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Self-Organized Criticality A perspective from applied mathematics Florin Spineanu National Institute of Laser, Plasma and Radiation Physics Bucharest, Romania
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Outline • Construction of a concept: • examples, l how h to t recognize i it in i nature t • difference when compared to organization into coherent structures • some history • Sand Pile • the statistics of a collection of sub-systems at marginal stability • Abelian (Dhar), (Dhar) exact. exact Almost irrepetable irrepetable. • Continuum : Dynamic Renormalization Group • sand sand-pile pile and friends • temperature gradient in plasma • Naive considerations of states in the atmosphere p
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Basic elements for a Self-Organized system For natural systems like • coast lines • river networks • stock market • sand flow in a hourglass • traffic • other systems exhibiting 1/f noise, etc. there are no space or time scales. There is self-similarity, scale-invariance.
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The criticality is known from phase transitions. However there is a difference: - for phase transition the self-similarity and scale-invariance appear as the property of the system at few isolated points in the space of parameters (for example the temperature approaching the value for condensation). The state just before changing from a phase to another phase is called criticality. - for many system the self-similarity and scale-invariance appear without changing a parameter. Then this is called self-organized criticality.
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Preliminaries: few concepts of statistical physics
Basic habitudes • correlations, cumulants • criticality • here it would be a branching to Phase Transitions (no such thing here) • exponents. Waiting for better. • A system consisting of a collection of sub-systems. They are independent but can interact by exchange of a fluctuating quantity. • Each sub-system has an instability with threshold. • There is a source acting randomly over the sub-systems. The whole system evolves to a state which looks like the critical state preceding a phase transition. But it does not make any phase transition. The Self-Organized Criticality is a magnificent idea in search for a formalism.
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Statistical order: correlations
Random two-dim. topography
At the critical point the correlation length is infinite (like in percolation). All scales are simultaneously present.
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Paradigmatic example: sand pile automaton It is a model of dissipative transport. It exhibits avalanches. There are three regimes for the behavior of the sand pile. 1. a short-time regime in which temporal behavior is dominated by isolated avalanches. 2. an intermediate hydrodynamic regime in which the avalanches interact and generate a rich temporal structure 3. an anti-correlated event regime, due to system-wide discharges. The basic element of a system that exhibits discharge events is the existence of a threshold instability.
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Critical profile and toppling in the sand-pile model
(Newman, Tokamak Plasma)
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Basic elements of the sand-pile model The first example is 1 + 1 dimensional. The length of the discretized line is L. The variable of the system is the height in each discrete site, n, H (n, t) as a function of time t. The operations are H (n, t + 1)
=
H (n, t) − Nf
H (n ± 1, t + 1)
=
H (n ± 1, t) + Nf
with the condition for toppling H (n, t) − H (n ± 1, t) > ∆ The boundary at n = 0 is kept closed and the boundary at n = L is open. Sand grains are deposited randomly on L with an input rate Jin (t)
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The stationarity is reached when the input is balanced by the output at the open end, L. The output is monitored by the function output current Jout (t) and by the instantaneous energy dissipation rate E (t) which is the total number of transport processes in all system L, on a time step. (There are two meanings of dissipation. One, loss of potential energy at topplings, if we imagine a gravitational field. Second, the term dissipation comes from the possibility that less grains reach the other sites).
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Critical profile and toppling in the sand-pile model
(Newman, Tokamak Plasma)
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The quantities of interest: 1. the size of the avalanche s=
Z
E (t) dt
2. the duration T of an avalanche In steady state Bak Tang Wiessenfeld find signatures of criticality which consists of power-law scaling in the distributions of the quantities that are monitored T −y P (T ) = T F Lσ s P (s) = s1−τ G LDf F. Spineanu – Cambridge 2011 –
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where P is the distribution function and the two exponents are σ Df
≡ ≡
dynamical exponent fractal dimension of the avalanches
The stationarity is hJin i = hJout i The output is a stochastic process which is characterized by its spectrum in frequency Z Z SJ (ω) = dt dτ exp (−iωτ ) Jout (t) Jout (t + τ ) which presents scaling SJ (ω) ∼ ω −β F (ωLσ ) where F is a cut-off function imposed by the finite-size effects. F. Spineanu – Cambridge 2011 –
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Single avalanche regime The scaling of the power spectrum SJ (ω) in the regime of single-avalanche. √ 1 SJ (ω, L) = ω −2 √ F ω L L
The scaling is due to random superposition of independent avalanches. Interacting avalanches regime (the hydrodynamic region) At time scales beyond the maximum duration of individual avalanches (small ω) the transport quantities exhibit 1/f noise (the original reason for the SOC). Sj (ω) = ω −1 Here it is possible to use continuum-field description.
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The discharge-event region System-wide discharge processes. The threshold nature of the dynamics provides a multitude of metastable states. In short, the system behaves like a filter which acts on an input signal (which is the random noise) and converts it into a highly correlated output Z t Jout (t) = dt′ G (t − t′ ) Jin (t′ ) −∞
where G is the delayed response function. The occurence of great events is a common feature of a wide variety of driven systems that possess threshold instabilities.
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A moment of reflection What the concept of Self-Organization at Criticality can offer? • stationary states for large collection of sub-systems, each being at marginal stability. Large-scale response of the whole system • statistical correlations are power-law, i.e. critical • the system regulates the extraction of free energy from a reservoir. Looks similar to the minimum rate of statistical entropy production. What can we do analytically? • nothing (let the computers work) • exact analytic in few cases (Abelian sand) • Dinamically driven Renormalization Group Approach ( a relative of Renormalization Group Approach)
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Abelian sand-pile; exact solution (Dhar) This is from Jenssen book. The system consists of N sites to each of which it is assigned an integer variable zi , i = 1, ..., N This defines a configuration. The space of configurations is a function space in which every point represents such a distribution {zi }. The dynamics is introduced according to the rules: 1. the addition rule (input). In a site i chosen at random we add an unit zi → zi + 1 2. define N threshold values zic 3. define the matrix ∆ of dimension N × N , with the elements: F. Spineanu – Cambridge 2011 –
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(a) diagonal ∆ii > 0, ∀i (b) non-diagonal ∆ij 6 0, ∀i 6= j (c) condition of conservation or positive output N X j=1
∆ij > 0, ∀i
(d) toppling rule if zi zj
>
zic , then
→ zj − ∆ij for j = 1, ..., N
The space of configurations contains a subset of stable states S = {C ≡ {zi } |1 6 zi 6 zic , ∀i} F. Spineanu – Cambridge 2011 –
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Define the operators ai
:
S→S ai
adds one unit at site i
↓ relaxation
after relaxation all zi will be again below zic . This is the reason for which S is invariant under ai . The main property of ai , i = 1, ..., N is the irrelevance of the order in which they act on a configuration. They commute: ai a j − aj ai = 0 A particular class of configurations within S: recurent configurations R⊆S F. Spineanu – Cambridge 2011 –
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defined by the property that one allways return to the same configuration after repetitive application of the operators ai . i R = {C ∈ S| ∃mi : am i C = C, mi = integer, ∀i}
Only the recurent configurations occur with probability other than zero in the stationary state of the system. On R the operators ai are invertible. If the operator ai applied on two configuration C1 and C2 gives the same result, ai C1 = ai C2 then the two configurations are identical: C 1 = C2 All the configurations belonging to R (recurent configurations)
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appear with equal probability in the final stationary state of the system. To prove this we introduce two probabilities probability that the system P (C, t) is in configuration C at time t ′ W C→C transition probability under random ai and the master equation P (C, t + 1)
=
P (C, t) − +
X
C ′ ∈R
X
C ′ ∈R
P (C, t) W (C → C ′ )
P (C ′ , t) W (C ′ → C)
The transition probabilities W (C → C ′ ) result from the events consisting of application of operators ai . Choosing the operator ai is
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done at random with the probability P (ai ) ≡ pi The other factor in the W is the probability that the chosen ai connects indeed the two states C and C ′ , P (ai C = C ′ ) Here we use the invertibility of ai to focus on the initial state C instead of C ′ (on which we have to sum over). 1 if C = a−1 C ′ i P (ai C = C ′ ) = 0 otherwise Then returning to the master equation we have P (C, t + 1) = P (C, t) −
N X i=1
pi P (ai C, t) − P
a−1 i C, t
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and at stationarity we must have P (C, t) = =
const
1 where |R| ≡ number of elements in R |R|
It is proved that P (C, t) =
1 1 = |R| det ∆
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Correlation functions for Abelian sand-pile The two-point correlations Gij
=
the average number of topplings at site j induced by avalanches released when a particle is added at site i
It is found that
−1
Gij = ∆ A series of results become accessible
ij
1. Height-height correlations 1 hzi zj i − hzi i hzj i ∼ 2d where d = dimension r
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2. the probability that an avalanche consists of n topplings 1
P (n) ∼
n3/2
3. the probability density for the avalanche sizes P (s) ∼
1 sτ
where s is the number of distinct sites toppled during an avalanche, and τ
=
or,
=
5 4 1.253 from DDRG
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A second moment of reflection Suppose we are lucky enough and an ensemble of localised centres of convection are spread at the surface, within the boundary layer. The threshold on the equivalent potential temperature defines a marginal stability for each centre (= sub-system). Heat/moisture input at random triggers a convective event. Instead of running-away, this sub-system will send a message (what? downdraft effects on temperature, turbulent eddies?) to neighbors and then returns to marginal stability as before. The others will do the same and we discover with satifaction an Abelian model. The the distribution of sizes s of propagated effects of that message quantity should obey P (s) = s−5/4
(1)
Is that OK for us?
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Leaving the discrete- for the continuum-field description of the system exhibiting Self-Organized Criticality The usual examples of systems with SOC are discrete (e.g. sand-pile) and have discrete transformations (e.g. toppling). The models are usually examined by computer simulation. At a limit of a very large number of sites we can try to use coarse-grained variables and equations of motion. The main obstacle: the SOC systems have a threshold which in any attempt to go to a continuum description will require a strong nonlinear behavior. However if we succeed to find a continuum version, we can try to apply the Renormalization Group Approach. The RGA has been applied to obtain statistical properties of systems close to the critical state before a phase transition. The properties are expressed as scaling laws with exponents that can be calculated by F. Spineanu – Cambridge 2011 –
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RGA. The critical state of the system is identified in this approach by finding the fixed point of the flow of the transformations of the group. The method does not work well for systems with SOC, due to the threshold-type nonlinearity. A version is Dynamically Driven Renormalization Group Approach. The method can be applied to mean-field versions of the systems. discrete system
+
discrete dynamics
֒→ mean-field
+
֒→ diff.equations
↑
↑
Dynamically-driven Renormalization group
→ exponents
Mean-field consists of the estimate of the “on-the-average” behavior of many interacting degrees of freedom.
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The hydrodynamic regime for an anisotropic 2D system (Hwa and Kardar) The system is described by a function similar to the height of the sand pile h (x, y, t) which has an equation of evolution that can be derived from general considerations based on symmetries. The continuity of h (x, y, t) is acceptable in the regime of overlapping avalanches. The fact that the elementary influence between the sites (in the discrete version) is nearest neighbor allows us to consider a Lagrangian derivative for h under the influence of the noise (source) η (x, y, t) ∂h + ∇ · j [h] = η (x, y, t) ∂t where j [h] is the current of h expressed as a functional of the
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configuration h at time t. The form of the current is very general 2 j [h] = −a1 ∇h − a2 ∇ h − ... 2 ek + λ1 h + λ2 h + ... b +...
This general expression identifies a particular direction along which we expect avalanches to be formed, b ek .
Since we are interested in large spatial scales, this means k→0
which makes that the terms containing derivatives of higher degree to be small. The final form of the current, removing the linear term in h due to symmetry, contains the first power of the gradient
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∇ ≡ ∇⊥ + ∂/∂lk and the second power in the amplitude, h2 , j [h]
∂h b ek = −ν⊥ ∇⊥ h − νk ∂lk λ ek + h2 b 2
Another reason for the absence of the linear term, which would have been of the type 1 − h τ is that this term would introduce a time scale τ and correspondingly a space scale, via the diffusion-like terms that contain spatial gradients, 2 1/2 τ l≃ ν or, this would destroy the scale-invariance, a characteristic of
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criticality. The source is similar to the sand pile case, a white Gaussian noise D ′ ′ ′ E ′ ′ η (x, y, t) η x , y , t = 2Dδ 2 x − x δ t − t where D is related to the grain-deposition rate D ∼ p2 .
The equation is ∂h (x, y, t) ∂t
=
∂2h + νk 2 ∂lk 2 λ∂ h − + η (x, y, t) 2 ∂lk
ν⊥ ∇2⊥ h
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Dynamical Renormalization-group analysis The steps before have led us to a continuous field theory for a variable h (x, y, t) that obeys a nonlinear differential equation of the Langevin type, i.e. with a random source η (x, y, t). The result we are looking for is NOT an exact solution of this equation but the statistical properties of the field h (x, y, t). These statistical properties are expressed as scaling relations of the following type D E 2 [h (x, y, t) − h (x′ , y ′ , t′ )] ≡ C (x − x′ , y − y ′ , t − t′ ) The scaling looks like
D C (x, y, t) = F νk
νk t , 2 xk
r
νk x⊥ ν⊥ xk
!
where F is a scaling function with limiting behavior. If we can treat the nonlinearity as small then we can calculate
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perturbatively the effect of the fluctuations on the physical parameters of the problem D νk ν⊥
≡
≡ ≡
average amplitude of the random source η effective diffusion coefficient in the direction of avalanches effective diffusion coefficient perpendicular on the avalanches
This operation is called renormalization and consists of replacing the bare physical parameters D, νk and ν⊥ with effective (i.e. manifest) values which contain the effect of smaller scale fluctuations. For example, the modification which leads from the bare average amplitude D to the effective (renormalized) one is expressed as a series of powers of a quantity involving the length in the direction of the avalanches (xk ) and the coefficient of the nonlinearity λ. The operator on h2 is ∇ and one would expect that the distance mixing both xk and x⊥ is used instead of only xk but the nonlinearity is due F. Spineanu – Cambridge 2011 –
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to superposition and interactions of avalanches and these take place in the direction xk . The expansion is made in the variable ε λ xk where
ε=4−d and this exponent arises from invariance of the equation to scaling of the spatial variables in d = 2 dimensions xk , x⊥ and ensures the non-dimensionality of the expansion parameter. The expansions are then re-summed if the system is renormalizable, and expressed as scaling formulas h iβ1 D R = D 1 + α1 λxεk h iβ2 νkR = νk 1 + α2 λxεk
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h iβ2 R ν⊥ = ν⊥ 1 + α3 λxεk The procedure for the calculation of the renormalized quantities. The first step is calculation of naive dimensions, by looking for the change made in the equation when a scaling of the variables is performed xk → bxk t → x⊥
→
h →
bz t bζ x ⊥ bχ h
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where the formal exponents are known as z
≡
dynamic scaling exponent
χ
≡
roughness exponent
ζ
≡
anisotropy exponent
The equation becomes bχ−z
∂h ∂t
=
2 νk bχ−2 ∂k2 h + ν⊥ bχ−2 ∂⊥ h
λ − b2χ−1 ∂k h2 2 +b−z/2−(d−1)ζ/2−1/2 η The scaling of η comes from the scaling of D.
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The naive scalings are derived from the terms → bz−2 νk
νk
→ bz−2ζ ν⊥
ν⊥
→ bχ+z−1 λ
λ
→ bz−2χ−(d−1)ζ−1 D
D The second step
The perturbative calculations. The Fourier transforms Z Z 1 1 2 h (x, y, t) = dωd k h (kx , ky , ω) exp [i (kx x + ky y − ωt)] 2 2π (2π) the correlations of the noise ′
′
hη (k, ω) η (k , ω )i = 2Dδ
2
′
k + k δ (ω + ω ′ )
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The formal solution of the equation h (k, ω) =
1 η (k, ω) 2 2 νk kk + ν⊥ k⊥ − iω
λ 1 1 1 − 2 − iω 2π 2 νk kk2 + ν⊥ k⊥ (2π)2
Z Z
dΩd2 qh (q, Ω) h (k − q, ω − Ω)
The factor is the bare (i.e. non-renormalized ) propagator G0 (k, ω) =
1 2 − iω νk kk2 + ν⊥ k⊥
The perturbative treatment should absorb the nonlinearity into a renormalized propagator GR (k, ω) such that the linear relationship to be formally maintained h (k, ω) = GR (k, ω) η (k, ω) Then F. Spineanu – Cambridge 2011 –
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GR (k, ω)
=
G0 (k, ω) 2 Z k 1 1 λ k 2 2DG20 (k, ω) −q +4 dΩd q ik i k 2 2π (2π)2 2 k ω k ω k ω ×G0 − q, − Ω G0 + q, + Ω G0 − − q, − − Ω 2 2 2 2 2 2 +...
The integrals can be calculated and, since the interest is for large 2 scales (kk → 0), it is retained to O k . !# " −ε kk 3π R 2 2 G (k, 0) = G0 (k, 0) + G0 (k, 0) − νk kk u 32 ε
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where ε=4−d and u≡
λ2 D 3/2 3/2
νk ν⊥
2Sd−1 (2π)d
is the effective coupling constant. The renormalized propagator can be written 1 G (k, ω) = R 2 R k 2 − iω νk k k + ν⊥ ⊥ R
R but only νk is renormalized (ν⊥ = ν⊥ , D R = D and λR = λ).
The next step is to use recursion relations.
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For the parallel diffusion coefficient we consider a fixed spatial scale kk−1 = bl0 where l0 is the microscopic cut-off length. The dimensionless renormalized diffusion coefficient is νkR (bl0 )
z−2
which will be noted νekR (b). We know the expression of the renormalized νk which can be used νekR (b)
= νkR (bl0 ) =
z−2
ε 3π (bl0 ) z−2 νk (bl0 ) 1+ u + ... 32 ε
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The rescaling operator is applied b
3π ∂ R z−2 ε z−2+ νek (b) = νk (bl0 ) u (bl0 ) ∂b 32
Now it is assumed renormalizability, which allows us to replace νk (bl0 )z−2 → νekR (b)
and u (bl0 )
ε
= ≡
R 2
λ DR 2Sd−1 2−d (bl ) 0 3/2 d 3/2 (2π) R νkR ν⊥
u eR (b)
due to the possibility to replace λ, D, ν⊥ with ”renormalized ” values, which are the same.
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It is introduced a length defined by l ≡ ln b and the equations of rescaling become d R 3π R R νe = νek z − 2 + u e dl k 32 d R R νe⊥ = νe⊥ (z − 2ζ) dl
d eR eR λ = λ (χ + z − 1) dl
d eR e R [z − 2χ − (d − 1) ζ − 1] D =D dl
From the requirement that the renormalized parameters should be dimensionless in the thermodynamic limit b → ∞ or l → ∞ F. Spineanu – Cambridge 2011 –
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it results that the derivative in the Left hand sides of the scaling equations are zero d R νek → 0 dl which gives the system z − 2ζ
=
0
χ+z−1
=
0
z − 2χ − (d − 1) ζ − 1
=
0
with the solution χ=
1−d 6 3 , z= , ζ= 7−d 7−d 7−d
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Remember that during the calculation of the one-loop correction To the propagator G we have introduced the effective coupling constant , u
λ2 D 2S d −1 u = 3/ 2 3/ 2 ν || ν ⊥ (2π )d
After scaling and assuming the renormalization, we get
u (bl0 ) = ε
(λ ) D R 2
R
(ν ) (ν ) R 3/ 2
||
⊥
R 3/ 2
2 S d −1 4−d ~ R (b ) ( ) bl ≡ u (2π )d 0
After introducing the variable l=ln(b) we will apply as for the others, the operator ~ R b using the fact that only d/dl on the expression of u d R
()
Will appear, the others not being affected by renormalization. We get the equation:
dl
ν~||
d ~R ~R ⎡ 9π ~ R ⎤ u = u ⎢(4 − d ) − u ⎥ dl 64 ⎦ ⎣ The equation can be solved below d = 4 and identifies a stable fixed point
u~ *
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The SOC for the temperature in plasma This is from Spineanu & Vlad Comments Plasma Phys.Controlled Fusion 18 (1997) 115.The equation h i ∂ ∂T ∂T ∂ ∂T ∂T ∂T = χ + α − g T + χ + α x x x y y ∂t ∂x ∂x ∂x ∂y ∂y ∂y − gy T + +p + η
(2) where χx , χy are thermal diffusion coefficients corresponding to the radial x and poloidal y directions, gx and gy are the components of the prescribed marginal stability profiles and αx , αy are coefficients. The source is p and the random perturbation by η. The Fourier transform Z Z 1 1 d i(kx x+ky y−̟t) T (x, y, t) = d̟d k T (k , k , ̟) e (3) x y d 2π (2π) The noise is supposed gaussian with the correlation: hη (k, ̟) η (k′ , ̟ ′ )i = 2Dδ d (k + k′ ) δ (̟ + ̟ ′ )
(4)
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The equation becomes:
= 1 1 + 2π (2π)d
1 (2π)d
RR
R
−i̟ +
χx kx2
+
χy ky2
T (k, ̟) =
dd q (−iαx gx (q) − iαy gy (q)) T (k − q, ̟) +
dd qdΩ (−αx kx qx − αy ky qy ) T (q, Ω) T (k − q, ̟ − Ω) + +p (k) δ (̟) + η (k, ̟) (5)
We develop a standard statistical treatment in order to determine the correlations of the fluctuating function T (x, t) induced by the random perturbation η (x, t). We start by defining the generating functional of the correlation
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functions of T (x, t) : Z=
Z
Z i D [T (k, ̟)] D Te (k, ̟) exp (iSde ) exp i dd kd̟ Te (−k, −̟) η (k, ̟) h
η
(6) where Te (k, ̟) is the conjugate field to T (k, ̟), D is the functional measure and Sde is the deterministic part of the action. The averaging over η is performed using the two-point correlation and we obtain: Z h i h i Z = D [T (k, ̟)] D Te (k, ̟) exp iS T, Te (7)
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with: h h i R 1 1 d 2 λ 2 e e iS T, T = i 2π (2π)d d kd̟ − −i̟ + χx kx + χy ky T (−k, −̟) T (k, ̟) + R d 1 + (2π)d d q (−iαx gx (q) − iαy gy (q)) Te (−k, −̟) T (k − q, ̟) + R d 1 1 +λ 2π (2π)d d qdΩ (−αx kx qx − αy ky qy ) Te (−k, −̟) T (q, Ω) T (k − q, ̟ − Ω) + +p (k) δ (̟) Te (−k, −̟) + i +iD Te (−k, −̟) Te (k, ̟)
(8)
where we have introduced a parameter λ in front of the nonlinear term. To this action one can add two terms of linear interaction with external currents: h i h i R d 1 1 λ λ e e iSJ T, T = iS T, T + i 2π (2π)d d kd̟ [T (k, ̟) J (−k, −̟) i +Te (k, ̟) Je (−k, −̟)
(9)
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In order to simplify the calculations we adopt gx (x) = g x = const. and gy (x) = 0. Then the quadratic part in the action becomes: i h iSJλ=0 T, Te = h R 1 1 2 d 2 e = i 2π (2π)d d kd̟ − −i̟ + iαx g x + χx kx + χy ky T (−k, −̟) T (k, ̟) + +iD Te (−k, −̟) Te (k, ̟) + i +T (k, ̟) J (−k, −̟) + Te (k, ̟) Je (−k, −̟) + p (−k) δ (̟) (10)
We can now perform the functional integrals and obtain from the linear part of the equation: h h i n R d (0) d k d̟ J (−k, −̟) G Z λ=0 J, Je = exp 2i (2π) d 2π 20 (k, ̟) J (k, ̟) + io ′ (0) +Je (k, ̟) G11 (k, ̟) J (−k, −̟)
(11)
where we have introduced notations for the matrix of Green
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functions: (0) G20
(k, ̟) =
(0)
2iD 2
(̟ − αx g x kx ) +
G11 (k, ̟) =
χx kx2
+
2 2 χy ky
1 −i̟ + iαx g x kx + χx kx2 + χy ky2
(12)
(13)
and the current has been modified to include the source term: ′ e J (k, ̟) = Je (k, ̟) + p (k) δ (̟).
We perform now a perturbative treatment of the nonlinearity, replacing the field by the corresponding functional differentiations with respect to the external currents. The renormalization procedure is applied to the vertex functions, whose generating functional is obtained by the Legendre transform from W = ln Z. The first three
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vertex functions in the lowest order are: (0)
Γ11 = −i̟ + iαx g x kx + χx kx2 + χy ky2 (0)
(0)
Γ02 = −2D
(14)
Γ21 = λ (αx kx qx + αy ky qy ) The correlations of the field T are the Green functions obtained by functional derivatives of the functional W = ln Z , as for example: 2 δ Z 1 G20 (k, ̟) = hT (k, ̟) T (−k, −̟)i = Z δJ (k, ̟) δJ (−k, −̟) J=0 (15) The diagrams contributing to the two-point vertex functions are obtained from the diagrams of the two-point Green functions omitting the external lines. We want to calculate the one-loop approximation to Γ02 which requires to calculate the corresponding
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term of G20 . This gives the following expression: n R dd q dΩ (0) 2 1 2 L = G11 (k, ̟) λ − 26 m (α k q + α k q ) × x x x y y y (2π)d 2π o (0) (0) (0) ×G20 (−q, −Ω) G20 (−k + q, −̟ + Ω) G11 (k, ̟)
(16)
where m is the multiplicity of the graph, m = 28 . We are interested in the longwave and low frequency part of this correction, the hydrodynamic limit: ̟ → 0 and kx,y → 0. This corresponds physically to the regime of overlapping avalanches, i.e. to the excitation of the threshold nonlinearity by high amplitude noise, which is the case we consider here. As explained before the two external propagator lines are omitted in the calculation of the correction to Γ02 . The Ω integrations are performed with the theorem of the residues. This gives the following
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result: L = −2D 2 αx2 kx2
1 χ3x
χx χy
(d−1)/2
2Sd−1 1 3π 1 −ǫ kx d−1 2π 4 ǫ (2π)
(17)
where we have introduced explicitly an infrared cut off in the k-integral and denoted ǫ = 4 − d . Sd−1 is the surface of the sphere in d − 1 dimensions.From this formula we obtain the one-loop approximation of the vertex 1 −ǫ Γ02 (k, ̟) = −2D 1 + Bd α2 kx2 kx (18) ǫ where Bd = (3π/4) 2Sd−1 / (2π)d and the effective coupling constant is: 1/4 χx 1 α = D 1/2 αx 3/2 (19) χy χx Similar calculations lead to corrections to the other vertex functions. F. Spineanu – Cambridge 2011 –
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54
In particular, for the vertex Γ11 the correction which contains also a factor kx2 represents a renormalization of the thermal conductivity coefficient χx . Due to the anisotropy of our problem (since we have taken the reference value of the gradient on the poloidal direction zero) the thermal conductivity χy is not renormalized. At the fixed point of the renormalization transformation one can calculate the dynamic scaling exponents. The scaling consists in the transformation: x → bx from which the other parameters change: t → bz t, y → bζ y and T → bχ T . The correlations of the fluctuating temperature are: D E t y 2 [T (x, t) − T (x′ ,t′ )] ≡ C (x − x′ , t − t′ ) ∼ x2χ F , ζ (20) z x x The calculations give corrections having the same structure as in the approach of Hwa and Kardar leading to the following values for the
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exponents: χ=
1−d 6 3 ,z= ,ζ= 7−d 7−d 7−d
(21)
F. Spineanu – Cambridge 2011 –
What conclusion? We need a large number of centres of convection, each at marginal stability. Random input of heat/moisture at the surface should generate large-scale effect, as an avalanche, but with output not larger than, in average, the input. N
∑AM
The delay is essentially involved. Instead of
i
ij
Aj
j 1 j=1
We should say
t
N
∑ A (t ) ∫ dτG (t − τ )A (τ ) i
j =1
ij
j
−∞
reflecting the finite-time propagation of the effect of j on i.
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Natural convection as a heat engine Renno Ingersoll Thermodynamics around a streamline T ds 1 2 |v| + Cp T + Lv r + gz −d 2 −f · dl =
0
where
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T
is the absolute temperature
s
specific entropy
Cp
heat capacity at constant pressure
Lv
latent heat of vaporization of water per unit mass
r
water vapor mixing ratio
f
the frictional force per unit mass
dl
incremental distance along the streamline
Along a closed cycle, the formula above can be integrated I I T ds − f · dl = 0 the net heat input
=
the friction
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On the other hand
I
T ds =
I
pdV
where V is the specific volume and the equation is the first law of thermodynamics. This equality is simply the statement that the heat is equal with the mechanical work: dQ and dW = pdV ds = T I
T ds = =
net work done by the heat engine cycle total mechanical energy available for convection over one cycle
Then Total Convective Available Potential Energy
=
T CAP E =
I
T ds
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The Carnot cycle in convective circulation hot adiabat
entropy s2
updraft convection
cold adiabat
entropy s1
downdraft convection
hot isotherm
temperature Th
bottom of convective column
cold isotherm
temperature Tc
∼top of convective column
The relation T CAP E −
I
f · dl = 0
leads to the introduction of a coefficient of friction H f · dl energy in friction µ≡ = w2 energy in the vertical velocity Then w= µ
−1
T CAP E
1/2
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The dissipation of the mechanical energy associated with the convective motion is due to the turbulent viscosity νturb = l (δw) where l
characteristic length of the eddy
(m)
from instability
δw
the scale of the velocity change in eddy
(m/s)
from instability
The coefficient of dissipation 2
µ
∼ ∼
νturb ∇ w lpath w2 (δw)2 lpath w2 l
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can be estimated on the basis of the approximations for δw and lpath δw lpath
∼
∼
2w = velocity change across the fluid 4l = scale of the unstable layer
Convective quasi-equilibrium The approximative equality between: the rate of production of Total Convective Available Potential Energy rate of conversion of TCAPE to kinetic energy by convection rate of dissipation of kinetic energy by mechanical friction
The T CAP E is generated by the large-scale forces. Introduce the thermodynamic efficiency η of the heat engine R R T ds − out T ds in R η= T ds in F. Spineanu – Cambridge 2011 –
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-
(T CAP E)
3/2
is -
-
proportional with the square-root
µ1/2
of the dissipation coefficient
proportional with the efficiency
η , i.e.
the difference between the temperatures
Th − Tc
at the root of updrafts and root of downdrafts total heat input
Fin
inverse proportional to the surface of convective updrafts
σ
The input flux in the cycle ηFin ≈ ρσ (T CAP E) The continuity equation ρσw ≈ ρR (1 − σ) wR F. Spineanu – Cambridge 2011 –
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wR ≈ σw where it has been assumed that σ≪1 and ρR ≈ ρ Then 2
from where
ηFin ≈ ρwR µw 3 wR ηFin ≈ ρµ σ2
1/2 3/2 −1/2 µ δp Fin σ≈ η ρgτR ρ
Numerical data F. Spineanu – Cambridge 2011 –
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η
0.1
Fin ≈ F L ≈ δp
ρ g
W m2
155
8 × 104 (P a) 1
kg m3
9.81
m s2 6
efficiency heat input flux thickness of the radiating layer
air density gravity acceleration
τr
1.3 × 10 (s)
time scale of radiation
µ
16
viscosity coefficient
it results σ ≈ 5 × 10−4
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Regulation of the moist convection (Raymond 1995) In the tropical oceanic atmosphere, only the boundary-layer is subject to instability. If we assume that there is fast adjustment such as to get quasi-equilibrium, we need to explain how effectively the instability developing in the boundary layer is suppressed by convection. A possibility: reduce the equivalent potential temperature in the boundary layer. (With the conservation of the vertically integrated enthalpy of the atmosphere). Raymond: θe and rt evolves through a succesion of relaxation processes, controlled by the surface fluxes, within the convective layer. The latter is the defined by the height at which a non-entraining surface parcel reaches the neutral buoyancy. The relaxation is toward the saturated values, θes and rts . The convection is controlled by the difference in the buoyancy
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between the boundary layer parcel and the environmental air just above the boundary layer. Therefore there is a threshold which is manifested as a value of the equivalent potential temperature θethres . The system makes oscillations such that the boundary layer is forced to remain close to the threshold value for convection. The budget of entropy in the boundary layer
=
∂θeb ∂θeb + (uh · ∇h ) θeb + we ∂t ∂z Qeb mean radiative tendency of θeb Fes + surface flux/depth of the boundary layer b
Here we ≡ entrainment volume flux which means the volume per area per time at which the boundary layer incorporates the immediately overlying air. This has the
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dimension of velocity [we ] =
m s
The derivative is approximated we
∂θeb δθeb → we ∂z b
where δθeb ≡ decrease of θe across the top of the boundary layer The mean radiative tendency of the equivalent potential temperature is Qeb , Qb Qeb = θeb θb
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where Qb
≡
radiative heating rate dθ dt rad
=
where θ is the potentail temperature, θb ≡ the potential temperature evaluated at the boundary layer. The surface flux of θe is Fes = Cd Ue (δθe ) Cd δθe Ue
≈
10−3
≡
θes,s − θeb q ≡ |uh |2 + w2 ≡ effective wind speed F. Spineanu – Cambridge 2011 –
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θes,s ≡ saturated equiv. pot. temperature at the sea surface The definition of the threshold. The threshold equivalent potential temperature for convection θethreshold is the minimum value of θeb for which an ascending parcel can pass through the stable layer just above the cloud base. Soundings. A boundary layer parcel experiences a small region of negative buoyancy between 880 and 950 mb, with positive buoyancy above. (See picture from Raymond). Little or no buoyancy exists for parcels above the boundary layer. (Therefore if we want to trigger convection we can restrict to the boundary layer).
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The dominant role of the boundary layer
Boundary layer (Raymond 1995)
Buoyancy (Raymond 1995)
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The dominant role of the boundary layer (cont)
The equivalent potential temperature (1)
The equivalent potential temperature (2)
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An example of threshold value of the equivalent potential temperature θethreshold ∼ 350 K The energy barriers were 0.7 J/kg or, in a different case, 7 J/kg The energy barrier is translated into minimum vertical velocity of the parcel. The minimum vertical velocities in the two cases are 1.2 m/s and, in the other case 4 m/s Corresponding to these values of the vertical velocity, the horizontal
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values were 2 m/s and respectively 8 m/s which shows that there were large differences in eddies. The convective deficit defined as I ≡ θethreshold − θeb The threshold equivalent potential temperature for convection can be approximated by the saturated equivalent potential temperature averaged through a layer just above the cloud base.
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Sugestion: self-organized criticality may be the concept adequate for the state before major convective events
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Conclusions
F. Spineanu – Cambridge 2011 –
