FDEPS 2012, Lecture 1

Two-Dimensional Turbulence in
the Atmosphere
Ted Shepherd
Department of Meteorology
University of Reading• Two-dimensional Euler equations∂ω+ ¶ (y ,w ) = 0, w = Ñ2y , ¶ ( f , g) = f x gy - f y gx , u = - y y , v = y x∂t• Conservation of energy E and enstrophy Z in nonlinearinteractions∂ω
¶ 1−ψ
- y ¶ (y ,w ) = 0 Þ - Ñ×(y Ñy t ) + ( 2 | Ñy |2 ) - ¶ ( 12 y 2 ,w ) = 0∂t¶t∂ω¶ 1 221ω+ w¶ (y ,w ) = 0 Þ( 2 w ) + ¶ (y , 2 w ) = 0∂t¶tdE ddZ d2211
⇒
=|
Ñy|dx
dy=0,=
òòòò22 w dx dy = 0dt dt
dt dt• The wavenumber spectra are related by Z(k)=k2E(k)• Evolution of E(k) is constrained by conservation of both
E and Z: prohibits a direct (downscale) energy cascade• Spreading of an initially localized energy spectrumdd2(k - k 0 ) E ( k ) dk = ò[ k 2 E ( k ) - 2 k k0 E ( k ) + k 02 E (k )] dk > 0òdtdt⇒

dk E ( k ) dk < 0ò
dt

• Hence energy moves
mainly to smaller k,i.e. to larger spatialscales• Similarly, enstrophy
is expected to move
mainly to larger k

Vallis (2006)• The classical picture of two-dimensional turbulence(after Kraichnan 1967 Phys. Fluids)– Argued to be relevant to the atmosphere byCharney (1971 JAS)
Energy and (potential)
enstrophy injection is bybaroclinic instability

Vallis (2006)

J. Atmos. Sci. (1983)• Motivation: Understand how to parameterize nonlinear
interactions with unresolved scales in a climate model• Two distinct wavenumber regimes identified:– n<8: Stationary, zonally anisotropic, seasonallydependent, no power law, upscale energy cascade– n>8: Transient, isotropic, universal, n -3 energy
spectrum, downscale enstrophy cascade• Questions:– What is the reason for the upscale energy cascade?
– Why is the n-3 energy spectrum so clean?

Kinetic energy spectra from FGGE data

Total

Zonal

n=8

n=8January

July
Transient

Meridional
Stationary

Horizontal axis is spherical harmonic index (n=1-32)

Boer & Shepherd (1983 JAS)Boer & Shepherd (1983 JAS)• The spectral fluxes can be decomposed into stationary(dash-dot), transient (dashed), and mixed stationarytransient (dotted) components
The mixed component
Energy flux
is a very significant
part of both the energy
and the enstrophy
fluxesEnstrophy flux

However these fluxes
are clearly not wellresolved by this data

Shepherd (1987 JAS)• Assuming 2-D turbulence, Leith(1971 JAS) represented theinteractions with unresolved
scales as an effective diffusion
with a negative spectral range,
giving zero energy loss (right)• Applying this to the FGGE data gave estimated total energy
and enstrophy fluxes which were consistent with theory
Energy flux
Enstrophy flux

Boer & Shepherd (1983 JAS)• Using higher-resolutionanalyses (here ECMWF
truncated to T60), the“Leith function” can be
estimated for n=0-32 (toppanel; note factor of 10
difference in positive andnegative C.I.’s)• Lower panels show thecorresponding energy andenstrophy interactions withthe scales smaller thann=32• Note energy “backscatter”
Koshyk & Boer (1995 JAS)• The T799 ECMWF operational analysis from January
2008 appears to resolve the fluxes at 250 hPa– Baroclinic excitation occurs over n=10-30– Well defined downscale enstrophy flux

Burgess, Erler & Shepherd (JAS, in press)• The upscale kinetic energy cascade mainly occurs in theupper troposphere, as two distinct peaks

Burgess, Erler & Shepherd (JAS, in press)• Schneider & Walker (2006 JAS) argue that the limited
transient upscale energy cascade is no accident– Atmosphere adjusts towards weak nonlinearity (i.e.
most-unstable scale equals energy-containing scale)• A more extensive upscale energy cascade is found in
the ocean (Scott & Wang 2005 JPO; Schlosser & Eden2007 GRL)
Why is the mixed (stationary-transient) component of
the atmospheric energy flux upscale?
• The textbook arguments for an upscale energy cascade
have lots of loopholes (see Holloway 2010 J. Turb.)– Moreover they are not relevant to this situation, which
involves spectrally non-local wavenumber triads

• In general, some disturbances will extract energy from alarge-scale flow (downscale energy flux; the “Orr effect”),
and some will give energy up (upscale energy flux)

Disturbance
growthDisturbancedecay

Shepherd (1987 JFM)

• An initially random collection of disturbances evolving
linearly in the presence of pure strain (Kraichnan 1976JAS) or pure shear (Shepherd 1985 JAS) will exactly
conserve its energy, implying zero net energy exchangewith the background flow
• Nonlinearity leads to net disturbance growth
Nonlinear

LinearNonlinear

LinearCummins & Holloway (2010 JFM)

• In the real atmosphere, the transient planetary waves
extract energy from the zonal mean flow, while thesynoptic-scale waves give energy up to the mean flow
– cf. Lorenz & Hartmann (2001 JAS)
• The synoptic-wave contribution dominates; this is whatgives the net upscale energy transfer
Gain of stationary zonal KE from eddy KE
Gain of eddy KE from stationary zonal KE
Loss of eddy KE to stationary zonal KE

FGGE analysis in Shepherd (1987 JAS)• The net transfer of kinetic energy from the eddies backto the mean flow is reflected in the Lorenz energy cycle
But the KE�KZ term
is not particularly wellconstrained
theoretically(and anyway the totalenergy transfer isfrom the mean flow
to the eddies!)

Oort’s calculation, inLorenz (1967)

• Also reflected in the fact that horizontal eddy momentum
fluxes are directed into the jet cores, i.e. upgradientShading is theeddy horizontal
momentum flux
convergence(DJF conditions)Contours arezonal wind
Vallis (2006)

Perfect alignment in SH
where jet is eddy driven

Not perfect alignment in NH wherejet involves Hadley circulation• Since horizonal momentum flux is the negative of horizontal
Eliassen-Palm flux, momentum fluxes into the jet arise from
Rossby-wave propagation out of the jet, as expected frombaroclinic instability (Held & Hoskins 1985 Adv. Geophys.)

Vallis (2006)• This phenomenon (originally explained by G.I. Taylor in1917!) is seen in laboratory rotating-tank experiments– A prograde jet emerges from random stirring,surrounded on either side by retrograde jets (seen indistortion of dye)Was proposed
as analogue of“moving flame”
effectWhitehead(1975 Tellus)• But if the jets are not eddy-driven, then it ’s not clear thatthe eddies have to maintain them• In numerical simulations with an imposed jet and random
forcing, the sign of the eddy-mean energy transfer
depends on the parameter regime
Gain of eddy KE fromstationary zonal KE

Loss of eddy KE tostationary zonal KE
Loss of stationary
zonal KE to eddy KE

Barotropic numerical simulations in Shepherd (1987 JFM)

Why is the n -3 energy spectrum so clean?
• Ironically, the n -3 spectrum long proved elusive in ideal
2-D turbulence simulations, but seems to be robust at
sufficiently high resolution (30,000 x 30,000) with nolarge-scale damping (Vallgren & Lindborg 2011 JFM)
• Yet it has been very robust in
atmospheric observations
and models!
• Figure shows spectrum from
ECMWF operational analysis
at T106
Trenberth & Solomon(1993 MWR)• An n-3 spectrum (for eddy KE) is also found (though lesscleanly) in an idealized GCM with wave-wave
interactions suppressed
Eddy KE
Dashed: Nonlinear
simulation
Solid: Wave-wave
interactions
suppressed(scaled)
O’Gorman &Schneider (2007GRL)• However, the wave-wave interactions strongly affect thewave-mean interactions, which are spectrally non-local(Shepherd 1987 JAS; Huang & Robinson 1998 JAS)
Zonal mean zonal windNonlinear
simulation

Wave-wave
interactions
suppressedO’Gorman &Schneider (2007GRL)• Upper tropospheric aircraft observations revealed a k -5/3
energy spectrum at scales from about 5-500 km
So for theparameterization
problem we
actually need tounderstand thedynamics of thisrange

Nastrom & Gage(1985 JAS)• The Gage-Nastrom spectrum (blue) is reproduced in highresolution GCMs (here AFES T639 at 45°N and 200 hPa)

Hamilton, Takahashi & Ohfuchi (2008 JGR)• The origin of the Gage-Nastrom spectrum has been amatter of considerable controversy– Some (e.g. Lilly 1983 JAS) have argued for an
inverse cascade of balanced (low Froude number)energy from the mesoscale (2-D turbulence)– However, evidence appears to be consolidatingaround a forward (downscale) cascade of
unbalanced energy, uninhibited by the potential
enstrophy constraint (e.g. Waite & Bartello 2004
JFM; Lindborg 2006 JFM)
– Cho & Lindborg (2001 JGR) inferred a forwardcascade from analysis of third-order structure
functions using the aircraft data
– Consistent with spontaneous generation of smallscale unbalanced motion from balanced flow (Waite& Bartello 2006 JFM; Skamarock 2004 MWR)• Also consistent with analysis of high-resolution GCMresults (Koshyk & Hamilton 2001 JAS), and with theexistence of a k -5/3 energy spectrum in aircraft
observations around 21 km (Bacmeister et al. 1996 JGR)
• Even low-resolution GCMs exhibit an unbalanced
spectrum, which emerges at sufficiently high altitudes

CMAM results from Shepherd, Koshyk & Ngan (2000 JGR)• The ECMWF operational analysis at T799 from the IPY
period (here January 2008) reveals a shallow mesoscalekinetic energy spectrum, emerging above 230 hPa– N.B. n=20 corresponds to a wavelength of λ=2000
kmBurgess, Erler & Shepherd (JAS, in press)• The power-law scaling in these spectra is remarkable!
• The mesoscale spectral slope is considerably steeper
than -5/3, but the divergent kinetic energy has a -5/3 slope

Burgess, Erler & Shepherd (JAS, in press)• The spectral break reflects the dominance of thedivergent (unbalanced) component of the flow at thesmaller scales; the divergent component grows withaltitude, and the spectral break moves upscale

Burgess, Erler & Shepherd (JAS, in press)• In T1279 forecasts, the mesoscale spectrum extends tohigher wavenumbers and the slope is closer to -5/3
– The dissipation range begins around n=200!

Burgess, Erler & Shepherd (JAS, in press)

Summary• The upscale energy cascade seen in atmosphericobservations is associated with wave-mean interactions,and the direction of the “cascade” is not inevitable– Depends on relation between themidlatitude/subtropical jet and the baroclinic zone,
and the role of low-frequency, barotropic eddies– Baroclinic excitation occurs over n=10-30• The origin of the n -3 spectral range remains unclear, but
seems robustly associated with geostrophic turbulence• The n-3 spectral range gives way to a n -5/3 range around
500 km, which appears to be associated with adownscale cascade of unbalanced energy– Seen in recent operational ECMWF analyses– Implications of this for subgridscale parameterization
remain to be exploredReference1• Boer, G. J., Shepherd, T. G., 1983: Large-Scale TwoDimensional Turbulence in the Atmosphere, J. Atmos. Sci., 40,164-184• Burgess, B. H., Erler, A. R., Shepherd, T. G., in press: The
troposphere-to-stratosphere transition in kinetic energy 2spectra and nonlinear spectral fluxes as seen in ECMWF
analyses, J. Atmos. Sci., 49pp
• Cummins, P. F., Holloway, G., 2010: Reynolds stress and eddy
viscosity in direct numerical simulations of sheared
twodimensional turbulence, J. Fluid Mech., 657, 394-412• Hamilton, K., Takahashi, Y. O., Ohfuchi, W., 2008: Mesoscale
spectrum of atmospheric motions investigated in a very fineresolution global general circulation model, J. Geophys. Res.,113, D18110• Koshyk, J. N., Boer, G. J., 1995: Parameterization of Dynamical
Subgrid-Scale Processes in a spectral GCM, J. Atmos. Sci., 52,965-976

Reference2• Lorenz, E. N., 1967: The Nature and Theory of the General
Circulation in the Atmosphere, W.M.O., Geneva, 161pp• Nastrom, G. D., Gage, K. S., 1985: A Climatology ofAtmospheric Wavenumber Spectra of Wind and Temperature
Observed by Commercial Aircraft, J. Atmos. Sci., 42, 950-960• O'Gorman, P. A., Schneider, T., 2007: Recovery ofatmospheric flow statistics in a general circulation model
without nonlinear eddy-eddy interactions, Geophys. Res.
Lett., 34, L22801• Shepherd, T. G., 1987: A Spectral View of Nonlinear Fluxesand Stationary-Transient Interaction in the Atmosphere., J.
Atmos. Sci., 44, 1166-1179• Shepherd, T. G., 1987: Rossby waves and two-dimensional
turbulence in a large-scale zonal jet, J. Fluid Mech., 183, 467509• Shepherd, T. G., Koshyk, J. N., Ngan, K., 2000: On the natureof large-scale mixing in the stratosphere and mesospher, J.
Geophys. Res., 105, 12433-12446

Reference3• Trenberth, K. E., Solomon, A., 1993: Implications of Global
Atmospheric Spatial Spectra for Processing and Displaying
Data, J. Climate, 6, 531-545• Vallis, G. K., 2006: Atmospheric and Oceanic Fluid Dynamics,
Cambridge University Press, 770pp• Whitehead, J. A., 1975: Mean flow generated by circulation
on a β-plane: An analogy with the moving flame experiment,
Tellus, 27, 358-364