Dynamical Analysis of the Snow Equation
Pouya Bastani - Ralf Wittenberg
pbastani@math.sfu.ca - ralfw@math.sfu.ca
Department of Mathematics - Simon Fraser University
Abstract
Kuramoto-Sivashinsky Equation
The Snow Equation is a mathematical model recently proposed by T. Tiedje
et al. to describe the ablation hollows or suncups that form on the surface
of snowfields in summer. This poster outlines the main ingredients of the
model partial differential equation (PDE) and shows how different factors
contribute to the formation of suncup patterns. The numerical schemes
used to study the dynamics of the PDE are described along with our present
numerical results and observations.
Numerical Results and Observations
• θ = 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
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
• As θ increases from 0 to π2 :
– chaotic and moving patterns become stationary
– characteristic length and amplitude grow
– edges become sharper (computation is harder)
– number of peaks in the power spectrum grows
– large modes decay less rapidly
KS Fourier Space Evolution
• On a periodic domain [0, L], Fourier representation of u(x, t) is:
• The so-called Snow Equation is a mathematical model of snow surface
growth based on solar radiation, recently proposed by T. Tiedje et al. [1]
• 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
h(x, t) =
∞
X
k=−∞
b
hk (t)eikx
2nπ
k=
L
• Fourier representation of the KS equation:
X
∂b
hk
k ′(k − k ′)b
hk ′b
hk−k ′
= (k 2 − k 4)b
hk −
∂t
′
k
• PDE is an infinite system of ODE’s
• hxx term destabilizes large scales, hxxxx damps small-scale modes, hhx
provides nonlinear coupling between scales
The Model
• h(x, y, t): surface height of snow relative to a horizontal reference plane
• Dispersion relation: γ = k 2 − k 4, reality condition: b
hk = b
h∗−k
• Nonlinearity in real space ⇒ coupling of modes in Fourier space
• ∂h
∂t ∝ rate of absorption of solar radiation (light-snow interaction)
• Snow-Light Interaction:
Future Work
– 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
– diffusion of light in the snow
• 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 quantities, such as characteristic length,
characteristic amplitude, Lyapunov exponent, ...
• Assumptions:
• Analysis of the 2D snow equation (mixed-derivative contribution)
– weak surface topography: |∇h| ≪ 1
– vertical incident light
Numerical Schemes
• Limitations:
– not accurate when suncups become deep
– neglects the fact that on sloping surfaces suncups tilt toward the sun
• Periodic discrete real domain ⇔ discrete periodic Fourier domain
• Discrete Fourier Transform (DFT) and Inverse DFT:
bk = ∆x
Fk {u} = u
Snow Equation
N
X
j=1
1
−1
−ikx
j
Fj {b
uj e
u} = uj =
2π
xj = j∆x
α2 + β 2 = 1
• ∇ is the Gradient, ∇2 is the Laplacian, ∇4 is the Biharmonic operator
∇4 ≡ (∂xx + ∂yy )2 = ∂xxxx + 2∂xxyy + ∂yyyy
• Snow equation in one dimensional periodic domain
ht = −hxx − hxxxx − cos θ (hx)2 + sin θ (hx)2xx
k=−N/2+1
u
bk eikxj
x ∈ [0, 2π]
• Real space differentiation ⇔ Fourier space multiplication:
• Non-dimensionalized snow equation:
∂th = −∇2h − ∇4h − α|∇h|2 + β∇2|∇h|2
N/2
X
π
θ ∈ [0, ]
2
• Similar equations were developed in other surface growth models by T.
Tiedje et al. at MBE Lab, UBC
hb′(k) = ikb
h(k)
• The snow equation is a ‘stiff’ PDE; Finite Difference Methods (FDMs)
require extremely small time stepping to guarantee numerical stability
• Stiff because the magnitude of the fourth-order linear term is large for
small-scale modes: b
ht = −k 4b
h + ...
References
• Integrating Factor [2]: Transform the PDE to solve the linear part exactly
• Allows 5 to 10 times larger time steps than FDMs
1. T. Tiedje, K.A. Mitchell, B. Lau, E. Nodwell, Radiation Transport Model
for Ablation Hollows on Snowfields , J. Geophys. Res. 111 (2006)
• Exponential Time Differencing Fourth-Order Runge-Kutta (ETDRK4) [3]
2. L.N. Trefethen, Spectral Methods in Matlab, SIAM (2000)
• Incorporated adaptive time stepping into ETDRK4
3. A.K. Kassam, L.N. Trefethen, Fourth-Order Time-Stepping For Stiff
PDEs, SIAM J. Sci. Comp. 26 (2005), pp. 1214-1233