(w)m(e)i (presen)m(t)g(in)e(this)g(pap)s(er)h(w)m(as)g(announced)h(in)e([7].)324 1119 y Fs(4)161 b(Sharp)53 b(F)-13 b(ron)l(ts)324 1338 y Fr(The)26 b(scalar)e(function)g Fq(\022)k Fr(is)c(con)m(v)m(ected)k (b)m(y)e(the)f(\015o)m(w,)i(therefore)f(the)f(lev)m(el)f(curv)m(es)j (mo)m(v)m(e)324 1458 y(with)g(the)i(\015o)m(w.)43 b(A)28 b(sharp)g(fron)m(t)g(forms)g(when)h(t)m(w)m(o)g(of)e(these)j(lev)m(el)d (curv)m(es)k(collapse)c(on)324 1578 y(a)j(single)g(curv)m(e.)45 b(W)-8 b(e)31 b(de\014ne)i(t)m(w)m(o)e(lev)m(el)g(curv)m(es)i(to)d(b)s (e)i(t)m(w)m(o)f(distinct)f(time-dep)s(enden)m(t)324 1699 y(arcs)k(\000)587 1714 y Fp(+)646 1699 y Fr(\()p Fq(t)p Fr(\),)g(\000)879 1714 y Fk(\000)938 1699 y Fr(\()p Fq(t)p Fr(\))g(that)f(mo)m(v)m(e)h(with)g(the)g(\015uid)f(and)h (collapse)f(at)g(\014nite)h(time)e(in)m(to)h(a)324 1819 y(single)e(arc)i(\000.)43 b(More)33 b(precisely)-8 b(,)33 b(supp)s(ose)h(the)f(arcs)g(are)g(giv)m(en)f(b)m(y)512 2140 y(\000)573 2155 y Fk(\006)659 2140 y Fr(=)c Fo(f)p Fr(\()p Fq(x)906 2155 y Fp(1)946 2140 y Fq(;)17 b(x)1045 2155 y Fp(2)1084 2140 y Fr(\))28 b Fo(2)g Fq(R)1319 2099 y Fp(2)1386 2140 y Fr(:)g Fq(x)1496 2155 y Fp(2)1563 2140 y Fr(=)g Fq(f)1715 2155 y Fk(\006)1774 2140 y Fr(\()p Fq(x;)17 b(t)p Fr(\))p Fq(;)g(x)2083 2155 y Fp(1)2150 2140 y Fo(2)29 b Fr([)p Fq(a;)17 b(b)p Fr(])p 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3512 y Fr(\()p Fq(x)556 3527 y Fp(1)596 3512 y Fq(;)17 b(x)695 3527 y Fp(2)734 3512 y Fq(;)g(t)p Fr(\))28 b(=)992 3445 y Fq(@)5 b(f)1096 3460 y Fk(\006)p 992 3489 165 4 v 998 3581 a Fq(@)g(x)1109 3596 y Fp(1)1166 3512 y Fr(\()p Fq(x)1259 3527 y Fp(1)1299 3512 y Fq(;)17 b(t)p Fr(\))22 b Fo(\001)g Fq(u)1544 3527 y Fp(1)1583 3512 y Fr(\()p Fq(x)1676 3527 y Fp(1)1716 3512 y Fq(;)17 b(x)1815 3527 y Fp(2)1854 3512 y Fq(;)g(t)p Fr(\))22 b(+)2101 3445 y Fq(@)5 b(f)2205 3460 y Fk(\006)p 2101 3489 V 2137 3581 a Fq(@)g(t)2275 3512 y Fr(\()p Fq(x)2368 3527 y Fp(1)2408 3512 y Fq(;)17 b(t)p Fr(\))65 b Fq(at)g(x)2796 3527 y Fp(2)2864 3512 y Fr(=)28 b Fq(f)3016 3527 y Fk(\006)3075 3512 y Fr(\()p Fq(x)3168 3527 y Fp(1)3207 3512 y Fq(;)17 b(t)p Fr(\))p Fq(:)86 b Fr(\(5\))324 3759 y(This)29 b(holds)g(in)g(particular)e(for)i(lev)m(el)g(curv)m(es)i(of)e (scalar)g(functions)g(g\(x,t\))h(that)f(satisfy)324 3879 y(\()p Fq(@)413 3894 y Fl(t)465 3879 y Fr(+)22 b Fq(u)g Fo(\001)f(r)773 3894 y Fl(x)817 3879 y 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Fo(2)g Fr([0)p Fq(;)17 b(T)d Fr(\).)470 4521 y(When)37 b(\(2\),)f(\(3\),)h(\(4\),)f(\(5\))g(and)g(\(6\))f (hold,)h(then)h(w)m(e)f(sa)m(y)h(that)f(the)g(\015uid)g(forms)f(a)324 4641 y Fg(sharp)j(fron)m(t)32 b Fr(at)h(time)e(T.)470 4761 y(The)d(standard)f(candidates)f(for)g(a)h(singularit)m(y)d(for)i (MHD)h(and)f(QG)g(are)h(describ)s(ed)324 4882 y(b)m(y)34 b(the)f(de\014nition)f(giv)m(en)h(for)g(a)f(sharp)i(fron)m(t.)44 b(W)-8 b(e)34 b(in)m(v)m(estigate)f(the)g(p)s(ossible)g(forma-)324 5002 y(tion)e(of)h(a)h(sharp)g(fron)m(t.)1918 5251 y(5)p eop %%Page: 6 6 6 5 bop 470 548 a Fr(The)45 b(follo)m(wing)c(assumption)i(will)e(allo)m (w)h(us)j(to)e(rule)g(out)h(formation)d(of)i(sharp)324 668 y(fron)m(ts.)h(W)-8 b(e)33 b(sa)m(y)g(that)g(the)g(\015uid)f(has)h Fg(con)m(trolled)i(v)m(elo)s(cit)m(y)h(gro)m(wth)c Fr(if)f(w)m(e)j(ha)m (v)m(e)424 796 y Fc(Z)523 823 y Fl(T)479 1022 y Fp(0)595 932 y Fq(sup)p Fo(fj)p Fq(u)p Fr(\()p Fq(x)973 947 y Fp(1)1011 932 y Fq(;)17 b(x)1110 947 y Fp(2)1150 932 y Fq(;)g(t)p Fr(\))p Fo(j)27 b Fr(:)h Fq(x)1432 947 y Fp(1)1499 932 y Fo(2)h Fr([)p Fq(a;)17 b(b)p Fr(])p Fq(;)g(f)1876 947 y Fk(\000)1935 932 y Fr(\()p Fq(x)2028 947 y Fp(1)2068 932 y Fq(;)g(t)p Fr(\))27 b Fo(\024)h Fq(x)2372 947 y Fp(2)2440 932 y Fo(\024)g Fq(f)2593 947 y Fp(+)2652 932 y Fr(\()p Fq(x)2745 947 y Fp(1)2785 932 y Fq(;)17 b(t)p Fr(\))p Fo(g)p Fq(dt)27 b(<)h Fo(1)p Fq(:)141 b Fr(\(7\))324 1186 y(If)32 b(\(7\))g(fails,)f(then)j(w)m(e)f(sa)m(y)h(that)e(the)h (\015uid)f(has)h Fg(uncon)m(trolled)j(v)m(elo)s(cit)m(y)g(gro)m(wth)o Fr(.)324 1374 y Fg(Lemma)h(1.)49 b Fb(L)-5 b(et)29 b Fq(\022)j Fb(b)-5 b(e)28 b(a)h(smo)-5 b(oth)28 b(solution)g(of)g(Eq.1)h (de\014ne)-5 b(d)27 b(for)i Fq(t)f Fo(2)g Fr([0)p Fq(;)17 b(T)d Fr(\))p Fb(.)42 b(Assume)324 1494 y(ther)-5 b(e)35 b(is)f(a)h Fa(sharp)40 b(fr)-6 b(ont)35 b Fb(at)g(time)g(T.)f(Then)339 1787 y Fc(\022)440 1860 y Fq(d)p 422 1904 86 4 v 422 1995 a(dt)518 1787 y Fc(\023)17 b(\022)681 1792 y(Z)781 1818 y Fl(b)736 2017 y(a)815 1927 y Fr([)p Fq(f)890 1942 y Fp(+)949 1927 y Fr(\()p Fq(x)1042 1942 y Fp(1)1082 1927 y Fq(;)g(t)p Fr(\))22 b Fo(\000)h Fq(f)1369 1942 y Fk(\000)1428 1927 y Fr(\()p Fq(x)1521 1942 y Fp(1)1561 1927 y Fq(;)17 b(t)p Fr(\)])p Fq(dx)1811 1942 y Fp(1)1850 1787 y Fc(\023)2006 1927 y Fr(=)83 b Fq( )t Fr(\()p Fq(a;)17 b(f)2413 1942 y Fp(+)2472 1927 y Fr(\()p Fq(a;)g(t)p Fr(\))p Fq(;)g(t)p Fr(\))22 b Fo(\000)h Fq( )t Fr(\()p Fq(a;)17 b(f)3165 1942 y Fk(\000)3224 1927 y Fr(\()p Fq(a;)g(t)p Fr(\))p Fq(;)g(t)p Fr(\))2006 2130 y(+)83 b Fq( )t Fr(\()p Fq(b;)17 b(f)2403 2145 y Fk(\000)2463 2130 y Fr(\()p Fq(b;)g(t)p Fr(\))p Fq(;)g(t)p Fr(\))22 b Fo(\000)h Fq( )t Fr(\()p Fq(b;)17 b(f)3136 2145 y Fp(+)3195 2130 y Fr(\()p Fq(b;)g(t)p Fr(\))p Fq(;)g(t)p Fr(\))p Fq(:)-98 b Fr(\(8\))324 2331 y(Pro)s(of:)53 b(T)-8 b(ak)m(e)40 b(the)e(deriv)-5 b(ativ)m(e)38 b(of)f(the)i(stream)e(function)h(with)f (resp)s(ect)j(to)d Fq(x)3256 2346 y Fp(1)3334 2331 y Fr(along)324 2452 y(an)32 b(arc)h(\000)683 2467 y Fk(\006)742 2452 y Fr(\()p Fq(t)p Fr(\))509 2641 y Fq(@)5 b( )t Fr(\()p Fq(x)725 2656 y Fp(1)766 2641 y Fq(;)17 b(f)858 2656 y Fk(\006)917 2641 y Fr(\()p Fq(x)1010 2656 y Fp(1)1049 2641 y Fq(;)g(t)p Fr(\))p Fq(;)g(t)p Fr(\))p 509 2685 775 4 v 820 2777 a Fq(@)5 b(x)931 2792 y Fp(1)1321 2708 y Fr(=)27 b Fq(u)1480 2723 y Fp(2)1519 2708 y Fr(\()p Fq(x)1612 2723 y Fp(1)1652 2708 y Fq(;)17 b(f)1744 2723 y Fk(\006)1803 2708 y Fr(\()p Fq(x)1896 2723 y Fp(1)1936 2708 y Fq(;)g(t)p Fr(\))p Fq(;)g(t)p Fr(\))22 b Fo(\000)2301 2641 y Fq(@)5 b(f)2405 2656 y Fk(\006)p 2301 2685 165 4 v 2307 2777 a Fq(@)g(x)2418 2792 y Fp(1)2475 2708 y Fq(u)2531 2723 y Fp(1)2570 2708 y Fr(\()p Fq(x)2663 2723 y Fp(1)2703 2708 y Fq(;)17 b(f)2795 2723 y Fk(\006)2854 2708 y Fr(\()p Fq(x)2947 2723 y Fp(1)2987 2708 y Fq(;)g(t)p Fr(\))p Fq(;)g(t)p Fr(\))216 b(\(9\))324 2956 y(b)m(y)33 b(com)m(bining)e(\(9\))h(and)h(\(5\))f(w)m(e)i(obtain)1190 3241 y Fq(@)5 b( )t Fr(\()p Fq(x)1406 3256 y Fp(1)1447 3241 y Fq(;)17 b(f)1539 3256 y Fk(\006)1598 3241 y Fr(\()p Fq(x)1691 3256 y Fp(1)1731 3241 y Fq(;)g(t)p Fr(\))p Fq(;)g(t)p Fr(\))p 1190 3285 775 4 v 1501 3376 a Fq(@)5 b(x)1612 3391 y Fp(1)2002 3308 y Fr(=)2115 3241 y Fq(@)g(f)2219 3256 y Fk(\006)p 2115 3285 165 4 v 2151 3376 a Fq(@)g(t)2289 3308 y Fr(\()p Fq(x)2382 3323 y Fp(1)2422 3308 y Fq(;)17 b(t)p Fr(\))849 b(\(10\))324 3556 y(Expression)43 b(\(8\))f(follo)m(ws) f(from)g(in)m(tegrating)g(\(10\))h(with)g(resp)s(ect)h(to)f Fq(x)3044 3571 y Fp(1)3126 3556 y Fr(b)s(et)m(w)m(een)j(a)324 3676 y(and)32 b(b.)324 3864 y Fg(Theorem)37 b(1.)49 b Fb(L)-5 b(et)49 b(u\(x,t\))h(b)-5 b(e)49 b(a)g(diver)-5 b(genc)g(e-fr)g(e)g(e)47 b(velo)-5 b(city)49 b(\014eld,)j(with)d(c)-5 b(ontr)g(ol)5 b(le)-5 b(d)324 3984 y(velo)g(city)34 b(gr)-5 b(owth.)45 b(Then)34 b(a)h Fa(sharp)40 b(fr)-6 b(ont)35 b Fb(c)-5 b(annot)34 b(develop)g(at)h(time)g(T.)470 4172 y Fr(Pro)s(of:)43 b(Assume)33 b(there)h(is)e(a)g(sharp)h(fron)m(t)g(at) f(time)f(T.)i(W)-8 b(e)33 b(de\014ne)1077 4452 y Fq(A)p Fr(\()p Fq(t)p Fr(\))28 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Fo(g)p Fq(ds)1918 5251 y Fr(6)p eop %%Page: 7 7 7 6 bop 324 548 a Fr(and)320 784 y(~)324 811 y Fq(b)p Fr(\()p Fq(t)p Fr(\))28 b(=)g Fq(b)22 b Fo(\000)771 675 y Fc(Z)870 701 y Fl(T)826 901 y(t)942 811 y Fq(sup)p Fo(fj)p Fq(u)p Fr(\()p Fq(x)1320 826 y Fp(1)1358 811 y Fq(;)17 b(x)1457 826 y Fp(2)1497 811 y Fq(;)g(s)p Fr(\))p Fo(j)27 b Fr(:)h Fq(x)1790 826 y Fp(1)1857 811 y Fo(2)g Fr([)p Fq(a;)17 b(b)p Fr(])p Fq(;)g(f)2233 826 y Fk(\000)2293 811 y Fr(\()p Fq(x)2386 826 y Fp(1)2425 811 y Fq(;)g(s)p Fr(\))28 b Fo(\024)g Fq(x)2741 826 y Fp(2)2808 811 y Fo(\024)h Fq(f)2962 826 y Fp(+)3021 811 y Fr(\()p Fq(x)3114 826 y Fp(1)3154 811 y Fq(;)17 b(s)p Fr(\))p Fo(g)p Fq(ds)470 1083 y Fr(There)31 b(is)d(con)m(trolled)h(v)m(elo)s(cit)m(y)g(gro)m (wth,)h(therefore)g(there)g(exists)g Fq(t)2955 1047 y Fk(\003)3023 1083 y Fo(2)e Fr([0)p Fq(;)17 b(T)d Fr(\))28 b(suc)m(h)324 1204 y(that)33 b(~)-50 b Fq(a)p Fr(\()p Fq(t)p Fr(\))28 b Fo(2)g Fr([)p Fq(a;)17 b(b)p Fr(])33 b(and)1228 1177 y(~)1232 1204 y Fq(b)p Fr(\()p Fq(t)p Fr(\))28 b Fo(2)g Fr([)p Fq(a;)17 b(b)p Fr(])33 b(for)f(all)f Fq(t)d Fo(2)g Fr([)p Fq(t)2233 1167 y Fk(\003)2273 1204 y Fq(;)17 b(T)d Fr(\).)470 1324 y(W)-8 b(e)33 b(tak)m(e)g(the)g(deriv) -5 b(ativ)m(e)33 b(of)f(A\(t\))g(with)g(resp)s(ect)i(to)e(time)334 1561 y Fq(dA)p Fr(\()p Fq(t)p Fr(\))p 334 1605 235 4 v 408 1696 a Fq(dt)606 1628 y Fr(=)c Fq(sup)p Fo(j)p Fq(u)p Fo(j)20 b(\001)i Fq(\016)t Fr(\()1124 1602 y(~)1128 1628 y Fq(b;)17 b(t)p Fr(\))22 b(+)g Fq(sup)p Fo(j)p Fq(u)p Fo(j)f(\001)h Fq(\016)t Fr(\()q(~)-50 b Fq(a;)17 b(t)p Fr(\))22 b(+)2113 1493 y Fc(Z)2210 1501 y Fp(~)2212 1519 y Fl(b)q Fp(\()p Fl(t)p Fp(\))2169 1718 y(~)-36 b Fl(a)q Fp(\()p Fl(t)p Fp(\))2371 1561 y Fq(@)p 2354 1605 93 4 v 2354 1696 a(@)5 b(t)2456 1628 y Fr([)p Fq(f)2531 1643 y Fp(+)2590 1628 y Fr(\()p Fq(x)2683 1643 y Fp(1)2723 1628 y Fq(;)17 b(t)p Fr(\))22 b Fo(\000)g Fq(f)3009 1643 y Fk(\000)3069 1628 y Fr(\()p Fq(x)3162 1643 y Fp(1)3201 1628 y Fq(;)17 b(t)p Fr(\)])p Fq(dx)3451 1643 y Fp(1)3491 1628 y Fq(:)324 1917 y Fr(where)35 b Fq(sup)p Fo(j)p Fq(u)p Fo(j)27 b Fr(=)i Fq(sup)p Fo(fj)p Fq(u)p Fr(\()p Fq(x)1380 1932 y Fp(1)1418 1917 y Fq(;)17 b(x)1517 1932 y Fp(2)1557 1917 y Fq(;)g(t)p Fr(\))p Fo(j)29 b Fr(:)g Fq(x)1842 1932 y Fp(1)1911 1917 y Fo(2)h Fr([)p Fq(a;)17 b(b)p Fr(])p Fq(;)g(f)2289 1932 y Fk(\000)2348 1917 y Fr(\()p Fq(x)2441 1932 y Fp(1)2481 1917 y Fq(;)g(t)p Fr(\))29 b Fo(\024)h Fq(x)2789 1932 y Fp(2)2858 1917 y Fo(\024)f Fq(f)3012 1932 y Fp(+)3072 1917 y Fr(\()p Fq(x)3165 1932 y Fp(1)3204 1917 y Fq(;)17 b(t)p Fr(\))p Fo(g)33 b Fr(and)324 2038 y Fq(\016)t Fr(\()p Fq(z)t(;)17 b(t)p Fr(\))28 b(=)g Fq(f)755 2053 y Fp(+)814 2038 y Fr(\()p Fq(z)t(;)17 b(t)p Fr(\))22 b Fo(\000)h Fq(f)1188 2053 y Fk(\000)1247 2038 y Fr(\()p Fq(z)t(;)17 b(t)p Fr(\).)470 2158 y(Using)31 b(the)i(de\014nition)d(of)h(the)i(stream)e (function,)g(the)i(mean)e(v)-5 b(alue)31 b(theorem)g(and)324 2278 y(\(8\),)j(it)f(is)h(easy)h(to)e(c)m(hec)m(k)k(that)1533 2231 y Fl(dA)p Fp(\()p Fl(t)p Fp(\))p 1533 2256 170 4 v 1586 2313 a Fl(dt)1742 2278 y Fq(>)30 b Fr(0)k(for)g Fq(t)c(>)g(t)2288 2242 y Fk(\003)2328 2278 y Fr(.)48 b(This)34 b(con)m(tradicts)h(\(6\))e(b)m(y)i(the)324 2399 y(dominated)c(con)m(v)m(ergence)k(theorem.)324 2602 y Fg(Ac)m(kno)m(wledgmen)m(ts)h(1.)49 b Fb(This)40 b(work)g(was)g (initial)5 b(ly)40 b(supp)-5 b(orte)g(d)40 b(by)h(the)g(A)n(meric)-5 b(an)324 2723 y(Institute)35 b(of)g(Mathematics.)324 3055 y Fs(References)373 3274 y Fr([1])48 b(J.)c(T.)g(Beale,)h(T.)f (Kato,)h(and)f(A.)f(Ma)5 b(jda.)76 b(Remarks)43 b(on)g(the)h(breakdo)m (wn)h(of)524 3395 y(smo)s(oth)k(solutions)g(for)h(the)g(3D)f(Euler)h (equations.)96 b Fb(Comm.)49 b(Math.)j(Phys.)p Fr(,)524 3515 y(94:61{64,)31 b(1984.)373 3719 y([2])48 b(R.E.)25 b(Ca\015isc)m(h,)i(I.)d(Klapp)s(er,)h(G.)f(Steele.)30 b(Remarks)24 b(on)h(singularities,)e(dimension)524 3839 y(and)i(energy)h(disspation)e(for)g(ideal)f(h)m(ydro)s(dynamics)i(and)g (MHD.)30 b Fb(Comm.)c(Math.)524 3959 y(Phys.)p Fr(,)33 b(184:443-455,)d(1997.)373 4163 y([3])48 b(P)-8 b(.)39 b(Constan)m(tin,)h(C.)e(F)-8 b(e\013erman,)39 b(and)f(A.)h(J.)f(Ma)5 b(jda.)60 b(Geometric)37 b(constrain)m(ts)524 4283 y(on)30 b(p)s(oten)m(tially)f(singular)f(solutions)h(for)h(the)h(3-D)e(Euler)h (equations.)40 b Fb(Commun.)524 4404 y(Part.)35 b(Di\013.)f(Eq.)p Fr(,)f(21:559{571,)e(1996.)373 4607 y([4])48 b(P)-8 b(.)36 b(Constan)m(tin,)g(A.)f(J.)g(Ma)5 b(jda,)36 b(and)f(E.)h(T)-8 b(abak.)51 b(F)-8 b(ormation)33 b(of)h(strong)h(fron)m(ts)524 4727 y(in)28 b(the)h(2-D)e(quasigeostrophic)i(thermal)e(activ)m(e)h (scalar.)37 b Fb(Nonline)-5 b(arity)p Fr(,)29 b(7:1495{)524 4848 y(1533,)j(1994.)1918 5251 y(7)p eop %%Page: 8 8 8 7 bop 373 548 a Fr([5])48 b(P)-8 b(.)39 b(Constan)m(tin,)i(Q.)d(Nie)g (and)h(N.)g(Sc)m(horghofer.)62 b(Nonsingular)37 b(surface-quasi-)524 668 y(geostrophic)c(\015o)m(w)44 b Fb(Phys.)34 b(L)-5 b(ett.)36 b(A)p Fr(,)d(24:168-172.)373 872 y([6])48 b(D.)37 b(Cordoba.)55 b(Nonexistence)39 b(of)d(simple)f(h)m(yp)s(erb)s(olic)h (blo)m(w-up)h(for)f(the)h(quasi-)524 992 y(geostrophic)c(equation.)43 b Fb(A)n(nn.)34 b(of)h(Math.)p Fr(,)e(148\(3\),)e(1998.)373 1196 y([7])48 b(D.)29 b(Cordoba)g(and)g(C.)g(F)-8 b(e\013erman.)37 b(Beha)m(vior)29 b(of)g(sev)m(eral)g(2D)g(\015uid)f(equations)h(in)524 1316 y(singular)i(scenarios.)44 b Fb(submitte)-5 b(d)35 b(to)g(Pr)-5 b(o)g(c.)35 b(Natl.)g(A)-5 b(c)g(ad.)35 b(Sci.)f(USA)373 1519 y Fr([8])48 b(D.)30 b(Cordoba)g(and)g(C.)g (Marliani.)37 b(Ev)m(olution)29 b(of)g(curren)m(t)i(sheets)h(and)e (regularit)m(y)524 1640 y(of)j(ideal)e(incompressible)h(magnetic)g (\015uids)h(in)f(2D.)45 b Fb(Comm.)34 b(Pur)-5 b(e)35 b(Appl.Math)p Fr(,)524 1760 y(53\(4\):512-524,)30 b(2000.)373 1963 y([9])48 b(W.)33 b(E)g(and)g(C-H.)f(Sh)m(u.)55 b Fb(Phys.)35 b(Fluids)p Fr(,)d(1:49-58.)324 2167 y([10])48 b(H.)37 b(F)-8 b(riedel,)36 b(R.)h(Grauer,)g(and)g(C.)g(Marliani.)53 b(Adaptiv)m(e)37 b(mesh)f(re\014nemen)m(t)i(for)524 2287 y(singular)g(curren)m(t)h(sheets)i(in)d(incompressible)g(magnetoh)m (ydro)s(dynamic)f(\015o)m(ws.)524 2408 y Fb(J.)e(Comput.)g(Phys.)p Fr(,)d(134:190{198,)f(1997.)324 2611 y([11])48 b(K.)36 b(Ohkitani)f(and)h(M.)h(Y)-8 b(amada.)52 b(In)m(viscid)37 b(and)f(in)m(viscid-limit)31 b(b)s(eha)m(vior)36 b(of)g(a)524 2731 y(surface)e(quasi-geostrophic)e(\015o)m(w.)44 b Fb(Phys.)35 b(Fluids)p Fr(,)d(9:876-882.)324 2935 y([12])48 b(E.N.)34 b(P)m(ark)m(er.)324 3138 y([13])48 b(E.)33 b(Priest)g(and)g(V.S.)g(Tito)m(v.)54 b Fb(Phil.)34 b(T)-7 b(r)i(ans.)34 b(R.)h(So)-5 b(c.)34 b(L)-5 b(ond.)34 b(A)i Fr(,)c(351:1-37.)324 3342 y([14])48 b(A.)33 b(Pumir)e(and)i(E.D.)g (Siggia.)52 b Fb(Phys.)34 b(Fluids)h(A)p Fr(,)e(4:1472-1491.)1918 5251 y(8)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF