A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) 7th Warnemünde Turbulence Days, Vilm, 30.08.-03.09.2015.

A scale invariance criterion for subgrid-scale parametrizations of general circulation models

(and others)

7th Warnemünde Turbulence Days, Vilm, 30.08.-03.09.2015

Urs Schaefer-RolffsCollaborators: Rahel Knöpfel and Erich Becker

Cape Verde (http://visibleearth.nasa.gov)Horns Rev, Denmark (Christian Steiness)

100km 100 m

Why


(and others)


Overview

• Motivation

• The Scale Invariance Criterion

• Modeling

• Outlook


(and others)


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?

spec

tral

den

sity

(a.

u.)

horizontal wavelength (km)104 103 102 101 100 10-1 10-2

k-3

k-5/3

10-3 10-4

isotropic inertial rangeanisotropic inertial rangesynopticscales

plan

etar

y sc

ales

viscous scales

e = const. > 0

Motivation


(and others)


Overview

• Motivation


• Modeling

• Outlook


(and others)


• Fast check for consistence: Scale invariance criterion (Schaefer-Rolffs et al., 2015)

• Each term in a given equation of motion can be investigated separately

• Constraints for scale-invariant formulation of parameterization

The Scale Invariance Criterion

𝑦=𝜙 ( 𝑦∗ , 𝑧∗ ) , 𝑧=𝜓 ( 𝑦∗ , 𝑧∗ )

𝑒𝑐𝑎−

23𝑐𝑥

𝜕𝑡 𝑎+𝑒𝑐𝑎−

23𝑐𝑥

(𝐯 ⋅𝛁 )𝑎=ℱ 𝑎(𝑒23𝑐𝑥

𝑡 ,𝑒𝑐𝑥𝐱 ,𝑒𝑐𝑎𝑎 ,𝑒𝑐𝑏 𝑖𝑏𝑖)

Scale invariance of the model equations within a finite range of scales is preserved if the source function fulfills

Hence, a specific relation between the scaling factors exist in the regime of scale invariance.

𝒢𝑎 (𝑡 , 𝐱 ,𝑎 ,𝑏𝑖 )≝ℱ 𝑎 (𝑡 , 𝐱 ,𝑎 ,𝑏𝑖 )


(and others)


Example 1: Classical Smagorinsky model (, )Example 2: Dynamic Smagorinsky Model (, )

Example 3: Vertical diffusion of horizontal momentum (, )

.

With for large scales !

The Scale Invariance Criterion


(and others)


Overview

• Motivation


• Modeling

• Outlook


(and others)


na nb nc nd

• Dynamic Smagorinksy Model (DSM): Spectra with slope for the first time without hyperdiffusion (Schaefer-Rolffs and Becker, 2013)

Modeling

na nb nc nd

na nb


(and others)


Schematic of the nonlinear interaction near the resolution scale

spec

tral

den

sity

(a.

u.)

horizontal wavelength (km)104 103 102 101 100 10-1 10-2

k-3

k-5/3

10-3 10-4

Isotropic inertial range

anisotropic inertial range viscous scales

synoptic scales

pla

neta

ry

sca

les

Test filter range Resolution scale

Modeling

• Problem at resolution scale: Are scale interaction relevant?


(and others)


na nb nc nd

• Dynamic Smagorinksy Model (DSM): Spectra with slope for the first time without hyperdiffusion (Schaefer-Rolffs and Becker, 2013)

Modeling

na nb nc nd

na nb


(and others)


latitude

pres

sure

(m

bar)

Horizontal dynamical mixing length (km) and zonal wind (m/s)

• Solution: Generalized DSM with filtering at larger scales than the resolution scale (Schaefer-Rolffs and

Becker, 2015)

Modeling


(and others)


HML dynamically, VML constant

HML and VML dynamically

HML and VML constant

horizontal wavenumber

Modeling


(and others)


• 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

• Energy spectra with transition to slope

Conclusions 𝒢𝑎 (𝑡 , 𝐱 ,𝑎 ,𝑏𝑖 )≝ℱ 𝑎 (𝑡 , 𝐱 ,𝑎 ,𝑏𝑖 )


(and others)


• Validation of the anisotropic scale ratio for the Primitive Equations

Compatibility with scale ratio of Stratified Turbulence, (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:

Outlook

A scale invariance criterion for subgrid-scale parametrizations of general circulation models (and others) 7th Warnemünde Turbulence Days, Vilm, 30.08.-03.09.2015.

Documents

regime of scale invariance

ma scale invariance

scale interaction relevant

overview motivation

nonlinear interaction

finite range of scales

rahel knpfel

erich becker cape verde