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We show that for any <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\epsilon &gt;0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03f5<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> there exist non-trivial explicit solutions, which are initially perturbations of size <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\epsilon $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03f5<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and grow to size 1 on a time scale <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\epsilon ^{-2}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>\u03f5<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:mrow>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Moreover, the (simplified) linearized problem around these non-trivial states exhibits improved upper bounds on the possible size of norm inflation for frequencies larger and smaller than <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\epsilon ^{-4}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>\u03f5<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>4<\/mml:mn>\n                    <\/mml:mrow>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00332-023-09958-2","type":"journal-article","created":{"date-parts":[[2023,9,19]],"date-time":"2023-09-19T11:01:47Z","timestamp":1695121307000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On Stability Estimates for the Inviscid Boussinesq Equations"],"prefix":"10.1007","volume":"33","author":[{"given":"Christian","family":"Zillinger","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,9,19]]},"reference":[{"issue":"4","key":"9958_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00030-022-00773-4","volume":"29","author":"D Adhikari","year":"2022","unstructured":"Adhikari, D., Ben, S., Oussama, P., Uddhaba, R., Wu, J.: Stability and large-time behavior for the 2D Boussineq system with horizontal dissipation and vertical thermal diffusion. 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