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Abstract For a smooth projective surface$$X$$satisfying$$H_1(X,\mathbb{Z}) = 0$$and$$w \in H^2(X,\mu _r)$$, we study deformation invariants of the pair$$(X,w)$$. Choosing a Brauer–Severi variety$$Y$$(or, equivalently, Azumaya algebra$$\mathcal{A}$$) over$$X$$with Stiefel–Whitney class$$w$$, the invariants are defined as virtual intersection numbers on suitable moduli spaces of stable twisted sheaves on$$Y$$constructed by Yoshioka (or, equivalently, moduli spaces of$$\mathcal{A}$$-modules of Hoffmann–Stuhler). We show that the invariants do not depend on the choice of$$Y$$. Using a result of de Jong, we observe that they are deformation invariants of the pair$$(X,w)$$. For surfaces with$$h^{2,0}(X) \gt 0$$, we show that the invariants can often be expressed as virtual intersection numbers on Gieseker–Maruyama–Simpson moduli spaces of stable sheaves on$$X$$. This can be seen as a$${\rm PGL}_r$$–$${\rm SL}_r$$correspondence. As an application, we express$${\rm SU}(r) / \mu _r$$Vafa–Witten invariants of$$X$$in terms of$${\rm SU}(r)$$Vafa–Witten invariants of$$X$$. We also show how formulae from Donaldson theory can be used to obtain upper bounds for the minimal second Chern class of Azumaya algebras on$$X$$with given division algebra at the generic point.
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Comparing local energy cascade rates in isotropic turbulence using structure-function and filtering formulations
Two common definitions of the spatially local rate of kinetic energy cascade at some scale$$\ell$$in turbulent flows are (i) the cubic velocity difference term appearing in the ‘scale-integrated local Kolmogorov–Hill’ equation (structure-function approach), and (ii) the subfilter-scale energy flux term in the transport equation for subgrid-scale kinetic energy (filtering approach). We perform a comparative study of both quantities based on direct numerical simulation data of isotropic turbulence at Taylor-scale Reynolds number 1250. While in the past observations of negative subfilter-scale energy flux (backscatter) have led to debates regarding interpretation and relevance of such observations, we argue that the interpretation of the local structure-function-based cascade rate definition is unambiguous since it arises from a divergence term in scale space. Conditional averaging is used to explore the relationship between the local cascade rate and the local filtered viscous dissipation rate as well as filtered velocity gradient tensor properties such as its invariants. We find statistically robust evidence of inverse cascade when both the large-scale rotation rate is strong and the large-scale strain rate is weak. Even stronger net inverse cascading is observed in the ‘vortex compression’$$R>0$$,$$Q>0$$quadrant, where$$R$$and$$Q$$are velocity gradient invariants. Qualitatively similar but quantitatively much weaker trends are observed for the conditionally averaged subfilter-scale energy flux. Flow visualizations show consistent trends, namely that spatially, the inverse cascade events appear to be located within large-scale vortices, specifically in subregions when$$R$$is large.
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- Award ID(s):
- 2103874
- PAR ID:
- 10511089
- Publisher / Repository:
- CUP
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 980
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2023.1066
