Scaling theory for vortices in the two-dimensional inverse energy cascade
Scaling theory for vortices in the two-dimensional inverse energy cascade
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DOI：
10.1017/jfm.2016.756
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发表时间：
2016-12
影响因子：
3.7
通讯作者：
B. H. Burgess;R. K. Scott
中科院分区：
文献类型：
--
作者：
B. H. Burgess;R. K. Scott
我们提出了一种新的二维逆能级联及其所包含的中尺度相干涡种群的相似理论。考虑到涡度守恒和规定的恒定能量注入速率的相似论点，$E\sim t$产生与涡度场、能量峰值和流函数有关的三个长度尺度，$L_{\UNICODE[STIX]{x1D714}}$，$L_{E}$和$L_{\UNICODE[STIX]{x1D713}}$，以及对它们的时间演变的预测，分别为$t^{1/2}$，$t$和$t^{3/2}$。因此，我们预测涡旋区随时间线性增长，$A\sim L{\unicode[stix]{x1D714}}^{2}\sim t$，而光谱峰值波数$k_{E}\equv2\unicode[stix]{x03C0}L_{E}^{-1}\sim t^{-1}$。我们建立了一个三段时间演化的涡数密度分布的理论框架，$n(A)\sim t^{\unicode[stix]{x1D6FC}_{i}}A^{-r_{i}}，~i\in 1，2，3$.在强迫标度($i=1$)的正上方，存在强迫平衡标度范围，其中在固定$A$处的涡旋数目是恒定的，并且涡旋‘自能’$E_{v}^{Cm}=(2{\mathcal{D}})^{-1}\int\overline{\unicode[STIX]{x1D714}_{v}^{2}}A^{2}n(A)\，\Text{d}A$在$A$-间隔$[\Unicode[stix]{x1D707}A_{0}(T)中守恒，A_{0}(T)]$与涡旋区的生长同步，$A_{0}(T)\sim t$。在此范围内，$\Unicode[stix]{x1D6FC}_{1}=0$和$n(A)\sim A^{-3}$。在距离强迫和最大涡旋足够远的中等尺度上($i=2$)，存在一个尺度不变的涡旋尺寸分布范围。我们预测在这个范围内涡旋拟能$Z_{v}^{cm}=(2{\mathcal{D}})^{-1}\int\overline{\unicode[STIX]{x1D714}_{v}^{2}}An(A)\，\Text{d}A$是守恒的，并且$n(A)\sim t^{-1}A^{-1}$.最后的范围($i=3$)在包含最大涡旋的尺度上延伸，保持$\unicode[STIX]{x1D70E}_{v}^{cm}=(2{\mathcal{D}})^{-1}\int\overline{\unicode[STIX]{x1D714}_{v}^{2}}n(A)\，\Text{d}A$。如果$OVERLINE{\UNICODE[STIX]{x1D714}_{v}^{2}}$在时间上是恒定的，则这相当于守恒涡旋数$N_{v}^{cm}=\int n(A)\，\Text{d}A$。这个区域代表了一个稀疏涡旋的“锋面”，它们实际上是点状的；在这个范围内，我们预测$n(A)\sim t^{r_{3}-1}A^{-r_{3}}$。允许使用随时间变化的$\overline{\unicode[stix]{x1D714}_{v}^{2}}$将导致对这些时间依赖关系的较小但显著的修正。高分辨率数值模拟验证了预测的涡旋和光谱峰值增长率，以及涡旋种群中三个尺度范围的理论图景。涡旋使能谱E(K)$变陡，超过经典的[k^-5/3}$在[k_{f}，k_{v}]$范围内的标度，其中$k_(V)$是与最大涡旋相关的波数，而在更大的尺度上斜率接近$-5/3$。尽管涡旋破坏了经典标度，但它们的数密度分布和演化揭示了更深更复杂的标度不变性，并从涡旋相互作用的角度提出了一种有效的逆级联理论。
We propose a new similarity theory for the two-dimensional inverse energy cascade and the coherent vortex population it contains when forced at intermediate scales. Similarity arguments taking into account enstrophy conservation and a prescribed constant energy injection rate such that $E\sim t$ yield three length scales, $l_{\unicode[STIX]{x1D714}}$ , $l_{E}$ and $l_{\unicode[STIX]{x1D713}}$ , associated with the vorticity field, energy peak and streamfunction, and predictions for their temporal evolutions, $t^{1/2}$ , $t$ and $t^{3/2}$ , respectively. We thus predict that vortex areas grow linearly in time, $A\sim l_{\unicode[STIX]{x1D714}}^{2}\sim t$ , while the spectral peak wavenumber $k_{E}\equiv 2\unicode[STIX]{x03C0}l_{E}^{-1}\sim t^{-1}$ . We construct a theoretical framework involving a three-part, time-evolving vortex number density distribution, $n(A)\sim t^{\unicode[STIX]{x1D6FC}_{i}}A^{-r_{i}},~i\in 1,2,3$ . Just above the forcing scale ( $i=1$ ) there is a forcing-equilibrated scaling range in which the number of vortices at fixed $A$ is constant and vortex ‘self-energy’ $E_{v}^{cm}=(2{\mathcal{D}})^{-1}\int \overline{\unicode[STIX]{x1D714}_{v}^{2}}A^{2}n(A)\,\text{d}A$ is conserved in $A$ -space intervals $[\unicode[STIX]{x1D707}A_{0}(t),A_{0}(t)]$ comoving with the growth in vortex area, $A_{0}(t)\sim t$ . In this range, $\unicode[STIX]{x1D6FC}_{1}=0$ and $n(A)\sim A^{-3}$ . At intermediate scales ( $i=2$ ) sufficiently far from the forcing and the largest vortex, there is a range with a scale-invariant vortex size distribution. We predict that in this range the vortex enstrophy $Z_{v}^{cm}=(2{\mathcal{D}})^{-1}\int \overline{\unicode[STIX]{x1D714}_{v}^{2}}An(A)\,\text{d}A$ is conserved and $n(A)\sim t^{-1}A^{-1}$ . The final range ( $i=3$ ), which extends over the largest vortex-containing scales, conserves $\unicode[STIX]{x1D70E}_{v}^{cm}=(2{\mathcal{D}})^{-1}\int \overline{\unicode[STIX]{x1D714}_{v}^{2}}n(A)\,\text{d}A$ . If $\overline{\unicode[STIX]{x1D714}_{v}^{2}}$ is constant in time, this is equivalent to conservation of vortex number $N_{v}^{cm}=\int n(A)\,\text{d}A$ . This regime represents a ‘front’ of sparse vortices, which are effectively point-like; in this range we predict $n(A)\sim t^{r_{3}-1}A^{-r_{3}}$ . Allowing for time-varying $\overline{\unicode[STIX]{x1D714}_{v}^{2}}$ results in a small but significant correction to these temporal dependences. High-resolution numerical simulations verify the predicted vortex and spectral peak growth rates, as well as the theoretical picture of the three scaling ranges in the vortex population. Vortices steepen the energy spectrum $E(k)$ past the classical $k^{-5/3}$ scaling in the range $k\in [k_{f},k_{v}]$ , where $k_{v}$ is the wavenumber associated with the largest vortex, while at larger scales the slope approaches $-5/3$ . Though vortices disrupt the classical scaling, their number density distribution and evolution reveal deeper and more complex scale invariance, and suggest an effective theory of the inverse cascade in terms of vortex interactions.