Statistical state dynamics of jet/wave coexistence in beta-plane turbulence

Navid ConstantinouScripps Institution of Oceanography

UC San Diego

APS March Meeting 2016

Petros J. Ioannou National and Kapodistrian

University of AthensBrian F. Farrell

Harvard University

NASA/Goddard Space Flight Center

Can this jet/wave coexistence regime exist merely as a consequence of the underlying dynamics and in the absence of topography? Question:

large-scale jets and planetary-scale coherent waves coexist in planetary turbulence

u = (u, v) = (��y�, �x�)stochastic

forcingdissipation

r · u = 0

� = (0,�)

β is the gradient of the planetary vorticity

⇣z

� = (� � u) · z = ��

barotropic vorticity equation on a β-plane

�t� + u·�� + �v = �r� + ���� �� �+�

� �

ξ injects energy at rate ε at typical scales ddomain size�f/L � 1

zero mean statistically homogeneous in space

white in time

this model exhibits large-scale structures

but it is a nonlinear stochastic system…

to try to understand it we will construct a dynamics that governs its statistics

(statistical state dynamics)

Statistical State Dynamics (2nd order closure — S3T)

Z(x, t)def= PK [�(x, t)] =

�

|kx|�Kky

�(k, t) eik·x

� �(x, t)def= (1 � PK) [�(x, t)] =

�

|kx|>Kky

�(k, t) eik·x�(x, t)

C(xa,xb, t)def= �� �(xa, t)� �(xb, t)� �(xa,xb,xb, t)

def= �� �(xa, t)� �(xb, t)�

�(xc, t)�

incoherent eddy motions

, ,

coherent mean flow low zonal wavenumber

bandpass filter, e.g. . .

|kx| = 0, 1

hierarchy of same-time n-point cumulants:

A def= �U·� � [��x � (�U)·�]��1 � r + ��

where

neglecting cumulants 3rd order and above we get a closed, autonomous, deterministic system for the evolution of 1st and 2nd cumulants

R(C)def= �·

�z

2�

��a��1

a + �b��1b

�C

�

xa=xb

= �· �u�� ��

�tZ = ��V � rZ + ��Z � PK [U·�Z + R(C)]

�tC = (I � PKa)Aa C + (I � PKb)Ab C + �Q S3T

S3T turbulent fixed states

zero mean flow + non-zero homogeneous 2nd-order eddy statistics

Ue = 0 , Ce(xa � xb) extensively studied in the literature

U

e(x) =⇣U

e(y), 0⌘

, C

e(xa � xb, ya, yb)

zonal jet mean flow + non-zero zonally homogeneous 2nd-order eddy statistics

the focus of this work

S3T turbulent fixed states

zero mean flow + non-zero homogeneous 2nd-order eddy statistics

Ue = 0 , Ce(xa � xb) extensively studied in the literature

U

e(x) =⇣U

e(y), 0⌘

, C

e(xa � xb, ya, yb)

zonal jet mean flow + non-zero zonally homogeneous 2nd-order eddy statistics

the focus of this work

perturbations ( ) about S3T turbulent equilibrium state obey the linearized S3T equations:

�Z, �C

we linearized about a turbulent state!

�t �Z = A(Ue) �Z + PK [R(�C)]

�t �C = (I � PKa) [Aa(Ue)�C + �AaCe] + (I � PKb) [Ab(Ue)�C + �AbC

e]

solve the eigenvalue problem (non-trivial — very high dimensionality)

stability of turbulent states

zonal jet S3T equilibria

U

e(x) =⇣U

e(y), 0⌘

, C

e(xa � xb, ya, yb)

U

e(x) =⇣U

e(y), 0⌘

, C

e(xa � xb, ya, yb)

for each of them

by varying we find a series of zonal jet S3T statistical equilibria�

for the specified forcing structure and

0 π

2π 3π

22π

y

-4

-3

-2

-1

0

1

2

3

4

Ue

(nondimensional Earth-like values)� = 10 , r = 0.15 , � = 10�2

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0ε

-1.5-1.0-0.50

0.51.01.52.02.5

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0ε

-6

-4

-2

0

-6

-4

-2

0

kx = 0kx = 1

kx = 0

kx = 1S3

T gr

owth

rate

sph

ase

spee

ds

S3T stability predicted for the 2-jet S3T equilibria

kx : zonal wavenumber

� = 1.86

-2

-4

-6

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0ε

-1.5-1.0-0.50

0.51.01.52.02.5

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0ε

-6

-4

-2

0

-6

-4

-2

0

kx = 0kx = 1

kx = 0

kx = 1S3

T gr

owth

rate

sph

ase

spee

ds

S3T stability predicted for the 2-jet S3T equilibria

kx : zonal wavenumber

� = 1.86

-2

-4

-6

-4 -2 0 2 4Ue

0

π

2

π

3π

2

2π

y

hydrodynamic stability Vs S3T stability of the jet at .

hydrodynamic stability of the

laminar jet

S3T stability of the

turbulent jet

grow

th ra

tes

phase speeds

growth rate: growth rate:

� = 1.86

STABLE Vs UNSTABLE

energy of the deviation from zonal jet equilibrium over

energy of the zonal jet

growth and equilibration of S3T wave instability

� = 1.86

energy of the deviation from zonal jet equilibrium over

energy of the zonal jet

growth and equilibration of S3T wave instability

� = 1.86

conclusions

Planetary turbulence may bifurcate to a state in which coherent large-scale waves coexist with jets

These large-scale waves are equilibrated external Rossby waves destabilized by the turbulence

This work provides a new mechanism for understanding planetary scale waves in the atmosphere and may even provide explanation for the existence of the ovals that are embedded in the turbulent jets of the outer planets (e.g. Jupiter)

Constantinou, Farrell & Ioannou (2016) Statistical state dynamics of jet/wave coexistence in barotropic beta-plane turbulence, J. Atmos. Sci., doi:10.1175/JAS-D-15-0288.1, in press.

thanks