<p>The statistical features of homogeneous, isotropic, two-dimensional stochastic turbulence are discussed. We derive some rigorous bounds for the mean value of the bulk energy dissipation rate <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3526_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation> and enstrophy dissipation rates <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3526_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\chi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation> for 2D flows sustained by a variety of stochastic driving forces. We show that <Equation ID="Equ26"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3526_Article_Equ26.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="183" /> </MediaObject> <EquationSource Format="TEX">$ \varepsilon \rightarrow 0 \hspace{0.5cm}\text{ and } \hspace{0.5cm} \ \chi \lesssim \mathcal {O}(1)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> <mspace width="14.22636pt" /> <mi mathvariant="normal">and</mi> <mspace width="14.22636pt" /> <mspace width="4pt" /> <mi>χ</mi> <mo>≲</mo> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </Equation>in the inviscid limit, consistent with the dual-cascade in 2D turbulence.</p>

Statistical Estimates for 2D stochastic Navier-Stokes Equations

期刊论文
发表日期：

2025年10月12日

查看全文

Anuj Kumar,
Ali Pakzad

摘要

The statistical features of homogeneous, isotropic, two-dimensional stochastic turbulence are discussed. We derive some rigorous bounds for the mean value of the bulk energy dissipation rate $\varepsilon $ $ε$ and enstrophy dissipation rates $\chi $ $χ$ for 2D flows sustained by a variety of stochastic driving forces. We show that $ \varepsilon \rightarrow 0 \hspace{0.5cm}\text{ and } \hspace{0.5cm} \ \chi \lesssim \mathcal {O}(1)$ $ε \to 0 and χ ≲ O (1)$ in the inviscid limit, consistent with the dual-cascade in 2D turbulence.