how turbulent is the atmosphere at large scales

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  • Quasi-linear approaches to large-scale atmospheric flows

    (or: how turbulent is the atmosphere?)

    Farid Ait-Chaalal(1), in collaboration with:

    Tapio Schneider(1,3) and Brad Marston(2)(1)ETH, Zurich, Switzerland, (2)Brown University, Providence, USA

    (3)Caltech, Pasadena, USA

  • The general circulation

    Superposition of a mean flow and turbulent eddies

    Source: EUMETSAT, https://www.youtube.com/watch?v=m2Gy8V0Dv78March 2013 brightness temperature (clouds)

    https://www.youtube.com/watch?v=m2Gy8V0Dv78

  • Relative vorticity (s-1) at 725 hPa in an idealized dry GCM

    The general circulation

  • FMS GFDL pseudospectral dynamical core

    Radiation: Newtonian relaxation of temperatures toward a fixed profile

    Convection: Relaxation of the vertical lapse rate toward 0.7 (dry adiabatic)

    Uniform surface, no seasonal cycle

    Run at T85 (256 x 128 in physical space) with 30 vertical sigma-levels

    600 days average after 1400 days spin-up

    (Held and Suarez, 1994; Schneider and Walker, 2006)

    An idealized dry general circulation model (GCM)

    Convenient to play with: We can change rotation rate, pole-to-equator temperature contrast, surface friction, convection, etc.

  • Contours: Zonal flow (m/s)

    Green line: Tropopause

    Sigm

    a30

    2010

    a

    60 30 0 30 60

    0.2

    0.8

    1

    0.5

    0

    0.5

    Latitude

    Sigm

    a

    40

    20

    10

    10

    b

    60 30 0 30 60

    0.2

    0.8

    1

    0.5

    0

    0.5

    Latitude

    Mid-latitude jet

    Surface westerlies

    Surface easterlies(trade winds)

    An idealized dry GCM: The mean zonal flow

  • Sigm

    a

    30 30

    20

    10

    20

    10

    10

    295

    320

    350

    a

    60 30 0 30 60

    0.2

    0.8

    30

    20

    10

    0

    10

    20

    30Colors: Eddy momentum flux (EMF) convergence

    Contours: Zonal flow (m s-1)

    Dotted lines: Potential temperature (K)

    Green line: Tropopause

    Eddy momentum flux (EMF)

    Friction on surface westerlies balances vertically averaged convergence of momentum

    Friction on easterlies (trade winds) balances vertically averaged divergence of momentum

    (Held 2000, Schneider 2006)

    u0v0 cosEM

    F co

    nver

    genc

    e (1

    0-6 m

    s-2 )

    Eddy zonal wind

    Eddy meridional wind

    Overbar:zonal-time mean

    Eddy momentum flux

    An idealized dry GCM: The mean zonal flow

    a = a+ a0

  • Sigm

    a

    53

    1

    3

    1

    5

    3

    1

    3

    1

    a

    60 30 0 30 60

    0.2

    0.8

    30

    20

    10

    0

    10

    20

    30

    Colors: Eddy momentum flux (EMF) convergence (10-6 m s-2)

    Contours: Mass stream function(1010 kg s-1)

    Dotted lines: Potential temperature (K)

    Green line: Tropopause

    Ferrel cell(Coriolis torque on the upper branch balances locally EMF convergence)

    Hadley cell(Coriolis torque on the upper branch balances locally EMF divergence)

    (Held 2000, Schneider 2006, Walker and Schneider 2006, Korty and Schneider 2007, Levine and Schneider 2015, etc)

    An idealized dry GCM: The mean meridional flow

    Stre

    amfu

    ncti

    on (

    1010

    kg

    s-1 )

    Eddy momentum flux

  • Heating the poles and cooling the equator

    Warm pole

    Cold tropics

    Near surface temperature

    Near surface relative vorticity

    Westerlies

    Easterlies

    (Ait-Chaalal and Schneider, 2015)

  • Heating the poles and cooling the equator

    Reversed insolation

    Latitude

    Sigm

    a

    22 2

    10

    20

    40 40

    60 30 0 30 60

    0.2

    0.810

    5

    0

    5

    10

    Latitude

    Sigm

    a

    295

    320

    350

    e

    60 30 0 30 60

    0.2

    0.81

    0

    1

    e

    Earth-Like

    EMF

    (m2 s-

    2 )St

    ream

    func

    tion

    (10

    10 k

    g s-

    1 )

    Latitude

    Sigm

    a

    30

    20

    10 5

    5 5

    60 30 0 30 60

    0.2

    0.8

    40

    30

    20

    10

    0

    10

    20

    30

    40

    Latitude

    Sigm

    a

    295

    320

    350

    f

    60 30 0 30 60

    0.2

    0.86

    0

    6

    Contours: Zonal mean flow (m/s) Dotted lines: Potential temperature (K) Green line: Tropopause

    (Ait-Chaalal and Schneider, 2015)

    EMF

    (m2 s-

    2 )St

    ream

    func

    tion

    (10

    10 k

    g s-

    1 )

  • Large-scale eddies and the general circulation

    Large-scale motion in the atmosphere is controlled by eddymean-flow interactions (e.g., Held 2000, Schneider 2006).

    Atmospheric flows look linear from macroturbulent scalings and do not exhibit nonlinear cascades of energy over a wide range of parameters (Schneider and Walker 2006, Schneider and Walker 2008, Chai and Vallis 2014)

    What happens if we retain eddy-mean flow interactions and neglect eddy-eddy interactions, in other words if we make a quasi-linear (QL) approximation?

  • Why is the QL approximation interesting?

    QL dynamics ~ closing the equations for statistical moments at the second order

    Is it possible to build statistical models to solve climate based on QL dynamics as a closure strategy?

    "More than any other theoretical procedure, numerical integration is also subject to the criticism that it yields little insight into the problem. The computed numbers are not only processed like data but they look like data, and a study of them may be no more enlightening than a study of real meteorological observations. An alternative procedure which does not suffer this disadvantage consists of deriving a new system of equations whose unknowns are the statistics themselves...."

    Edward Lorenz, The Nature and Theory of the General Circulation of the Atmosphere (1967)

  • The QL approximation

    Take for example the meridional advection of a scalar (zonal mean/eddy decomposition)

    a = a+ a0

    @a

    @t= v@a

    @y v@a

    0

    @y v0 @a

    @y v0 @a

    0

    @y@a

    @t= v@a

    @y v@a

    0

    @y v0 @a

    @y v0 @a

    0

    @ybecomes

    Equation for the mean flow:

    Equation for the eddies: @a0

    @t= v@a

    0

    @y v0 @a

    @y (v0 @a

    0

    @y v0 @a

    0

    @y).

    QL

    @a

    @t= v@a

    @y v0 @a

    0

    @y.

    Removing eddy-eddy interactions in the GCM:

    Eddy-eddy interactions

    (OGorman and Schneider 2007; Ait-Chaalal et al., 2015)

    @a

    @t= v@a

    @y= v@a

    @y v@a

    0

    @y v0 @a

    @y v0 @a

    0

    @y

  • The QL approximation conserves invariants consistent with the order of truncation, for example zonal momentum and energy (Marston et al., 2014). In the literature

    Stochastic structural stability (S3T) theory to study coherent structures in stable flows: Farrell, Ioannou, Bakas, Krommes, Parker, etc

    Cumulant expansions of second order (CE2): Marston, Srinivasan, Young, etc

    Some attempts to recover atmospheric statistics from linearized GCMs with a stochastic forcing: Whitaker and Sardeshmuck, 1998; Zhang and Held 1999; Delsole 2001Here: we look at unstable planetary baroclinic flows with large-scale forcing and dissipation.

    The QL approximation

  • Full

    The QL approximation: Mean zonal flow

    Contours: Zonal flow (m/s)

    Green line: Tropopause

    Sigm

    a

    30

    2010

    a

    60 30 0 30 60

    0.2

    0.8

    1

    0.5

    0

    0.5

    Latitude

    Sigm

    a

    40

    20

    10

    10

    b

    60 30 0 30 60

    0.2

    0.8

    1

    0.5

    0

    0.5

    (Ogorman and Schneider, 2007)

    QL

  • Eddy Momentum Flux Divergence

    Colors: Eddy momentum flux (EMF)

    Contours: Zonal flow (m/s)

    Dotted lines: Potential temperature (K)

    Green line: Tropopause

    The QL approximation: The eddy momentum flux

    EMF

    (m2 s-

    2 )EM

    F (m

    2 s-

    2 )

    Full

    Sigm

    a

    30

    2010

    a

    60 30 0 30 60

    0.2

    0.850

    0

    50

    Latitude

    Sigm

    a

    40

    10

    10

    b

    60 30 0 30 60

    0.2

    0.8 20

    10

    0

    10

    20

    (Ait-Chaalal and Schneider, 2015)

    QL

  • Eddy Momentum Flux Divergence

    Colors: Eddy kineticenergy (EKE)

    Contours: Zonal mean flow (m/s)

    Dotted lines: Potential temperature (K)

    Green line: Tropopause

    EKE

    (m2 s-

    2 )EK

    E (m

    2 s-

    2 )

    Full

    Sigm

    a

    30

    20

    10

    a

    60 30 0 30 60

    0.2

    0.8 100

    200

    300

    Latitude

    Sigm

    a

    10

    10

    40

    b

    60 30 0 30 60

    0.2

    0.8 150

    250

    350

    (Ait-Chaalal and Schneider, 2015)

    QL

    0.5 (u02 + v02)

    The QL approximation: The eddy kinetic energy

  • How is large-scale eddy decay captured in the QL model?

    Why is the eddy momentum flux not maximum in the upper troposphere in the QL model ?

    Why are weak momentum fluxes associated with high EKE in the QL model?

    The QL approximation: Summary

  • 5/29/13 7:28 PMMac App Store - GCM

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