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  <front>
    <journal-meta>
      <journal-id journal-id-type="publisher-id">Physical Oceanography</journal-id>
      <journal-title-group>
        <journal-title xml:lang="en">Physical Oceanography</journal-title>
      </journal-title-group>
      <issn publication-format="print">1573-160X</issn>
    </journal-meta>
    <article-meta>
      <article-id pub-id-type="publisher-id">20250303</article-id>
      
      <article-categories>
        <subj-group subj-group-type="toc-heading" xml:lang="en">
          <subject>Analysis of observations and methods of calculating Hydrophysical fields in the ocean</subject>
        </subj-group>
        <subj-group subj-group-type="article-type">
          <subject>Research Article</subject>
        </subj-group>
      </article-categories>

      <title-group>
        <article-title xml:lang="en">Estimation of the Stationarity Interval of Wind Wave Field</article-title>
      </title-group>

      <contrib-group>
        <contrib contrib-type="author">
          <contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-0357-0773</contrib-id>
          <name>
            <surname>Saprykina</surname>
            <given-names>Ya. V.</given-names>
          </name>
          <address>
            <country country="RU">Russian Federation</country>
          </address>
          <email>saprykina@ocean.ru</email>
          <xref ref-type="aff" rid="aff1"/>
        </contrib>
        <contrib contrib-type="author">
          <contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-2452-1922</contrib-id>
          <contrib-id contrib-id-type="researcherid">C-7262-2014</contrib-id>
          <name>
            <surname>Divinsky</surname>
            <given-names>B. V.</given-names>
          </name>
          <address>
            <country country="RU">Russian Federation</country>
          </address>
          <email>divin@ocean.ru</email>
          <xref ref-type="aff" rid="aff1"/>
        </contrib>
        <contrib contrib-type="author">
          <contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-0045-0282</contrib-id>
          <name>
            <surname>Shtremel</surname>
            <given-names>M. N.</given-names>
          </name>
          <address>
            <country country="RU">Russian Federation</country>
          </address>
          <email>shtremel@ocean.ru</email>
          <xref ref-type="aff" rid="aff1"/>
        </contrib>
        <contrib contrib-type="author">
          <contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-7013-339X</contrib-id>
          <name>
            <surname>Likutova</surname>
            <given-names>O. A.</given-names>
          </name>
          <address>
            <country country="RU">Russian Federation</country>
          </address>
          <email>olga.ku-ocean@yandex.ru</email>
          <xref ref-type="aff" rid="aff1"/>
        </contrib>
      </contrib-group>

      <aff id="aff1">
        <institution>Shirshov Institute of Oceanology, Russian Academy of Sciences</institution>
        <addr-line>Moscow</addr-line>
        <country country="RU">Russia</country>
      </aff>

      <pub-date date-type="pub" iso-8601-date="2025-06-30" publication-format="electronic">
        <day>30</day>
        <month>06</month>
        <year>2025</year>
      </pub-date>
      <volume>32</volume>
      <issue>3</issue>
      <fpage>297</fpage>
      <lpage>311</lpage>

      <permissions>
        <copyright-statement xml:lang="en">Copyright © 2025, Ya. V. Saprykina, B. V. Divinsky, M. N. Shtremel, O. A. Likutova</copyright-statement>
        <copyright-year>2025</copyright-year>
        <copyright-holder xml:lang="en">Ya. V. Saprykina, B. V. Divinsky, M. N. Shtremel, O. A. Likutova</copyright-holder>
        <ali:free_to_read xmlns:ali="http://www.niso.org/schemas/ali/1.0/"/>
        <license>
          <ali:license_ref xmlns:ali="http://www.niso.org/schemas/ali/1.0/">https://creativecommons.org/licenses/by-nc/4.0/</ali:license_ref>
        </license>
      </permissions>
      <self-uri xlink:href="https://physical-oceanography.ru/repository/issues/2025/03/03/" xlink:title="Article page">https://physical-oceanography.ru/repository/issues/2025/03/03/</self-uri>
      <abstract xml:lang="en">
        <p><bold>Purpose.</bold> The wind wave field is non-stationary due to variability of wave formation factors such as wind action and the processes of non-linear wave interactions. However, for many practical applications, it is assumed that the wind wave process, described by some time record of waves, is quasi-stationary. Globally, there is no generally accepted length of a wave record for which the conditions of homogeneity and stationarity of the wave field would be valid. Therefore, the main purpose of the work is to estimate the interval of wave field stationarity based on the data of field contact measurements carried out in the deep-water and coastal zones of the Black Sea in different years and seasons.</p>
        <p><bold>Methods and Results.</bold> The data from two long-term field experiments in the Black Sea were analyzed. Waves in the open sea were measured using a Directional Waverider buoy, and in the coastal zone – a Spoondrifter Spotter buoy and a resistive string wave gauge, installed on a specialized marine trestle offshore. Spectral analysis methods were applied to the data. The spectral peak width, defined as the peakedness parameter, was used to characterize the homogeneity of wind waves. Wave records containing high values of the peakedness parameter and characterized by a narrow-band spectral distribution, were classified as the cases of quasi-stationary waves homogeneous in their spectral composition close to regular stationary waves. As a result, the characteristic time intervals were obtained for which the wave field could be assumed to be homogeneous and quasi-stationary.</p>
        <p><bold>Conclusions.</bold> Regardless of the conditions of wave formation, the interval of wave stationarity in the Black Sea can be taken equal to 8–12 minutes. These estimates are the same for the deep-water and coastal parts of the sea, and qualitatively correspond to the theoretical ones.</p>
      </abstract>

      <kwd-group>
        <kwd>wind waves</kwd>
        <kwd>wave parameters</kwd>
        <kwd>wave record length</kwd>
        <kwd>wave field homogeneity</kwd>
        <kwd>wave spectrum</kwd>
        <kwd>stationarity interval</kwd>
      </kwd-group>

      <funding-group>
        <funding-statement xml:lang="en">The study was carried out with the support of RSF, grant No. 24-27-00082.</funding-statement>
      </funding-group>
    </article-meta>
  </front>

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    <p>Article text not included.</p>
  </body>

  <back>
    <ref-list>
      <title>REFERENCES</title>
      <ref id="B1">
        <label>1.</label>
        <element-citation publication-type="book">
          <person-group person-group-type="author">
            <name><surname>Longuet-Higgins</surname><given-names>M.S.</given-names></name>
            <name><surname>Cartwright</surname><given-names>D.E.</given-names></name>
            <name><surname>Smith</surname><given-names>N.D.</given-names></name>
          </person-group>
          <article-title>Observation of the Directional Spectrum of Sea Waves Using the Motions of a Floating Buoy</article-title>
          <source>Ocean Wave Spectra: Proceedings of a Conference</source>
          <publisher-loc>Englewood Cliffs, N.J.</publisher-loc>
          <publisher-name>Prentice-Hall</publisher-name>
          <year>1963</year>
          <fpage>111</fpage>
          <lpage>132</lpage>
        </element-citation>
      </ref>
      <ref id="B2">
        <label>2.</label>
        <element-citation publication-type="report">
          <person-group person-group-type="author">
            <name><surname>Hasselmann</surname><given-names>K.</given-names></name>
            <name><surname>Barnett</surname><given-names>T.</given-names></name>
            <name><surname>Bouws</surname><given-names>E.</given-names></name>
            <name><surname>Carlson</surname><given-names>H.</given-names></name>
            <name><surname>Cartwright</surname><given-names>D.</given-names></name>
            <name><surname>Enke</surname><given-names>K.</given-names></name>
            <name><surname>Ewing</surname><given-names>J.A.</given-names></name>
            <name><surname>Gienapp</surname><given-names>H.</given-names></name>
            <name><surname>Hasselmann</surname><given-names>D.E.</given-names></name>
            <etal/>
          </person-group>
          <source>Measurements of Wind-Wave Growth and Swell Decay during the Joint North Sea Wave Project (JONSWAP)</source>
          <publisher-loc>Hamburg</publisher-loc>
          <publisher-name>Deutsches Hydrographisches Institut</publisher-name>
          <year>1973</year>
          <size units="pages">95</size>
          <comment>(Erganzungsheft zur Deutschen Hydrographischen Zeitschrift. Reihe A, Nr. 12)</comment>
        </element-citation>
      </ref>
      <ref id="B3">
        <label>3.</label>
        <element-citation publication-type="book">
          <person-group person-group-type="author">
            <name><surname>Divinsky</surname><given-names>B.V.</given-names></name>
          </person-group>
          <article-title>Results of Measuring of Surface Waves Parameters with Help of Buoy in Gelendzhik Area</article-title>
          <source>Dynamical Processes in the Sea Nearshore Zone</source>
          <person-group person-group-type="editor">
            <name><surname>Kos’yan</surname><given-names>R.D.</given-names></name>
            <name><surname>Podymov</surname><given-names>I.S.</given-names></name>
            <name><surname>Pykhov</surname><given-names>N.V.</given-names></name>
          </person-group>
          <publisher-loc>Moscow</publisher-loc>
          <publisher-name>Scientific World</publisher-name>
          <year>2003</year>
          <fpage>70</fpage>
          <lpage>91</lpage>
          <comment>(in Russian)</comment>
        </element-citation>
      </ref>
      <ref id="B4">
        <label>4.</label>
        <element-citation publication-type="journal">
          <person-group person-group-type="author">
            <name><surname>Tayfun</surname><given-names>M.A.</given-names></name>
          </person-group>
          <article-title>Effects of Spectrum Band Width on the Distribution of Wave Heights and Periods</article-title>
          <source>Ocean Engineering</source>
          <year>1983</year>
          <volume>10</volume>
          <issue>2</issue>
          <fpage>107</fpage>
          <lpage>118</lpage>
          <pub-id pub-id-type="doi">10.1016/0029-8018(83)90017-3</pub-id>
        </element-citation>
      </ref>
      <ref id="B5">
        <label>5.</label>
        <element-citation publication-type="book">
          <person-group person-group-type="author">
            <name><surname>Efimov</surname><given-names>V.V.</given-names></name>
            <name><surname>Polnikov</surname><given-names>V.G.</given-names></name>
          </person-group>
          <source>Numerical Modeling of Wind Waves</source>
          <publisher-loc>Kiev</publisher-loc>
          <publisher-name>Naukova Dumka</publisher-name>
          <year>1991</year>
          <size units="pages">238</size>
          <comment>(in Russian)</comment>
        </element-citation>
      </ref>
      <ref id="B6">
        <label>6.</label>
        <element-citation publication-type="journal">
          <person-group person-group-type="author">
            <name><surname>Trubkin</surname><given-names>I.P.</given-names></name>
          </person-group>
          <article-title>Method and some Results of Directional Spectra Estimations of Random Waves Field in the Coastal Zone</article-title>
          <source>Ecological Systems and Devices</source>
          <year>2003</year>
          <issue>9</issue>
          <fpage>47</fpage>
          <lpage>53</lpage>
          <comment>(in Russian)</comment>
        </element-citation>
      </ref>
      <ref id="B7">
        <label>7.</label>
        <element-citation publication-type="journal">
          <person-group person-group-type="author">
            <name><surname>Vapnyar</surname><given-names>M.U.</given-names></name>
            <name><surname>Tatlyeva</surname><given-names>Z.I.</given-names></name>
          </person-group>
          <article-title>On the Nature of the Long-Term Distribution of Wave Heights in the Caspian Sea</article-title>
          <source>Trudy GOIN</source>
          <year>1989</year>
          <volume>188</volume>
          <fpage>132</fpage>
          <lpage>137</lpage>
          <comment>(in Russian)</comment>
        </element-citation>
      </ref>
      <ref id="B8">
        <label>8.</label>
        <element-citation publication-type="book">
          <person-group person-group-type="author">
            <name><surname>Jenkins</surname><given-names>G.M.</given-names></name>
            <name><surname>Watts</surname><given-names>D.G.</given-names></name>
          </person-group>
          <source>Spectral Analysis and Its Applications</source>
          <publisher-loc>San Francisco</publisher-loc>
          <publisher-name>Holden-Day</publisher-name>
          <year>1969</year>
          <size units="pages">541</size>
        </element-citation>
      </ref>
      <ref id="B9">
        <label>9.</label>
        <element-citation publication-type="journal">
          <person-group person-group-type="author">
            <name><surname>Divinskii</surname><given-names>B.V.</given-names></name>
            <name><surname>Kos’yan</surname><given-names>R.D.</given-names></name>
            <name><surname>Podymov</surname><given-names>I.C.</given-names></name>
            <name><surname>Pushkarev</surname><given-names>O.V.</given-names></name>
          </person-group>
          <article-title>Extreme Waves in the Northeastern Black Sea in February 2003</article-title>
          <source>Oceanology</source>
          <year>2003</year>
          <volume>43</volume>
          <issue>6</issue>
          <fpage>891</fpage>
          <lpage>893</lpage>
        </element-citation>
      </ref>
      <ref id="B10">
        <label>10.</label>
        <element-citation publication-type="journal">
          <person-group person-group-type="author">
            <name><surname>Goda</surname><given-names>Y.</given-names></name>
          </person-group>
          <article-title>Numerical Experiments on Wave Statistics with Spectral Simulation</article-title>
          <source>Report of the Port and Harbour Research Institute</source>
          <year>1970</year>
          <volume>9</volume>
          <issue>3</issue>
          <fpage>3</fpage>
          <lpage>57</lpage>
        </element-citation>
      </ref>
      <ref id="B11">
        <label>11.</label>
        <element-citation publication-type="book">
          <person-group person-group-type="author">
            <name><surname>Marple</surname><given-names>S.L.</given-names><suffix>Jr.</suffix></name>
          </person-group>
          <source>Digital Spectral Analysis with Applications</source>
          <publisher-loc>Englewood Cliffs, New Jersey</publisher-loc>
          <publisher-name>Prentice-Hall, Inc.</publisher-name>
          <year>1987</year>
          <size units="pages">492</size>
        </element-citation>
      </ref>
      <ref id="B12">
        <label>12.</label>
        <element-citation publication-type="journal">
          <person-group person-group-type="author">
            <name><surname>Saprykina</surname><given-names>Ya.V.</given-names></name>
            <name><surname>Kuznetsov</surname><given-names>S.Yu.</given-names></name>
            <name><surname>Andreeva</surname><given-names>N.K.</given-names></name>
            <name><surname>Shtremel</surname><given-names>M.N.</given-names></name>
          </person-group>
          <article-title>Scenarios of Nonlinear Wave Transformation in the Coastal Zone</article-title>
          <source>Oceanology</source>
          <year>2013</year>
          <volume>53</volume>
          <issue>4</issue>
          <fpage>422</fpage>
          <lpage>431</lpage>
          <pub-id pub-id-type="doi">10.1134/S0001437013040103</pub-id>
        </element-citation>
      </ref>
      <ref id="B13">
        <label>13.</label>
        <element-citation publication-type="journal">
          <person-group person-group-type="author">
            <name><surname>Bailard</surname><given-names>J.A.</given-names></name>
          </person-group>
          <article-title>An Energetics Total Load Sediment Transport Model for a Plane Sloping Beach</article-title>
          <source>Journal of Geophysical Research: Oceans</source>
          <year>1981</year>
          <volume>86</volume>
          <issue>C11</issue>
          <fpage>10938</fpage>
          <lpage>10954</lpage>
          <pub-id pub-id-type="doi">10.1029/JC086iC11p10938</pub-id>
        </element-citation>
      </ref>
      <ref id="B14">
        <label>14.</label>
        <element-citation publication-type="journal">
          <person-group person-group-type="author">
            <name><surname>Ruessink</surname><given-names>B.G.</given-names></name>
            <name><surname>Van Den Berg</surname><given-names>T.J.J.</given-names></name>
            <name><surname>Van Rijn</surname><given-names>L.C.</given-names></name>
          </person-group>
          <article-title>Modeling Sediment Transport beneath Skewed Asymmetric Waves above a Plane Bed</article-title>
          <source>Journal of Geophysical Research: Oceans</source>
          <year>2009</year>
          <volume>114</volume>
          <issue>C11</issue>
          <fpage>C11021</fpage>
          <pub-id pub-id-type="doi">10.1029/2009JC005416</pub-id>
        </element-citation>
      </ref>
      <ref id="B15">
        <label>15.</label>
        <element-citation publication-type="journal">
          <person-group person-group-type="author">
            <name><surname>Saprykina</surname><given-names>Y.</given-names></name>
          </person-group>
          <article-title>The Influence of Wave Nonlinearity on Cross-Shore Sediment Transport in Coastal Zone: Experimental Investigations</article-title>
          <source>Applied Sciences</source>
          <year>2020</year>
          <volume>10</volume>
          <issue>12</issue>
          <fpage>4087</fpage>
          <pub-id pub-id-type="doi">10.3390/app10124087</pub-id>
        </element-citation>
      </ref>
      <ref id="B16">
        <label>16.</label>
        <element-citation publication-type="journal">
          <person-group person-group-type="author">
            <name><surname>Wang</surname><given-names>L.</given-names></name>
            <name><surname>Zhou</surname><given-names>B.</given-names></name>
            <name><surname>Jin</surname><given-names>P.</given-names></name>
            <name><surname>Li</surname><given-names>J.</given-names></name>
            <name><surname>Liu</surname><given-names>S.</given-names></name>
            <name><surname>Ducrozet</surname><given-names>G.</given-names></name>
          </person-group>
          <article-title>Relation between Occurrence Probability of Freak Waves and Kurtosis/Skewness in Unidirectional Wave Trains under Single-Peak Spectra</article-title>
          <source>Ocean Engineering</source>
          <year>2022</year>
          <volume>248</volume>
          <fpage>110813</fpage>
          <pub-id pub-id-type="doi">10.1016/j.oceaneng.2022.110813</pub-id>
        </element-citation>
      </ref>
      <ref id="B17">
        <label>17.</label>
        <element-citation publication-type="journal">
          <person-group person-group-type="author">
            <name><surname>Elgar</surname><given-names>S.</given-names></name>
          </person-group>
          <article-title>Relationships Involving Third Moments and Bispectra of a Harmonic Process</article-title>
          <source>IEEE Transactions on Acoustics, Speech, and Signal Processing</source>
          <year>1987</year>
          <volume>35</volume>
          <issue>12</issue>
          <fpage>1725</fpage>
          <lpage>1726</lpage>
          <pub-id pub-id-type="doi">10.1109/TASSP.1987.1165090</pub-id>
        </element-citation>
      </ref>
    </ref-list>
  </back>
</article>
