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Abstract
We propose a new similarity theory for the twodimensional 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∼t yield three length scales, l_{ω}, l_{E} and l_{ψ}, 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∼l^{2}_{ω}∼t, while the spectral peak wavenumber k_{E} ≡ 2πl^{−1}_{E} ∼ t^{−1}. We construct a theoretical framework involving a threepart, timeevolving vortex number density distribution, n(A) ∼ t^{αi}A^{−ri}, i ∈ 1,2,3. Just above the forcing scale (i =1) there is a forcingequilibrated scaling range in which the number of vortices at fixed A is constant and vortex ‘selfenergy’ E_{v}^{cm} = (2D)^{−1}∫ω_{v}^{2}A^{2}n(A) dA is conserved in Aspace intervals [μA_{0}(t), A_{0}(t)] comoving with the growth in vortex area, A_{0}(t) ∼ t. In this range, α_{1} = 0 and n(A) ∼ A^{−3}. At intermediate scales (i = 2) sufficiently far from the forcing and the largest vortex, there is a range with a scaleinvariant vortex size distribution. We predict that in this range the vortex enstrophy Z_{v}^{cm} = (2D)^{−1}∫ ω_{v}^{2}An(A)dA is conserved and n(A) ∼ t^{−1}A^{−1}. The final range (i = 3), which extends over the largest vortexcontaining scales, conserves σ_{v}^{cm }= (2D)^{−1}∫ ω_{v}^{2}n(A)dA. If ω_{v}^{2} is constant in time, this is equivalent to conservation of vortex number N_{v}^{cm} =∫ n(A)dA. This regime represents a ‘front’ of sparse vortices, which are effectively pointlike; in this range we predict n(A) ∼ t^{r3−1}A^{−r3}. Allowing for timevarying ω_{v}^{2} results in a small but significant correction to these temporal dependences. Highresolution 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 ∈ [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.
Original language  English 

Pages (fromto)  742756 
Number of pages  15 
Journal  Journal of Fluid Mechanics 
Volume  811 
Early online date  16 Dec 2016 
DOIs  
Publication status  Published  25 Jan 2017 
Keywords
 Turbulence simulation
 Turbulence theory
 Vortex dynamics
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 1 Finished

Jets in the atmosphere and oceans: Structure and transport properties of jets in the atmosphere and oceans
18/05/15 → 17/05/18
Project: Standard