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Low order models of the stochastic convection (self - organization at criticality) Florin Spineanu and Madalina Vlad National Institute of Laser, Plasma and Radiation Physics Bucharest, Romania
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When the low order models can still be interesting? Answer: when nothing else works. Or, we dispose of too much freedom, the space of solutions is too large. Low order models may be successful if there are regularities in the correlations.
Figure 1: Cahalan and Corral.
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1
SOC - Self - Organization at Criticality
- long range power law decay of the spatial and temporal correlations. Scale-free correlations. - heavy tail power law distributions of the sizes of the events The large separation between the driving and the relaxation time. Universality: the collective properties are largely independent on the physics of the individual subsystems. Out of equilibrium systems that present the following particularities: • they have a threshold - like response. Highly nonlinear. • a very slow driving rate F. Spineanu – COST-2014 Milano –
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• globally stationary regimes, stationarity of the statistical properties • power distribution of event sizes; no scale, typical for criticality. But no phase transition. The threshold is necessary for the separation of time scales and for creation of metastable states. Models are of the types: • algorithmic, like Bak - Sneppen: a set of rules • analytic: like the Kardar Parisi Zhang equation describing gradient drive v = −∇h for a scalar field h that may represent the local accumulation of a quantity (dust falling randomly on the bottom of a river).
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DRNG. All models require statistical analysis. The result of the statistical analysis are exponents for correlations (space or times).
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The SOC of the atmospheric convection processes Natural constituents and scenario of the convection
What is the natural development of a convection scenario? What are the natural constituents? F. Spineanu – COST-2014 Milano –
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Two processes are important (Fraedrich I) in the interaction between the ensemble of convective columns and the environment; 1. vertical flux of mass ; in particular the downdraft warms the environment air through adiabatic compression 2. exchange of heat between the column and the environmental air, lateral mixing. Observation: ”dynamical motion of tropical cumulo-nimbi, which travel faster than the vertically averaged tropospheric horizontal wind would allow” Citation: Cruz 1973. ”The radar observed progression of one hot tower is a sequence of growing deep cumuli one ahead of the other in the direction of the cloud motion”.
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Questions regarding the large scale evolution of the convection (propagation and clusterization) The organization of the convection into clusters • can be due to waves (later : gravitational waves invoked by Bretherton in formation of clusters) • or can be due to the propagation of a front that corrects the execess of water vapors. A maximum, strongly localized perturbation, that circulates across a region inducing instability and precipitation. It cleans the excess of water vapor. The Gierer Meinhardt model can be useful. This is another kind of avalanche: a front of perturbation that cleans the background where excess of vapors have been accumulated. The apparent manifestation is squall lines where the first convective motions are manifested.
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3
The Bak Sneppen model (random neighbor and K = 2)
At every time step (update) it is chosen the site with the minimum barrier xi and in addition K − 1 other sites and the fittness parameters are updated. The K − 1 other sites are chosen at random. (de Boer Derrida) few characteristics of the model random neighbor model. • all of the barriers are distributed uniformly in the interval 2 ≤ bi ≤ 1 3 and at the limit of infinite number of sites N →∞ F. Spineanu – COST-2014 Milano –
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(thermodynamic limit) only a vanishing fraction of barriers are in the interval 0 ≤ bi ≤ 23 . • the definition of an avalanche is: – fix a threshold barrier λ;
– count the number of active sites, which means count the number of sites that have a barrier less than λ. – an avalanche of temporal duration T is is said to occur when there are active sites for consecutive T temporal steps – duration of the avalanches has a probability that has the scaling form 1 Paval (T ) ∼ 3/2 T at large T . • there is a scaling low governing the time separation of the moments when the same site will be again the minimum from all F. Spineanu – COST-2014 Milano –
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the sites. If a site has been the minimum barrier at time 0 then it will be again minimum barrier after a time interval t with a probability 1 Pt (t) ∼ 3/2 t This is the all-return probability which refers to the situation that between t0 = the moment when the site i was the minimum; and t0 + t = the moment when the site i is again the minimum, it may happen that the site i has been possibly the minimum again. • the first return probability S (t) is defined as the probability that, if a given site i is the minimum at time t0 it will be again minimum - but for the first time - at time t0 + t. This means that between t0 and t0 + t the site i has not been minimum. The probability S (t) for the random neighbor model, scales as S (t) ∼
1 t3/2
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We will use the semantic substitution: species → sites. 3.0.1
Case with K = 2
This means that besides the sites with the smallest value xi only just another one xj is changed by replacing xj with a new, random, value. The site xj is chosen at random. We fix a real value for a parameter λ. Consider the number n of sites that have the value xi less than λ. Define Pn (t) as the probability that at time t there are n sites that have value xi lower than λ.
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This probability verifies the following master equation Pn (t + 1) =
N X
Mn,m Pm (t)
m=0
where Mn+1,n
=
Mn,n
=
Mn−1,n
=
Mn−2,n
=
n−1 λ −λ N −1 2
2
n−1 2λ (1 − λ) + 3λ − 2λ N −1 n−1 2 2 (1 − λ) + −3λ + 4λ − 1 N −1 n−1 (1 − λ)2 N −1 2
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M0,0
=
(1 − λ)2
M1,0
=
2λ (1 − λ)
M2,0
=
λ2
Results Now let us collect the results
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A1. xl < λ A.
x′k
>λ A2. xl > λ B1. xl < λ
B.
x′k
λ
A1.1
x′l > λ
A1.2
x′l < λ
A2.1
x′l > λ
A2.2
x′l < λ
B1.1
x′l > λ
B1.2
x′l < λ
B2.1
x′l > λ
B2.2
x′l < λ
2 n−1 (1 − λ) N −1 n−1 N −1 λ (1 − λ) 2 N −n (1 − λ) N −1 N −n N −1 λ (1 − λ) n−1 N −1 λ (1 − λ) n−1 2 N −1 λ N −n N −1 λ (1 − λ) N −n 2 N −1 λ
n−2 n−1
n−1 n
n−1 n
n n+1
At the limit N → ∞.
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We introduce these results in the expression for Pn (t + 1), Pn (t + 1)
=
λ2 Pn−1 (t) +2λ (1 − λ) Pn (t) 2
+ (1 − λ) Pn+1 (t) This is the result.
3.0.2
Avalanches in the Bak Sneppen model
It is defined an λ-avalanche: ”an evolution taking place between two successive times where the number n of sites lower than λ vanishes”. To make practical this definition one considers that an avalanche has started t temporal steps ago. This can be considered time 0. One defines the probability Qn (t) of having n sites with barriers
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xi < λ, conditioned by the situation that at t there was NO site less than λ. The probability Qn (t) verifies the same equation like Pn (t) but with the constraint M0,n → 0. The probability that an avalanche starts at 0 and ends at t is q (t) This is calculated by first obtaining the probabilities Qn (t) that, with an avalanche started at time 0, there are at moment t a number of n sites that are still under λ. Q1 (1)
=
2λ (1 − λ)
Q2 (1)
=
λ2
Qn (1)
=
0 for n ≥ 3 F. Spineanu – COST-2014 Milano –
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The result is Qn (t) =
2n (2t + 1)! λt+n−1 (1 − λ)t−n+1 (t + n + 1)! (t − n + 1)!
and since q (t)
=
(1 − λ)2 Q1 (t − 1) 2
(1 − λ) + Q2 (t − 1) N −1
we calculate the probability that an avalanche has duration t: q (t) =
(2t)! t+1 λt−1 (1 − λ) (t + 1)!t!
For large t the probability of an avalanche of duration t, q (t) has the
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asymptotic form t
(1 − λ) [4λ (1 − λ)] 1 √ q (t) ∼ λ π t3/2 with the limit at λ → 1/2 given by q (t) ∼
1 τ 3/2
which is specific to SOC.
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Atmospheric
(Bak Sneppen)
convection
probability of the duration τ
probability (Ole Peter)
q (t) ∼
1 τ 3/2
of an avalanche
of the duration D
N (D) ∼
1 D 1.42
of a precipitation event
probability of the first return
S (t) ∼
1 t3/2
at site i probability (Ole Peter) of the size M of a
N (M ) ∼
precipitation event
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1 M 1.36
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The Gierer - Meinhardt model of clusters and spikes in clusters (slow activator fast inhibitor). Spotty - spiky solutions
The model of the activator - inhibitor type. The equations ∂A ∂t ∂H τ ∂t
=
A2 ε ∆A − A + H
=
D∆H − H + A2
2
A, H > 0
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where A ≡
H
≡
concentration of activator at point (x, y, t) concentration of inhibitor at point (x, y, t)
we will name • the function A is activity at the location (x, y) • the function H is degree of chances for instability: potential instability. When the degree of chances of instability (potential instability) is larger than the local activity (H > A) the activity has not yet started. The activity A is low. The so-called inhibitor H does NOT suppress the activity but is just a measure in which this one has not yet started.
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When the potential instability is lower than the activity, H < A this means that the activity has started in that point and the growth of the activity is very efficient. The degree of fitness is the ”barrier” of the sub-system against mutation. It is then the inverse of the amount of vapor,
= =
”barrier” against mutation 1 fitness = CW V 1 H
is the inverse of the inhibitor. If there is large CWV (vapor) H in a point then this means that there is low fitness, low ”barrier” against mutation and the chances of instability are large. However when the
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vapor H is measured there is no or low activity A in that point. The operation of update consists of conversion of some vapor H into activity A: the vapor H decreases and the activity A increases, the barrier is increased as required by the Bak - Sneppen dynamics. The activity A may increase but not sufficiently such that A > H. Then what we get is just another landscape of A (x, y) and H (x, y) but not spiky solution. After the update we have a smaller H (less vapors) in the updated locations, which means higher barriers. The activity has also increased to a certain extent in those points. The standard Bak - Sneppen random dynamics has been realised by a random factor of conversion from the vapor content H to the activity A. After a sequence of such updates the activity A will be higher than initially, and with all points being at comparable degrees of activity. F. Spineanu – COST-2014 Milano –
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In the same time, the vapor has decreased in almost all points (i.e. the barriers have increased for those points).
4.1
Spikes in the solutions of the Gierer Mainhardt model
The system is treated in the work Metastable spikesolutions. ∂A ∂t ∂H τ ∂t
=
A2 ε ∆A − A + H
=
D∆H − µH + A2
2
for x∈Ω , t>0 and boundary conditions∂n A = 0 , ∂n H = 0 for x ∈ ∂Ω F. Spineanu – COST-2014 Milano –
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Conclusions The low order model may be successful for the parameterization problem of the convection in a grid cell. The basic model is Bak Sneppen system, a SOC reference algorithm, that appears to be quite adequate for a certain regime of convection. The exponents seem to be close to the observed ones. We still have to ask the adequate questions to compare the Bak Sneppen system with the atmospheric convection. The continuous version that seems to provide the same behavior is Giere Meinhardt. Exact solutions can be obtained, numerically. One of the results is the existence of a regime of the front that is propagating and cleans the excess of water vapor.
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