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Cloud expansion
Geometrical aspects of the interaction between expanding clouds and environment Florin Spineanu, Madalina Vlad, Dragos Palade National Institute of Laser, Plasma and Radiation Physics Bucharest, Romania
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Cloud expansion
The expansion of a convective column It is important for parameterization It involves entrainment and detrainment. These are not well known, due to the diversity of situations. We focus here on geometry of the contact between the cloud and the environment. Three aspects will be studied: • The interface cloud/environment resulted from fingering instability • The interface cloud/environment affected by CUSP singularities • The breaking of the rising column
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This is a contribution to the study of the morphology of the rising convective columns, in view of a better description of the entrainment and detrainment. We examine technical methods for the description of the interface of expanding clouds and reveal the role of fingering instability which increases the effective length of the periphery of the cloud. Assuming Laplacian growth we give a detailed derivation of the time-dependent conformal transformation that solves the equation of the fingering instability. For the phase of slower expansion, the evolution of complex poles with a dynamics largely controlled by the Hilbert operator (acting on the function that represents the interface position) leads to cusp singularities but smooths out the smaller scale perturbations. We review the arguments that the rising column cannot preserve its integrity (seen as compacity in any horizontal section), because of the penetrative downdrafts or to the incomplete repulsion of the static
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environmental air through momentum transfer. Then we propose an analytical framework which is adequate for competition of two distinct phases of the same system. Each of the methods can be further developed. They offer parameters that allows to adapt them to applications.
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Cloud expansion
Many observational data support the idea that a rising column is not homogeneous in horizontal plane the vertical velocity has fluctuations.
(Malkus)
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Horizontal front of expansion (Sivashinsky) Sivashinsky.
(Sivashinsky)
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The slowing down and the dissipation of the cloud mass at the end Fingering?.
(Proccacia)
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Formulation of a simple model for interface instabilities The two fluids (cloud and external environment) are assimilated with two different phases separated by a moving interface Γ (t), a curve in the physical plane of coordinates [X (t) , Y (t)]. In a simplified representation, the interface is a line that extends between −∞ and +∞, i.e. it separates two regions of the plane. The region I is the inside of the cloud, enclosed by Γ (t) and the region II is outside, the environment. It is assumed that the velocity of the expanding fluid (the cloud) is the gradient of a scalar function P (X, Y ). This corresponds to the observation of many natural systems where the interface expands along the normal and with a speed proportional with the curvature. The 2D equation for the scalar function is Laplace: ΔP = 0 in the free (environment) space, outside the region enclosed by Γ (t). The physical reason for expansion is the input of cloud air from below the current height
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level. This is represented as an asymptotic condition for velocity: somewhere very far inside the cloud (Y → −∞), the velocity of the cloud air is a constant directed toward the interface ∇P = eY for Y → −∞
(1)
P = 0 at the interface Γ
(2)
vn = − (∇P )n at the interface Γ
(3)
In addition
The subscript n means projection of the vector ∇P on the normal at the interface Γ. We still need an initial condition for the system.
The time-dependent complex conformal transformation The idea is to map the physical plane onto the complex F. Spineanu – Bucharest Meteo 2015 –
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plane through a time-dependent conformal transformation. At every moment of time t, the line separating the two fluids is mapped through a conformal transformation to the fixed unit disk in a reference complex plane. The evolution of the interface is then a sampling through the family of condormal transformations. The lower semi-plane in the complex plane is mapped to the space below the interface Γ (t), where we have the Laplace equation for P . In order to get a complex variable from P , we add a new real function Ψ (Z) which will be taken as the imaginary
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part of a complex function W (Z) = P (Z) + iΨ (Z)
(4)
Then Ψ and P are harmonically conjugated. The conformal map is time-dependent such as to follow the motion of the interface f : C → C, Z ≡ f (z, t) where z = x + iy is in the lower half of the complex plane Since we assume that f (z, t) makes a conformal map between the lower half complex plane to the region under should have no singularities or zeros in the lower Γ (t), ∂f ∂z semi-plane. All of them must be in the upper semi-plane. F. Spineanu – Bucharest Meteo 2015 –
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Cloud expansion
Translating Eq.(1) it is expected that very at large distances on the plane, relative to the interface, the variables will have very close values f (z, t) ∼ z for z → x − i∞
(5)
which corresponds to constant velocity of the incoming air, the source being the air rising from below the current height. The system is rewritten: the new function (instead of the pressure) is W (z) and the new variables (instead of (x, y) )
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are z and z. ∂W = 0 (6) ∂z ∂W = −i for z → −i∞ at large distance from Γ ∂z ReW = 0 at z = x − i0 (at the mapped interface) and
Re n
∂W
∂f + ∂z ∂f ∂t
=0
(7)
∂z
here n is a complex number associated to the normal versor at the interface.
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The solution is W (z) = −iz
(8)
= −if −1 (Z, t) from where the pressure is
P (X, Y ) = Im f
−1
(X + iY, t)
(9)
The normal at the interface is n=
∂f ∂z −i ∂f ∂z
at z = x − i0
(10)
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then the equation describing the Laplacian growth is ∂f (z, t) ∂f (z, t) = 1 at z = x − i0 Im ∂z ∂t
(11)
(for z on the real x axis, just below) which is Polubarinova - Galin equation. Input data consists of taking a number of zeros and poles inside the unit disk of the complex plane. A complex function is generated and this represents the mapping between the plane where the interface exists to the complex plane with a centered disk representing one phase. The conformal transformation is dependent on time F. Spineanu – Bucharest Meteo 2015 –
Cloud expansion
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in a way that should reflect the time evolution of the interface between the two phases (or, more generally, two media with distinct properties). The prototype is the Hale Shaw system exhibiting the fingering instability. Very important: this approach to the study of the cloud/environment exterior interface dynamics has a useful particularity. Asymptotically there is a flow radially outward, normalized to 2π for the choice of f adopted by Ponce-Mineev. Then we can study indeed the expansion of the cloud into the environment, as a geometrical effect.
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A class of solutions in the infinite case finf (z, t) = z − it − i
N +1
αl log [z − ζl (t)]
(12)
αl ≡ αl + iαl are N + 1 complex constants
(13)
l=1
where
and (α , α ) are real. ζl ≡ ξl + iηl are N + 1 singularities simple poles of
∂f ∂z
(14)
that move in time.
What we need to examine here: • how to prescribe the zeros and the poles in the plane, F. Spineanu – Bucharest Meteo 2015 –
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inside the unit disk. • how the evolution keeps the zeros and the poles inside the disk. • how evolves the length of the circumference of the interface deformed by the effects of the presence of the singularities. • what is the effect of changing the number of poles/zeros. In particular is-there a good prescription that should be observed such that introducing (spontaneously born) new poles to reproduce even closer the formation of singularities at the interface (similar to what is seen in the reality)
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Fingering −17.5
−18
−18.5
Imag(f)
−19
−19.5
−20
−20.5
−21
4
6
8
10 xl
12
14
16
(imaginary part of f)
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Fingering −17.5
−18
−18.5
Imag(f)
−19
−19.5
−20
−20.5
−21 −5
0
5
10
15
20
Real(f)
(interface)
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Fingering 20
15
Real(f)
10
5
0
−5
4
6
8
10 xl
12
14
16
(real part of f)
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CUSP singularity of the interface The large scale dynamical structuring of the interface: cusp singularities and the possibility of air-enclosure at expansion We are here interested in the phase of the cloud expansion where the convection flux coming from lower levels is progressively reduced, the column reaching a regime of quasi-stationarity. On a large spatial scale (of the circumference of the expanding cloud) the interface has a dynamics of the expansion of a front. Qualitatively, it bears some resemblance to what results from the physical processes underlying the explosions. A model equation for the latter case has been developed by Sivashinsy. It treats the advancement of a front of a flame in a chanel. The front is unstable and it generates singularities in finite time. The singularities are of the cusp type. The first step in an analytic development is to extend the space
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coordinates to complex variables. The nature of processes that take place during time evolution is: (1) clusterization of poles along vertical direction in the complex plane, and (2) dynamical evolution of the poles towards the real axis. This produces singularities that look like cusps. We note that this evolution implicitely renders the interface piecewise smoother since it collects the pole singularities on a line parallel to the imaginary axis, just above the real-space position of the cusp. This corresponds to the situation where the flux of cloud air is more reduced.
Pole decomposition The problem is restricted to a single space variable, relative to which we measure the spatial position of the interface. The nature and the positions of the wrinkles of the interface can be associated to the
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singularities of the interface - function in the plane of the complexified spatial variable. A singular profile will occur in finite time but for short time the equation is integrable which means that it has the Painleve property and the formal solution can be written as an expansion in a set of order one poles. The individual contributions 1/ (x − zk ) have the particularity that they are eigenfunctions of the Hilbert transform. The eq. is essentially constructed on the ground of Burgers equation ∂u ∂2u ∂u +u = Λ [u] + ν 2 ∂t ∂x ∂x
(15)
The Hilbert transform acts by Λ [u] : u (t, k) → |k| u (t, k)
(16)
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where the Fourier transform of the function u has been introduced
+∞ u (t, x) = dk exp (ikx) u (t, k) (17) −∞
In the real form of the equation the operator Λ first makes the Fourier transform, then multiplies the Fourier transform of the function by |k| and returns to real variables. It will be shown later that this has the effect of pushing the poles towards the real axis. The operator Λ produces advection in the complex plane, in the imaginary direction, towards the real axis. This is because the solution expressed in terms of poles will contain terms like 1 pα = x − zα
(18)
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and the operator Λ is applied on them 1 Λpα = Λ x − zα =
sign [Im (z)] i
(19) ∂ pα (x) ∂x
The multiplication with |k| and is equivalent to the operator of spatial displacement ∂ (20) k→i ∂x We conclude that the operator Λ actually produces a spatial displacement in the vertical direction, parallel to the imaginary axis, towards the real axis (from above and from below since the operator multiplies with |k|). This approaches the poles to the real axis and raises the effect of the singularity, i.e. the real function which is the solution becomes even more perturbed there.
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The typical profiles are quasicusps, which are cusps with the tip rounded due to the presence of ν. In Thual this is explained by showing that the singularities never reach the real axis, but remains at a distance whose expression is given by ν. This will produce wrinkles. The nonlinear operator of Hilbert transform is the difference with respect to the Equation of Burgers ∂u ∂2u ∂u +u =ν 2 ∂t ∂x ∂x
(21)
where u≡
∂φ ≡ slope of the profile of the scalar function φ ∂x
(22)
where φ ≡ flame front displacement. The solution of the
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Sivashinsky equation is u (t, x) = −2ν
2N
1 x − zα (t) α=1
(23)
where zα (t) are 2N poles placed symmetrically, as complex conjugated pairs (this is why u (x, t) results real ) relative to the real axis, with the equations of motion 1 dzα = −2ν − isign [Im (zα )] dt zα − zβ
(24)
β=α
If the spatial domains is periodic u (t, x) = −ν
2π α=1
cot
x − zα 2
(25)
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and the equations of motion of the poles in the periodic case are dzα zα − zβ = −ν − isign [Im (zα )] cot (26) dt 2 β=α
Lee and Chen have found this application of the pole dynamics in the case of the Equation of Benjamin Ono. The poles tend to attract themselves horizontally, parallel with the real axis. Then there is a tendency to place themselves on a line parallel to the imaginary axis (this is the analog of the same phenomenon for the Burgers equation). The poles tend to repel each other vertically and the repulsion between two poles aligned on a vertical (y) line becomes infinite when they come too close. In the case of the Burgers’ equation this is translated in real space by formation and coalescence of shocks.
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Cusp singularity of the interface Time evolution of the singularities ζ(t) 0.7
0.6
yp = Imag(ζ)
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6 0.8 xp = Real(ζ)
1
1.2
1.4
(time evolution of positions of the poles)
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Cusp singularity of the interface The profile of the interface function h(x,t) 1
0
h(x,t)
−1
−2
−3
−4
−5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
xl
(interface)
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Cusp singularity of the interface
(cusps)
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The overshooting can lead to enclosing a mass of environmental air a situation that is hardly accessible to conformal transformation.
(Swallow)
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Breaking of the rising column competition of phases; coupled lattice maps. The coupled lattice maps are described by the set of equations for a discretized variable q (x, t) which represents the nonconserving order parameter at the points of a regular lattice q (i, t + 1)
=
f [q (i, t)] ⎡ ⎤ p +γ ⎣ q (j, t) − pq (i, t)⎦
(27)
j=n.n
where i ≡ (i1 , i2 , ...) is a set of integers that specifies the position of a point in the lattice. In our case the dimension of the problem is d = 2 (plane), i ≡ (i1 , i2 ) = (ix , iy ). The sum in the square paranthesis extends over the nearest neighbors (n.n) of the point i of the lattice. They are in number of p in general and in d = 2 they are p = 4. The
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square paranthesis is actually a discretization of the Laplace operator and this term represents the diffusion. The nonlinearity of the dynamics of the order parameter q (x, t) is introduced by the term f [q (i, t)] = −q 3 + (1 + ε) q + c
(28)
We recognize easily the meaning of choosing this nonlinearity. The “potential” f has two extrema, corresponding to non-dynamical equilibria. The order parameter can take one or another of these two equilibria values, and they are associated to the two states. The dynamical equation for q (x, t) is a discretized form of the Landau-Ginzburg equation.
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Breaking of the rising column competition of phases.
(Kapral)
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Progressive loss of compacity
Figure 1: Progressive loss of conectiveness.
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Breaking of rising column
(time evolution of positions of the poles)
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Breaking of the rising column
(breaking)
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Breaking of the rising column
(breaking)
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Cloud expansion
Breaking of rising column
(time evolution of positions of the poles)
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Breaking of the rising column
(breaking)
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Cloud expansion
Breaking of the rising column
(breaking)
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Breaking of rising column
(time evolution of positions of the poles)
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Breaking of the rising column
(breaking)
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Breaking of the rising column
(breaking)
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