| 30 Jun 2026
Rethinking fractal scaling of sea ice deformation: the dominant role of signed-gradient cancellation
Abstract. We use synthetic sea ice velocity fields to assess the robustness and physical interpretation of first-order scaling diagnostics, commonly associated with spatial localization in the sea ice modeling community. The synthetic fields are constructed to reproduce a wide range of RADARSAT Geophysical Processor System (RGPS)-derived deformation statistics while providing full control over signal-to-noise ratio, divergence-to-shear ratio, lead density, orientation and spatial localization. Results show that spatial scaling arises from the cancellation of signed velocity gradients in the coarse-graining procedure rather than from localization. For instance, dirac-delta deformation field mimicking idealized fracture lines does not show scaling in the absence of signed-gradient cancellation, and smoothly varying (non-localized) deformation fields yield scaling exponents comparable to observations when they contain both positive and negative velocity gradients. We further demonstrate that the standard fractal diagnostic is not invariant under rotation: a pure shear (non-divergent) deformation field exhibits different scale dependence when expressed in a rotated frame. These conclusions derived from synthetic deformation fields when applied to real sea ice deformation from RGPS and sea ice models from the Sea Ice Rheology Experiment (SIREx) show that observed scaling is similarly dominated by cancellation between positive and negative strain rates rather than geometric localization, and therefore are largely determined by PDFs of deformations. These findings reveal key limitations in current scale-invariance-derived diagnostics and call for physically interpretable and robust metrics.
Summary
Sea ice moves non-uniformly under the influence of wind, currents, and internal stress. The degree of non-uniformity, or deformation, may depend on the spatial scale over which it is calculated. Previous studies have found a power-law relationship between spatial scale and deformation with a negative exponent (i.e., larger deformation at smaller scales). This paper re-examines the reason for such a relationship. Previous studies have asserted that "localization" is responsible, in which the deformation originates from highly localized (small-scale) discontinuities or non-uniformities in the sea-ice motion. This paper finds that "signed-gradient cancellation" is responsible, in which positive and negative velocity gradients tend to cancel each other more and more at larger and larger scales. The paper also claims that standard invariant measures of deformation are actually not invariant under rotation.
Comments
The Introduction presents a good overview of sea-ice deformation and scaling.
Sections 2.2, 2.3, and 3.1 are full of notational problems (and a few math errors), making it difficult to understand or evaluate the main analysis of the paper. Details are given in the Math and Notation sections below.
The analysis is based on synthetic velocity fields defined on a grid shown in Figure 2. On lines 159-160, the authors write: "Note that off-diagonal terms of the deformation matrix (du/dy and dv/dx) have a larger stencil than the diagonal terms and therefore introduce asymmetry into the x- and y-directions." Yes, I can see that in Figure 2. The question is, why use such a strange stencil that introduces asymmetry when a very simple square stencil with no asymmetry could be used instead? Consider a square with velocity components (u,v) defined at the corners. All four velocity derivatives can be calculated for the square without asymmetry. Why not use this very simple and obvious stencil? Doesn't the stencil of Figure 2 introduce an undesirable property into the calculations and results?
Section 3.4 is titled "Scaling is not invariant under rotation". The Abstract also makes that claim (lines 9-10). The quantity under consideration here is the total deformation, which I'll call eps_tot, given by equation (5). I can assure you that eps_tot is invariant under a rotation of the coordinate system -- that is a mathematical fact. If a particular gridding scheme of the velocity field (u,v) leads to non-invariance of eps_tot under rotation, that is the fault of the gridding scheme. The authors need to make very clear that their result is due to the discrete spatial representation of (u,v) and not due to some property of eps_tot. If I am mis-interpreting the authors' argument then that argument needs to be clarified.
At lines 250-251, the authors say that the total deformation (eps_tot) at the smallest scale (DELTAx) and largest scale (N) can be derived from the PDFs of the velocity gradients alone, i.e. the spatial arrangement of the velocity gradients doesn't matter. In Marsan et al. (2004), which claimed that eps_tot vs. scale followed a power law, they randomly rearranged the velocity gradients and re-did the calculation of eps_tot vs. scale, finding non-power-law behavior. From this they concluded that the spatial arrangement of the velocity gradients does matter. How should the results of the present study be reconciled with those of Marsan et al. (2004)?
I'm wondering if the authors have simply re-discovered the Divergence Theorem and labelled it "signed-gradient cancellation". A version of the Divergence Theorem states:
Integral over a region (du/dx) dx dy =
Integral around the boundary (u) dy.
The left-hand side is the average value of du/dx times the area of the region. The right-hand side is an integral around the boundary of the region. In other words, when calculating the average value of du/dx over a region, all the interior values cancel, leaving only values on the boundary. This can be seen in a square grid, for example where the vertices are labeled A to I:
A B C
D E F
G H I
Suppose u(x,y) is defined at the 9 vertices.
There are 4 grid cells: ADEB, BEFC, DGHE, and EHIF.
The segment BE is common to the first 2 grid cells, the segment DE is common to the first and third grid cells, etc. The finite difference terms of du/dx from the interior segments (such as BE) all cancel, leaving only a sum of u around the outer boundary. That looks like "signed-gradient cancellation" to me. Another way to look at it is that the integral around ADEB (counterclockwise) traces the segment EB while the integral around BEFC (also counterclockwise) traces BE, i.e. in the opposite direction, resulting in cancellation. All the interior segments cancel because they are traced in opposite directions from adjoining cells. Anyway, I would be curious to know if the authors think that signed-gradient cancellation is something other than a consequence of the Divergence Theorem.
Math
Lines 115-116. The maximum shear strain rate, epsII, is not the determinant of the matrix L in equation (2). Suppose the eigenvalues of L are denoted by e1 and e2, with e1 > e2. Then epsII = e1 - e2. This is not the same as the determinant of L. The divergence epsI = e1 + e2 is the trace, which is correctly written.
Figure 1 caption, last line. The number of boxes is given as:
(N/n - 1)^2 for n = 1, 2, ..., 10. This is not correct.
The correct expression is: (N/p - 1)^2 where p = 2^(n-1).
In equation (8), the denominator on the right hand side should be 4*DELTAy, not 2*DELTAy. In equation (9), the denominator on the right hand side should be 4*DELTAx, not 2*DELTAx. If the authors used 2*DELTA in the calculations then they should re-do the calculations.
Notation
Line 103. L^n = 2^n * DELTAx. It took me a long time to realize that the "n" on the left-hand side is a superscript while the "n" on the right-hand side is an exponent. This is confusing. Why can't the "n" on the left-hand side be placed in the more usual subscript position to avoid confusion with exponents?
Equation (1). The quantity L is not defined.
Line 144. Apparently Nx = 1024. Why does it need the subscript x?
Line 149. In the equation for u(x), the variable x is used as both the independent variable and the dummy variable of integration.
Line 169. s = k * DELTAx for integer k, and DELTAx = 10 km. Now go four lines down to equation (10). Several problems here:
(i) The summation is over index k, but apparently the right-hand side depends on k through the definition of s.
(ii) eps0, defined on line 171, should be a function of two variables, but it is written as a function of one variable.
(iii) The argument of eps0 is k*s which is apparently k*k*DELTAx. This seems very strange.
(iv) The upper limit of the summation is Nx*Nx/s. Note that Nx is dimensionless (value = 1024) while s has the dimension of length through DELTAx, so the upper limit of the summation is not dimensionless.
Equation (11). Same problems as equation (10). Also, "N" on the left-hand side is not defined.
Equation (12). Same problems as equation (11).
Line 191. Here, s = 2^n. Now s is dimensionless. Is this the same s as previously introduced?
Minor Comments
RGPS data are available "from 1997 to 2008" (line 92) but "we use data from ... 1997 and 2008" (line 99). Just to clarify: only the first and last years of the available data are used, right?
Line 95. "model physics (EVP, VP, EAP, MEB)." These are better described as model rheologies. They should probably be spelled out somewhere (not necessarily here, it would be distracting).
Line 125. How is the weighting done in the "weighted average"?
Line 138. What is the "weighting procedure"?
Line 125. "This correction" -- what correction?
Lines 126-127. How could the field contain only one strain rate invariant? Is this meant to say that the second invariant equals zero?
Line 142. What is meant by the "strain-rate intensity"?
Figure 1. In panel (c), it's very difficult to see the "circles with black contour". Why not just use solid black circles for the mean values?
Figure 2 caption. The vectors (u,v) are defined on the horizontal and vertical *midpoints* of each grid box, not the vertices.
Line 171. It would be helpful to explain that |eps0| is actually eps_tot for the case where u = function of x only, and v = constant, as in Figure 3.
Line 243. "To test this hypothesis" -- what hypothesis?
Lines 254-255. "The departure from linearity of <eps_tot>N" -- I'm not sure what this refers to. Linearity with respect to what?
Lines 259-260. Deformation intensity and mean are given as percentages, but eps_tot is a dimensional quantity. What is the percentage based on?