Scale Invariance Criterion for Subgrid-Scale Parametrizations
Explore the scale invariance criterion for subgrid-scale parametrizations in general circulation models and beyond, aiming for a better understanding of atmospheric energy cycles and the consistency of dynamical core and parameterizations. This study investigates the scale invariance of model equations within various scales, offering insights into the inertial range and the formulation of parameterizations for accurate energy transfer descriptions.
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A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) Why ??= ? Urs Schaefer-Rolffs Collaborators: Rahel Kn pfel and Erich Becker 100 m 100km Cape Verde (http://visibleearth.nasa.gov) Horns Rev, Denmark (Christian Steiness) 7th Warnem nde Turbulence Days, Vilm, 30.08.-03.09.2015
A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) Overview Motivation The Scale Invariance Criterion Modeling Outlook 7th Warnem nde Turbulence Days, Vilm, 30.08.-03.09.2015
A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) k-3 Motivation k-5/3 spectral density (a.u.) planetary scales synoptic scales viscous scales anisotropic inertial range isotropic inertial range 104 103 102 100 10-1 10-2 10- 3 10-4 101 horizontal wavelength (km) Aim: Better understanding of atmospheric energy cycle Requirement: Consistence of the dynamical core and parameterizations for a correct description of energy transfer Scale- invariant inertial range? Investigation of scale invariance up to now only for isotropic, incompressible, and three-dimensional turbulence Scale invariance in the real atmosphere? 7th Warnem nde Turbulence Days, Vilm, 30.08.-03.09.2015
A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) Overview Motivation The Scale Invariance Criterion Modeling Outlook 7th Warnem nde Turbulence Days, Vilm, 30.08.-03.09.2015
A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) The Scale Invariance Criterion 2 3???; 1 3???; = ?????? ? = ? ? = ????; ? = ? ? ,? , ? = ? ? = ????; ? = ? ? ,? ?? ??? + ? ? ? = ??,?,?,?? ?? ? + ? ? ? = ?? ,? ,? ,?? 2 3?? ?? ? ? 2 3???,????,????,?????? ???,?,?,?? ??? 2 3????? + ??? 2 2 3???,????,????,?????? 3??? ? ? = ? ? ??? + ? ? ? = ? ???,?,?,?? ??,?,?,?? Scale invariance of the model equations within a finite range of scales is preserved if the source Fast check for consistence: Scale invariance criterion (Schaefer-Rolffs et al., 2015) function fulfills ???,?,?,?? ??,?,?,?? Each term in a given equation of motion can be investigated separately Constraints for scale-invariant formulation of parameterization Hence, a specific relation between the scaling factors exist in the regime of scale invariance. 7th Warnem nde Turbulence Days, Vilm, 30.08.-03.09.2015
A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) The Scale Invariance Criterion 2? , ? ?????.) 2? , ? = ?????.) Example 1: Classical Smagorinsky model (? = ? Example 2: Dynamic Smagorinsky Model (? = ? 2? ? ? ; ?? = 0; ?? = ??= 2 2? ? ; ?? = ??; ?? = 2 ? = ?; ??= ? ? = ?; ??= ? ? 3?? 3?? 2 3?? 1 2 3?? 1 2? ??? ?2 2 2?2 2 3??? 3??? ??? ?2??? 3??? ? = ? 2??? 3??? ? = ? ? 2? ? ? ?? 2? ? = ?? ??= ? ??= ? 2??? , ? = ?????.) Example 3: Vertical diffusion of horizontal momentum (??= ?? 2??? ??? ; ? = ?; ??= ???? ???=? 1 3??? ?????2????? = ?2??? (3?? ??)???? 2?2(1 2??? ??? = ?? ???=3?? ?? 3?? ??)??? ??? ! ??= ? . . 2 With ??=2 3??for large scales ???=1 2??=1 2?? ??= ??? 7th Warnem nde Turbulence Days, Vilm, 30.08.-03.09.2015
A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) Overview Motivation The Scale Invariance Criterion Modeling Outlook 7th Warnem nde Turbulence Days, Vilm, 30.08.-03.09.2015
A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) Modeling na nb nc na nb nd nc na nb nd Dynamic Smagorinksy Model (DSM): Spectra with (Schaefer-Rolffs and Becker, 2013) 5 3 slope for the first time without hyperdiffusion 7th Warnem nde Turbulence Days, Vilm, 30.08.-03.09.2015
A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) Modeling Test filter range Resolution scale k-3 k-5/3 spectral density (a.u.) planetary scales synoptic scales Isotropic inertial range anisotropic inertial range viscous scales 104 103 102 100 10-1 10-2 10-3 10-4 101 horizontal wavelength (km) Schematic of the nonlinear interaction near the resolution scale Problem at resolution scale: Are scale interaction relevant? 7th Warnem nde Turbulence Days, Vilm, 30.08.-03.09.2015
A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) Modeling na nb nc na nb nd nc na nb nd Dynamic Smagorinksy Model (DSM): Spectra with (Schaefer-Rolffs and Becker, 2013) 5 3 slope for the first time without hyperdiffusion 7th Warnem nde Turbulence Days, Vilm, 30.08.-03.09.2015
A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) Modeling Horizontal dynamical mixing length (km) and zonal wind (m/s) pressure (mbar) latitude Solution: Generalized DSM with filtering at larger scales than the resolution scale (Schaefer-Rolffs and Becker, 2015) 7th Warnem nde Turbulence Days, Vilm, 30.08.-03.09.2015
A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) Modeling HML and VML constant HML dynamically, VML constant HML and VML dynamically horizontal wavenumber 7th Warnem nde Turbulence Days, Vilm, 30.08.-03.09.2015
A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) Conclusions ???,?,?,?? ??,?,?,?? Scale Invariance Criterion for fast check of consistence Each term of equation can be checked separately Constraints for new parametrizations Generalized Dynamic Smagorinsky Model with test filter independent from resolution scale 5 3 slope Energy spectra with transition to 7th Warnem nde Turbulence Days, Vilm, 30.08.-03.09.2015
A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) Outlook Validation of the anisotropic scale ratio for the Primitive Equations Compatibility with scale ratio of Stratified Turbulence, ? 1 3? 2 3? (Lindborg, J. Fluid. Mech., 2006) ? ? Investigation of the dynamical vertical mixing length What is the relevance of ??? (Necessary) improvement of the concept of the DSM according to thermodynamic equations of motion, implementation of dynamical tracers/moisture parameterizations in the KMCM Dynamical mixing length for temperature/H2O-concentration Self-consistent determination of Prandtl/Schmidt number Comparison with gravity wave (GW) spectra of the MLT from horizontal Radar and rocket measurements: 7th Warnem nde Turbulence Days, Vilm, 30.08.-03.09.2015
