Saturation of exponents and the asymptotic fourth state of turbulence

A recent discovery about the inertial range of homogeneous and isotropic turbulence is the saturation of the scaling exponents <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>ζ</a:mi><a:mi>n</a:mi></a:msub></a:math> for large <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>n</b:mi></b:math>, defined via structure functions of order <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>n</c:mi></c:math> as <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msub><d:mi>S</d:mi><d:mi>n</d:mi></d:msub><d:mrow><d:mo>(</d:mo><d:mi>r</d:mi><d:mo>)</d:mo></d:mrow><d:mo>=</d:mo><d:mrow><d:mo>〈</d:mo><d:msup><d:mrow><d:mo>(</d:mo><d:msub><d:mi>δ</d:mi><d:mi>r</d:mi></d:msub><d:mi>u</d:mi><d:mo>)</d:mo></d:mrow><d:mi>n</d:mi></d:msup><d:mo>〉</d:mo></d:mrow><d:mo>=</d:mo><d:mi>A</d:mi><d:mrow><d:mo>(</d:mo><d:mi>n</d:mi><d:mo>)</d:mo></d:mrow><d:msup><d:mi>r</d:mi><d:msub><d:mi>ζ</d:mi><d:mi>n</d:mi></d:msub></d:msup></d:mrow></d:math>. We focus on longitudinal structure functions for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msub><e:mi>δ</e:mi><e:mi>r</e:mi></e:msub><e:mi>u</e:mi></e:mrow></e:math> between two positions that are <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>r</f:mi></f:math> apart in the same direction as <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>u</g:mi></g:mrow></g:math>. In a previous work [], two of the present authors developed a theory for <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:msub><h:mi>ζ</h:mi><h:mi>n</h:mi></h:msub></h:math>, which agrees with measurements for all <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mi>n</i:mi></i:math> for which reliable data are available, and shows saturation for large <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mi>n</j:mi></j:math>. Here, we derive expressions for the probability density functions of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:msub><k:mi>δ</k:mi><k:mi>r</k:mi></k:msub><k:mi>u</k:mi></k:mrow></k:math> for four different states of turbulence, including the asymptotic fourth state defined by the saturation of exponents for large <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mi>n</l:mi></l:math>. This saturation means that the scale separation is violated in favor of strongly coupled quasiordered flow structures, which likely take the form of long and thin (worm-like) structures of length <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>L</m:mi></m:math> and thickness <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mrow><n:mi>l</n:mi><n:mo>=</n:mo><n:mi>O</n:mi><n:mo>(</n:mo><n:mi>L</n:mi><n:mo>/</n:mo><n:mi>R</n:mi><n:mi>e</n:mi><n:mo>)</n:mo></n:mrow></n:math>. Published by the American Physical Society 2024