{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:U3YAN2TIBPAMMXP42GKCBB3Y6J","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c0c688f5e723b8ce8b92549b6d40e2d3f741886f64f079c3fbd508191c4cd1f9","cross_cats_sorted":["physics.ao-ph","physics.flu-dyn","physics.geo-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-06-26T13:06:24Z","title_canon_sha256":"f8c0a21b8d210282a2db2062f31eaacbb671940e5ef239cf6f01a66926709312"},"schema_version":"1.0","source":{"id":"1706.08885","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.08885","created_at":"2026-05-18T00:41:30Z"},{"alias_kind":"arxiv_version","alias_value":"1706.08885v1","created_at":"2026-05-18T00:41:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.08885","created_at":"2026-05-18T00:41:30Z"},{"alias_kind":"pith_short_12","alias_value":"U3YAN2TIBPAM","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_16","alias_value":"U3YAN2TIBPAMMXP4","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_8","alias_value":"U3YAN2TI","created_at":"2026-05-18T12:31:46Z"}],"graph_snapshots":[{"event_id":"sha256:98f617c6714d4a9f8421814e000481bce331ed4c248bddb2f5265a737f71f3da","target":"graph","created_at":"2026-05-18T00:41:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"An important feature of the planetary oceanic dynamics is that the aspect ratio (the ratio of the depth to horizontal width) is very small. As a result, the hydrostatic approximation (balance), derived by performing the formal small aspect ratio limit to the Navier-Stokes equations, is considered as a fundamental component in the primitive equations of the large-scale ocean. In this paper, we justify rigorously the small aspect ratio limit of the Navier-Stokes equations to the primitive equations. Specifically, we prove that the Navier-Stokes equations, after being scaled appropriately by the ","authors_text":"Edriss S. Titi, Jinkai Li","cross_cats":["physics.ao-ph","physics.flu-dyn","physics.geo-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-06-26T13:06:24Z","title":"The primitive equations as the small aspect ratio limit of the Navier-Stokes equations: rigorous justification of the hydrostatic approximation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.08885","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:4b90a25abe3688742c5c6d93a7ecca3bcefefc0f6cd351d6e3dd217c11597db4","target":"record","created_at":"2026-05-18T00:41:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c0c688f5e723b8ce8b92549b6d40e2d3f741886f64f079c3fbd508191c4cd1f9","cross_cats_sorted":["physics.ao-ph","physics.flu-dyn","physics.geo-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-06-26T13:06:24Z","title_canon_sha256":"f8c0a21b8d210282a2db2062f31eaacbb671940e5ef239cf6f01a66926709312"},"schema_version":"1.0","source":{"id":"1706.08885","kind":"arxiv","version":1}},"canonical_sha256":"a6f006ea680bc0c65dfcd194208778f24be25ec932f0ca7bcf9897a5fb33d6e0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a6f006ea680bc0c65dfcd194208778f24be25ec932f0ca7bcf9897a5fb33d6e0","first_computed_at":"2026-05-18T00:41:30.701870Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:41:30.701870Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"418pmSdr/oYu23cunTIIFEiflD2bBcjFfUUspPCygv9VNwcR9aRSi19rkC9vlmthGcy7TTLZ1XZqknidC4cKBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:41:30.702501Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.08885","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4b90a25abe3688742c5c6d93a7ecca3bcefefc0f6cd351d6e3dd217c11597db4","sha256:98f617c6714d4a9f8421814e000481bce331ed4c248bddb2f5265a737f71f3da"],"state_sha256":"6f66a10e8d88678fe2ac29e8ad043efbc052f9515d85309441585da9d43e6fc5"}