Dynamical Analysis of the
Snow Equation
Simon Fraser University
University of British Columbia
Department of Mathematics
Department of Physics
Pouya Bastani
Kevin Mitchell
Ralf Wittenberg
Tom Tiedje
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Outline
Introduction
The Model
Numerical Schemes
Numerical Results
Future Work
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Introduction
Suncups form on the surface of snow during the summer as a result of
melting and evaporation due to solar radiation
Two-dimensional patterns with a periodicity of 20-80 cm (characteristic
length) and typically 2-50 cm deep
Formed because hollows trap incident sunlight more efficiently and thus
melt faster than peaks, leading to instability
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The so-called “Snow Equation” is a mathematical model of snow surface
growth based on solar radiation, recently proposed by T. Tiedje et al.
T. Tiedje, K.A. Mitchell, B. Lau, E. Nodwell, Radiation Transport Model for
Ablation Hollows on Snowfields ,
J. Geophys. Res. 111 (2006)
Derived from the effect of the snow surface shape on the absorption and
diffusion of light in snow
Explains the shape, size, and dynamical behavior of suncups in terms of
the interaction of solar radiation with snow
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The Model
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Assumptions
h(x, y, t): surface height of snow relative to a horizontal reference plane
∂h
∂t
∝ rate of absorption of solar radiation (light-snow interaction)
Assumptions:
weak surface topography: |∇h| ≪ 1
vertical incident light
Limitations:
not accurate when suncups become deep
neglects the fact that on sloping surfaces suncups tilt toward the sun
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Light-Snow Interaction
Surface Topography:
Slope dependence: light at off-normal incidence to the snow
surface is more likely to be scattered out of snow
Curvature dependence: for concave surfaces light must travel
further through the snow to escape, and so more of it is absorebed.
Multiple reflections: radiation intensity increases proportionally to
the solid angle of snow subtended by the surrounding snow and its
albedo(The fraction of light scattered back into the atmosphere)
Flow of radiation in the snow
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The Snow Equation
Non-dimensionalized snow equation:
∂t h = −∇2 h − ∇4 h − α|∇h|2 + β∇2 |∇h|2
α2 + β 2 = 1
∇ is the Gradient, ∇2 is the Laplacian, ∇4 is the Biharmonic operator
Snow equation in one dimensional periodic domain
ht = −hxx − hxxxx − α(hx )2 + β(hx )2xx
x ∈ [0, L]
Similar equations were developed in other surface growth models by T.
Tiedje et al. at MBE Lab, UBC
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Kuramoto-Sivashinsky Equation
β = 0 gives the (integrated) Kuramoto-Sivashinsky (KS) Equation
ht = −hxx − hxxxx − (hx )2
Contexts: flame fronts, plasma ion waves, chemical phase turbulence, ...
An example of a deterministic dynamical system that exhibits complex
spatiotemporal phenomena and chaotic behaviour
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Fourier Space Evolution
On a periodic domain [0, L], Fourier representation of u(x, t) is:
h(x, t) =
∞
X
k=−∞
b
hk (t)eikx
1
b
hk (t) =
L
Z
L
h(x, t)e−ikx dx
k=
0
2nπ
L
P ′
2
4 b
b
hk′ b
hk−k′
KS in Fourier space: ∂t hk = (k − k )hk − k′ k (k − k ′ )b
Infinite system of ordinary differential equations
hxx term destabilizes large scales, hxxxx damps small-scale modes, hhx
provides nonlinear coupling between scales
Dispersion relation: γ = k 2 − k 4 , reality condition: b
hk = b
h∗−k
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Nonlinearity in real space ⇒ coupling of modes in Fourier space
Solutions with fundamental period less than L are unstable
Power Spectrum:
small modes (large scales): flat power spectrum
active scales: modes near the peak of the spectrum
large modes (small scales): decay rapidly
KS Power Spectrum
1
10
0
10
−10
10
0
|v|2
10
−20
10
−1
10
−30
−1
0
10
1
10
k
10
|v|2
10
−2
10
5
−3
h
10
0
−4
10
−5
−50
−5
−40
−30
−20
−10
0
x
10
20
30
40
10
−2
10
−1
0
10
10
1
10
k
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Numerical Schemes
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Finite Differences (FD)
Intuitive, and simple. Accuracy and numerical stability issues
θ-method for Heat Equation: ut = uxx
2 n+1
n
+ (1 − θ)δx2 unj
θδ
−
u
un+1
x uj
j
j
=
∆t
∆x2
δx2 unj = unj+1 − 2unj + unj−1
θ = 0: Euler Forward Difference Method - Explicit
stable if ∆t ≤ 0.5∆x2 , error = O(∆t + ∆x2 )
θ = 0.5: Crank-Nicolson Method - Implicit
unconditionally stable, error = O(∆t2 + ∆x2 )
θ = 1: Backward Difference Method - Implicit
unconditionally stable, error = O(∆t + ∆x2 )
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Spectral Methods
Periodic discrete real domain ⇔ discrete periodic Fourier domain
Discrete Fourier Transform (DFT) and Inverse DFT:
Fk {u} = u
bk = ∆x
N
X
−ikxj
uj e
u}
Fj−1 {b
j=1
xj = j∆x
1
= uj =
2π
N/2
X
k=−N/2+1
u
bk eikxj
x ∈ [0, 2π]
Real space differentiation ⇔ Fourier space multiplication:
ub′ (k) = ikb
uk
Fast Fourier Transform (FFT): O(N log N ) instead of O(N 2 ), N = 2n
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Integrating Factor (IF)
The snow equation is a “stiff" PDE requiring extremely small time
stepping to guarantee stability with FD Methods
Stiff because the magnitude of the fourth-order linear term is large in the
Fourier space for large modes: u
bt = −k 4 u
b + ...
Transform the PDE to solve the linear part exactly
bt = −(α + βk 2 )e−(k2 −k4 )t F {F −1 {ike(k2 −k4 )t U
b }2 }
U
Treat the nonlinear terms in real space
b = e−k2 t u
U
b
Allows 5 or 10 times larger time steps than FDM’s
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Exponential Time Differencing (ETD)
S. M. Cox, P. C. Matthews, Exponential Time Differencing for Stiff Systems,
J. Comp. Phys. 176 (2002), pp. 430-455
Developed the ETDRK4 (Exponential Time Differencing fourth-order
Runge-Kutta) method for solving stiff nonlinear PDEs
A.K. Kassam, L.N. Trefethen, Fourth-Order Time-Stepping For Stiff PDEs,
SIAM J. Sci. Comp. 26 (2005), pp. 1214-1233
Solved the problem of numerical instabilities and generalized to
nondiagonal operators
Modifications to the implementation by Kassam and Trefethen
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Adaptive Time Stepping
Achieve a predetermined accuracy with efficient time stepping
Calculate two independent solutions of the next time step by
independently taking one whole step and also two half steps
Estimate the error using these two solutions
Adapt the stepsize:
if error < threshold
increase time step accordingly and continue to the next time step
otherwise
decrease time step and repeat until desired accuracy is achieved
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Results and Observations
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Numerical Results
ht = −hxx − hxxxx − cos θ (hx )2 + sin θ (hx )2xx
π
θ ∈ [0, ]
2
x ∈ [−16π, 16π]
θ = π/4
θ = π/2
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Observations
Stationary or moving patterns, depending on the parameter θ
Characteristic length and amplitude depend on θ
14.5
14
13.5
13
char. len.
12.5
12
11.5
11
10.5
10
9.5
0
0.05
0.1
0.15
0.2
θ
0.25
0.3
0.35
0.4
0.45
Need larger domain size, or equivalently, smaller Fourier mode spacing,
to get more accurate results
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Future Work
Validity of the model
Generalization of the model to any angle of incident light
Field observations and comparison with numerical solutions
Further calculations of dynamical variables, such as characteristic,
amplitude, Lyapunov exponent, ...
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