Statistics of Atmospheric Circulations from Cumulant Expansions

Brad Marston & Florian SabouBrown University

Figure: NASA

Outline

• Prototypical problem: Barotropic Jet

• Direct Statistical Simulation (DSS) by Cumulant Expansions

• Two-layer primitive equations

• Stochastic Jet

• Conclusions

Barotropic Point JetM. R. Schoeberl & R. S. Lindzen, JAS 41, 1368 (1984)+ J. E. Nielson, JAS 43, 1045 (1986)

f(φ) = 2Ω sinφ

ω ≡ r · (∇× v)

∂ω

∂t+ (v · ∇)(ω + f) =

ωjet − ω

τ

Barotropic Point Jet

Statistics are much smoother in space than instantaneous flows & hence require fewer degrees of freedom to describe.

Statistics are also much stiffer in time than instantaneous flows & hence may be described by a fixed point or by a nearly fixed point.

Observations

Correlations are non-local in space.

qi = Ai +Bij qj + Cijk qjqk + fi(t)

fi(t) fj(t) = Γij δ(t− t)

qi = qi+ qi with qi = 0

ci ≡ qi = mi

cij ≡ qiqj = qiqj − qiqj = mij −mi mj

Equations of Motion and Statistics

Eddies sheared apartby mean flows

Mean Flowsand Eddies

q(m)

q(m)

q

CE2

q(m1) q(m2)

q(m1 +m2)

CE3

q(−m) q(m)

q

CE2

CE2:

CE3:

tau = 0 days

DNS: tau = 1.5625 days

CE: tau = 1.5625 days

DNS: tau = 3.125 days

CE: tau = 3.125 days

DNS: tau = 6.25 days

CE: tau = 6.25 days

DNS: tau = 25 days

CE: tau = 25 days

(a)

Me

an

Ab

so

lute

Vo

rtic

ity (

1/s

)

10 -5

-10

-8

-6

-4

-2

0

2

4

6

8

10

Latitude (degrees)

-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70

JBM, E. Conover, T. Schneider, JAS 65, 1955 (2008)

Flow only weakly non-linear.Mixing occurs even in absence of eddy-eddy interactions.

Two-Layer Primitive Equations (Held & Suarez 1978 & 1994)

P. OʼGorman & T. Schneider, GRL 34, 524 (2007)

T. Schneider and C. C. Walker, JAS 63, 1569 (2006)

Mea

n Po

tent

ial T

empe

ratu

re (C

)

−30

−25

−20

−15

−10

−5

0

5

10

15

Latitude−80 −60 −40 −20 0 20 40 60 80

Mean Potential Temperature vs. Latitude

DNSCE2

Zona

l Vel

ocity

−0.1

0

0.1

0.2

0.3

Latitude−80 −60 −40 −20 0 20 40 60 80

Mean Zonal Velocity vs. Latitude

DNSCE2

45 K equilibrium pole-to-equator gradient + 10 K stable stratification

60 K equilibrium pole-to-equator gradient + 10 K stable stratification

500.0 days Barotropic Mode

EQ

30N

60N

30S

60S

0 100 200 300 400 500DAYS

-10

-1

-0.1 0 0.1

1

10

DNS:

1000.3 days Barotropic Mode

EQ

30N

60N

30S

60S

0 1000DAYS

-10

-1

-0.1

0

0.1

1

10

CE2:

DNS:

CE2:

DNS:

JBM, arXiv:1008.2442 (to appear in Chaos)

CE2:

30 K equilibrium pole-to-equator gradient + 20 K stable stratification

1 Chua attractor

In electrical circuits field, a simple chaotic attractor, known as Chua attractor, has been observedwith an extremely simple autonomous circuit. It is third-order, reciprocal, and has only one non-linear element; a 3-segment piecewise-linear resistor [1]. A modified Chua-type attractor thatreplaces the 3-piecewise-linear function with a Heaviside function, can be obtained by varying theparameters in the equation of motion for the three variables.

0

2

4

x2

1

0

1y

2

1

0

1

2

z

Figure 1: Chua attractor with a Heaviside Term

The resulting attractor shows a basin of attraction and well defined 1st and 2nd moments ofdynamical variables, given that the initial conditions are within the basin. The numerical valuesfor the two parameters used in the simulation are A1 = −0.6 and A3 = −0.46. The equations ofmotion for the Chua-type attractor with a Heaviside non-linearity is given by Eq. [1].

x1 = x2

x2 = x3 (1)

x3 = A3x3 − x2 +A1x1 +H(x1)

The shape of the Chua attractor with a Heaviside term is shown in Fig. 1. From the equations ofmotion for the dynamical variables [1], we can derive the equations of motions (EOM) for the 1st

1

and 2nd cumulants. The EOM for the cumulants [2], contains terms such as H(x1), x1H(x1),x2H(x1) and x3H(x1). These terms can be expressed in terms of the 1st and 2nd cumulantsby assuming that the probability density function (PDF) of the dynamical variables is a Gaussianin three-dimensional space, equivalent to truncating the cumulants at second order (CE2).

c1 = c2

c2 = c3

c3 = A1c1 +A2c2 +A3c3 + H(x1)˙c11 = 2c12

˙c12 = c22 + c13 (2)

˙c13 = c23 +A1c11 +A2c12 +A3c13 + x1H(x1) − c1H(x1)˙c22 = 2c23

˙c23 = c33 +A1c12 +A2c22 +A3c23 − c2H(x1)+ x2H(x1)˙c33 = 2A1c13 + 2A2c33 + 2A3c33 − 2c3H(x1)+ 2x3H(x1)

The EOM terms containing the Heaviside function are evaluated symbolically in Mathematica,yielding error-functions and subsequently, the EOM themselves are solved numerically within thesame scientific software package. The results for the cumulants are compared with the resultsstemming from direct numerical simulation (DNS) in Table 1.

Table 1: The Chua attractor cumulant values given by EOM and DNS.

1st cumulant 2nd cumulant1.43 1.57 0.00 -1.50

DNS 0.00 0.00 1.51 0.000.00 -1.50 0.00 1.501.52 1.23 0.00 -1.23

CE2 0.00 0.00 1.23 0.000.00 -1.23 0.00 1.23

2

Chua Attractor

Stochastically Driven Jet

CE2:

DNS:

Zona

l Vel

ocity

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Latitude-80 -60 -40 -20 0 20 40 60 80

Zonal Mean Zonal Velocity

DNS (D=163,842) B = 0DNS (D=40,962) B = 0CE2 (L=100) B = 0CE2 (L=50) B = 0

Jet formation not due to inverse-cascade processes S. M. Tobias, K. Dagon, JBM, arXiv:1009.2684 (to appear in ApJ)

Conclusions

• Direct Statistical Simulation, by integrating out fast

modes, focuses on the slow modes of most interest.• Works for problems with non-trivial mean flows.• Accuracy can be systematically improved (eg. CE3)• DSS can lead to improved understanding.• DSS can be significantly faster than DNS.

• Offers a natural way to incorporate statistical models of subgrid physics.

Statistics of Atmospheric Circulations from Cumulant ExpansionsJ. B. Marston and F. Sabou

Large-scale atmospheric flows are not so nonlinear as to preclude their direct statistical simulation (DSS) by systematic expansions in equal-time cumulants. Such DSS offers a number of advantages: (i) Low-order statistics are smoother in space and stiffer in time than the underlying instantaneous flows, hence statistically stationary or slowly varying fixed points can be described with fewer degrees of freedom and can also be accessed rapidly. (ii) Convergence with increasing resolution can be demonstrated. (iii) Finally and most importantly, DSS leads more directly to understanding, by integrating out fast modes, leaving only the slow modes that contain the most interesting information. This makes the approach ideal for simulating and understanding modes of the climate system, including changes in these modes that are driven by climate change. The equations of motion for the cumulants form an infinite hierarchy. The simplest closure is to set the third and higher order cumulants to zero. We extend previous work (Marston, Conover, and Schneider 2008) along these lines to two-layer models of the general circulation which has previously been argued to be only weakly nonlinear (O'Gorman and Schneider, 2006). Equal-time statistics so obtained agree reasonably well with those accumulated by direct numerical simulation (DNS) reproducing efficiently the midlatitude westerlies and storm tracks, tropical easterlies, and non-local teleconnection patterns (Marston 2010). Low-frequency modes of variability can also be captured. The primitive equation model of Held & Suarez, with and without latent heat release, is investigated, providing a test of whether DSS accurately reproduces the responses to simple climate forcings as found by DNS.