## 1. Introduction

Declines in Arctic sea ice coverage, age, and thickness over the past few decades have been dramatic and appear to be accelerating, particularly during the warm season (Polyakov et al. 2012; Comiso 2012; Stammerjohn et al. 2012; Cavalieri and Parkinson 2012, and references therein). Record-breaking extent minima and abrupt changes in Arctic sea ice seasonality have been observed, including an abrupt increase in the amplitude of the seasonal cycle since 2007 (Stammerjohn et al. 2012; Livina and Lenton 2013). These changes in Arctic sea ice suggest scientifically important changes in the position, width, and area of the marginal ice zone (MIZ)—a dynamic and biologically active region that transitions from the dense inner pack ice zone to open ocean (e.g., Squire 1998; Wadhams 2000; Squire 2007; Weeks 2010; Barber et al. 2015). The width of the MIZ in particular is recognized as a fundamental length scale for climate dynamics and polar ecosystem dynamics (e.g., Wadhams 2000; Stroeve et al. 2016). The MIZ represents a region of intense air–sea ice interactions that have major effects on atmospheric boundary layer structure and meteorological processes (e.g., Shaw et al. 1991; Glendening 1994), and the MIZ provides a physical buffer that largely protects the more consolidated inner pack from the effects of ocean waves (e.g., Squire 2007). Increasing Arctic open water area may allow waves to evolve into swells that enhance sea ice breakup and further accelerate sea ice retreat (Thomson and Rogers 2014). The width of the MIZ and its variability are important drivers of marine habitat selection for a broad range of biota (Ribic et al. 1991; Perrette et al. 2010; Post et al. 2013; Williams et al. 2014), including Antarctic minke whales (Williams et al. 2014). Finally, changes in the MIZ also impact human accessibility to the Arctic, as broken ice in the MIZ is more navigable than dense inner pack ice (Stephenson et al. 2011; Schmale et al. 2013; Rogers et al. 2013).

Strong (2012) introduced an objective and automated method for identifying and measuring the width of the MIZ from satellite-retrieved sea ice concentrations. A follow-up application of the method to satellite data (Strong and Rigor 2013) revealed that the warm-season (July–September) Arctic MIZ widened over the past three decades by 39% while moving poleward and that the cold-season (February–April) Arctic MIZ narrowed by 15% over the same period. A representative warm-season sea ice configuration from early in the satellite record (Fig. 1a) shows a large region of inner pack ice (gray shading) surrounded by a narrow MIZ (white shading). More recently in the satellite era (Fig. 1b), the inner pack ice has retreated more rapidly than the marginal ice, leaving a markedly widened MIZ, particularly in the East Siberian and Beaufort Seas.

There are challenges associated with objective definition and automated analysis of MIZ width in part because of the nonconvex shape of the MIZ (e.g., Fig. 1b). In medical imaging, Jones et al. (2000) introduced a definition of the width of a nonconvex region as the arc length of a curve (streamline) along *ϕ* is the solution of Laplace’s equation (

The motivation of the present study is to provide a mathematically rigorous basis for defining MIZ width and MIZ spatial average width. The first novel component in the manuscript is articulation and illustration of three desirable mathematical properties for width: invariance with respect to translation and rotation, uniqueness at every point on the MIZ, and nonheuristic handling of both convex and nonconvex regions (section 2). The second novel component is the development of the mathematics of an annulus as an analytically tractable idealized MIZ geometry, enabling us to quantitatively investigate the response of various spatial averaging formulas to changes in MIZ shape (section 3). The results in sections 2 and 3 support a recommendation to define MIZ width using streamlines through the solution to Laplace’s equation within the MIZ and to average those results with respect to distance along the MIZ boundaries to yield the spatial average width. While the analyses in Strong (2012) and Strong and Rigor (2013) used these formulations, their mathematical justification was implicit and is here made rigorous. Finally, in section 4, we apply the streamline method to Arctic warm-season MIZ for 1979–2015 [a 3-yr extension of the results in Strong and Rigor (2013)], and we compare this extended time series to results from three alternative definitions that lack desirable mathematical properties or local width values but offer computational efficiency (one previously published, and two formulated for this study).

## 2. Definition of MIZ width

While the MIZ may be defined as the portion of the ice pack over which ocean waves significantly impact the dynamics of the sea ice cover (Wadhams 2000; Weeks 2010), several definitions of the MIZ based on sea ice concentrations have been used in recent studies founded on multidecade passive microwave satellite data (Strong 2012; Strong and Rigor 2013; Williams et al. 2014; Stroeve et al. 2016). Concentration-based definitions of the MIZ are also implemented operationally by the National Ice Center (NIC 2016). In most cases, concentration-based MIZ width is defined as the distance on the sphere between two concentration contours: typically 0.15, corresponding to the conventional ice edge (Comiso 2006); and 0.80, corresponding to what the World Meteorological Organization refers to as “close ice” (WMO 2009). We note that the methods for measuring width between the boundaries of the MIZ explored here do not require that the boundaries be defined by concentration thresholds, and they can applied in settings where the boundaries are defined by, for example, wave penetration or floe size (Williams et al. 2013).

Once the MIZ boundaries have been identified, a rationale is needed for defining the path along which the MIZ width is to be measured. Strong (2012) and Strong and Rigor (2013), for example, measured width along streamlines through the solution to Laplace’s equation within the MIZ, and we refer to this as the *streamline* definition. MIZ width has also been measured along meridians, and we refer to this as the *meridian* definition. Examples of the meridian definition in the literature often involve some form of time averaging prior to the width calculation. For example, Comiso and Zwally (1984) applied the meridian definition to MIZ boundaries identified in monthly mean concentrations, and Stroeve et al. (2016) averaged the latitude of the inner and outer edges of the MIZ at each meridian prior to calculating the distance between them. In medical imaging applications, Jones et al. (2000) note two other potential definitions: width defined as the shortest distance from a measurement point on one edge to the opposite edge, referred to here as the *shortest path* definition, and width defined as the distance along a straight line (geodesic on the sphere) that is orthogonal to the edge at the measurement point, referred to here as the *orthogonal geodesic* definition.

As a framework for considering the properties of these four definitions (streamline, meridian, shortest path, and orthogonal geodesic), we propose a set of three desirable mathematical properties for MIZ width:

Invariance: the width at every point in the MIZ is invariant with respect to translation and rotation of the ice field.

Uniqueness: the width at every point in the MIZ is uniquely defined.

Generality: the width at every point in the MIZ generalizes to nonconvex shapes without requiring heuristic or arbitrary rules.

Entries indicate which mathematical properties (rows) are satisfied by four width definitions (columns), as detailed in section 2.

### a. Invariance

We view width as an intrinsic property that does not change if a shape is moved or rotated on the sphere. The streamline definition yields widths that are invariant with respect to translation and/or rotation because the solution to Laplace’s equation provides width-defining streamlines that are determined by the shape of the MIZ and hence move with the MIZ, as illustrated by the idealized example in Figs. 2c,d. Results from the meridian method may change under translation and/or rotation as illustrated by the corresponding example in Figs. 2a,b (the shapes in Fig. 2 represent a single observation, but it could also represent MIZ boundaries diagnosed from time-averaged concentrations or boundaries constructed by time-averaging the latitude of the inner and outer MIZ edges). The shortest-path method and orthogonal-geodesic definitions are invariant with respect to translation and rotation because minimum-distance and orthogonality criteria are defined by the shape of, and hence move with, the MIZ.

(a) Dark gray shading represents a portion of an idealized inner pack ice region, and light gray shading represents a segment of an idealized MIZ. This panel is a polar stereographic projection with the pole indicated by the filled black circle near the lower edge of the panel, and the straight line indicates a meridian. The width of the MIZ through the red point as defined by the

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

(a) Dark gray shading represents a portion of an idealized inner pack ice region, and light gray shading represents a segment of an idealized MIZ. This panel is a polar stereographic projection with the pole indicated by the filled black circle near the lower edge of the panel, and the straight line indicates a meridian. The width of the MIZ through the red point as defined by the

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

(a) Dark gray shading represents a portion of an idealized inner pack ice region, and light gray shading represents a segment of an idealized MIZ. This panel is a polar stereographic projection with the pole indicated by the filled black circle near the lower edge of the panel, and the straight line indicates a meridian. The width of the MIZ through the red point as defined by the

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

### b. Uniqueness

We view width at every point in the MIZ as an intrinsic property that does not change depending on whether width is measured *from* the point or *to* the point. Stated differently, every point on one boundary of the MIZ is mapped to one and only one point on the other boundary of the MIZ by a width measurement path. The uniqueness property assures an unambiguous width for every point in the MIZ and is provided by the streamline definition. As illustrated in Fig. 3b, for every point in the MIZ (interior or boundary), the streamline definition provides exactly one width because every point is intersected by exactly one streamline. The width of the MIZ at point P in Fig. 3b, for example, is the arc length of the streamline intersecting P, and this value holds for all points along the streamline, including its point of intersection with the outer edge of the MIZ. The width at point P thus does not depend on whether the measurement is made from P or to P. The meridian definition also satisfies the uniqueness property because meridians do not intersect (except at the poles), and they uniquely map points between the two edges of the MIZ (unless curvature on an edge results in a meridian intersecting the same edge more than once; see discussion in section 2c).

(a) Dark gray shading represents a portion of an idealized inner pack ice region, and light gray shading represents a segment of an idealized MIZ in a stereographic projection with the origin at P so that geodesics intersecting P are projected as straight lines. (b) MIZ segment from (a) with the solution to Laplace’s equation (shading) and a subset of the streamlines along which width is measured (black solid and dashed curves). (c) As in (a), but with a modified outer edge of the MIZ. (d) MIZ segment from (c) with the solution to Laplace’s equation (shading) and a subset of the streamlines along which width is measured (black solid and dashed curves).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

(a) Dark gray shading represents a portion of an idealized inner pack ice region, and light gray shading represents a segment of an idealized MIZ in a stereographic projection with the origin at P so that geodesics intersecting P are projected as straight lines. (b) MIZ segment from (a) with the solution to Laplace’s equation (shading) and a subset of the streamlines along which width is measured (black solid and dashed curves). (c) As in (a), but with a modified outer edge of the MIZ. (d) MIZ segment from (c) with the solution to Laplace’s equation (shading) and a subset of the streamlines along which width is measured (black solid and dashed curves).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

(a) Dark gray shading represents a portion of an idealized inner pack ice region, and light gray shading represents a segment of an idealized MIZ in a stereographic projection with the origin at P so that geodesics intersecting P are projected as straight lines. (b) MIZ segment from (a) with the solution to Laplace’s equation (shading) and a subset of the streamlines along which width is measured (black solid and dashed curves). (c) As in (a), but with a modified outer edge of the MIZ. (d) MIZ segment from (c) with the solution to Laplace’s equation (shading) and a subset of the streamlines along which width is measured (black solid and dashed curves).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

As pointed out by Jones et al. (2000), seemingly natural or intuitive concepts such as the shortest distance and orthogonality to an edge work well in simple shapes, such as rectangles, but they can result in ambiguity and nonuniqueness when applied to regions with curved boundaries. To illustrate, consider the points *loss of reciprocity* or *loss of uniqueness* because there are at least two width measurement paths associated with point P. Similar ambiguities arise from the orthogonal-geodesic definition. For point Q in Fig. 3a, for example, the orthogonal geodesic across the MIZ is to point P, but the orthogonal geodesic from P returns to R rather than back to Q. These examples illustrate that the shortest-path and orthogonal-geodesic definitions do not satisfy the uniqueness property, meaning they yield results for a given point that may depend on the direction of measurement.

### c. Generality

The actual MIZ features curvature and concavity, which may render it nonconvex, as seen in the example in Fig. 1b. Such nonconvexity may require arbitrary or heuristic decision-making for methods based on lines (or geodesics on the sphere) as illustrated by the idealized example in Fig. 3c. For point P, the path PS in Fig. 3c indicates the geodesic along which the shortest-path definition and the orthogonal–geodesic method would be applied, and it could additionally represent a segment of a meridian used for the meridian definition. This geodesic crosses the MIZ (segment PQ), leaves the MIZ (segment QR), and then reenters the MIZ (segment RS), rendering the width at point P ambiguous and not uniquely defined in the absence of an arbitrary rule. For the same MIZ configuration, the streamline definition has a unique width at P along the streamline (bold curve, Fig. 3d) that terminates at the outer edge of the MIZ, and additional streamlines objectively handle the surrounding nonconvexity by establishing nonoverlapping width measurement paths that do not leave the MIZ (e.g., dashed curves, Fig. 3d). Every point along the MIZ outer edge in Fig. 3d is mapped to a unique point on the inner edge by a streamline that connects the pair of points without leaving the MIZ, and the black solid and dashed curves in Fig. 3d are illustrative examples.

In summary, only the streamline definition provides the three mathematical properties of invariance, uniqueness, and generality (Table 1). Each of the other three definitions considered provide only one of the three properties. The remainder of the manuscript thus proceeds using the streamline definition, but in section 4 we consider some computationally efficient alternatives that can approximate results from the streamline method when calculated as a spatial average over the MIZ.

## 3. Definition of MIZ spatial average width

Now that we have width defined for every point on the MIZ interior and boundaries based on the streamline definition (denoted

(a) Schematic indicating notation for the eccentric annulus model: *h*, and the annulus is denoted by *ϕ*), and (c) the imaginary part (*ψ*). (d) Shading indicates *ψ*) through that point.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

(a) Schematic indicating notation for the eccentric annulus model: *h*, and the annulus is denoted by *ϕ*), and (c) the imaginary part (*ψ*). (d) Shading indicates *ψ*) through that point.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

(a) Schematic indicating notation for the eccentric annulus model: *h*, and the annulus is denoted by *ϕ*), and (c) the imaginary part (*ψ*). (d) Shading indicates *ψ*) through that point.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

Shading indicates *ψ*) through that point. Each panel is an eccentric annulus example used to illustrate the effect of *h* and

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

Shading indicates *ψ*) through that point. Each panel is an eccentric annulus example used to illustrate the effect of *h* and

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

Shading indicates *ψ*) through that point. Each panel is an eccentric annulus example used to illustrate the effect of *h* and

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

To more rigorously investigate the properties of the spatial average width and support the recommendation to use

### a. Annulus model and Laplace’s equation

*ϕ*represents an idealized (smooth) sea ice concentration field within the MIZ modeled by the annulus with boundary conditions

*h*. Assume that the

*x*–

*y*plane in Fig. 4a represents a complex

*z*plane with

*a*is determined by the geometry of the annulus [

*ϕ*represents an idealized sea ice concentration field for the MIZ that transitions smoothly between its boundary conditions (sea ice concentrations 0.80 on the pack ice edge and 0.15 on the ocean edge), as shown for an actual MIZ configuration in Fig. 1c.

*ψ*through that point (example streamline curves representing the level sets of

*ψ*are shown by black contours in Figs. 1c,d, 4c). For the eccentric annulus model, we derived an explicit expression for

*ψ*,

### b. Candidate formulations for MIZ spatial average width

The streamline definition provides a unique width at every point on the MIZ (interior and boundaries), so there is potential flexibility in how the MIZ spatial average width can be defined. For example, width could be averaged with respect to area over the MIZ, meaning averaged over the white shaded region for the example in Figs. 1a,b. Alternatively, width could be averaged with respect to distance along a curve, such as the outer boundary of the MIZ, meaning the boundary between blue and white shading for the examples in Figs. 1a,b. We define and investigate the properties of various candidate formulations for MIZ spatial average width in this subsection, culminating in the recommendation to use width averaged with respect to distance along the perimeter

Average annulus width for six definitions of average width (definitions A–F). Columns indicate the associated formula, the equation number, and the average width for the examples in Figs. 5a–e.

*s*along the curve

*γ*is

*γ*. As a first specific case of Eq. (5), we consider width averaged with respect to arc length around the MIZ’s outer boundary

*ϕ*are circles (Fig. 4b).

For averages with respect to arc length along boundaries, we note that both the length

### c. Response to eccentricity

In this section, we examine how eccentricity affects results from the six candidate formulations for MIZ spatial average width. Our baseline case is the concentric annulus (Fig. 5a) whose average width

For various definitions of MIZ spatial average width, dependence on (a) eccentricity, (b) inner circle radius, (c) waviness on the outer edge quantified by *δ* in Eq. (14), and (d) waviness on the inner edge quantified by *δ* in Eq. (14).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

For various definitions of MIZ spatial average width, dependence on (a) eccentricity, (b) inner circle radius, (c) waviness on the outer edge quantified by *δ* in Eq. (14), and (d) waviness on the inner edge quantified by *δ* in Eq. (14).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

For various definitions of MIZ spatial average width, dependence on (a) eccentricity, (b) inner circle radius, (c) waviness on the outer edge quantified by *δ* in Eq. (14), and (d) waviness on the inner edge quantified by *δ* in Eq. (14).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

(a) MIZ width as a function of

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

(a) MIZ width as a function of

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

(a) MIZ width as a function of

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

*h*. The average with respect to distance along the inner and outer boundaries (

Probing further for a definition of average width that is invariant with respect to eccentricity, we consider averages with respect to distance along level sets of *ϕ* on the interior of the annulus [Eq. (10)]. The width averaged with respect to distance along circular level sets of *ϕ* is continuous and monotonic over the range *ϕ* (denoted *h*, yet it has order

Finally, we consider the average with respect to area

### d. Response to size of inner pack ice and total ice

Here we examine for zero and nonzero eccentricity how results from the candidate formulations for MIZ spatial average width respond to changes to the inner radius length relative to the outer radius length. A smaller inner radius will result in a larger average width for any reasonable definition. With zero eccentricity, all definitions yield

### e. Response to sinusoidal modulation of the boundaries

*δ*and

*f*are chosen to capture scales of variation salient in the observed examples (cf. Figs. 1a,b, 5d,e). The introduction of this waviness caused the area to increase by

Next we construct Fig. 5e by perturbing the radius of the inner circle as in Eq. (14) but with

The abovementioned analysis indicates that lengthening of one edge by waviness tends to modestly increase the average width measured along the wavy edge and more substantially decrease the average width measured along the nonwavy edge for all formulations. This effect is reduced by averaging the inner and outer boundary results (i.e., using *f* (not shown).

### f. Summary of sensitivity to MIZ shape

This subsection is a summary of findings in the preceding three subsections. As eccentricity increases, width averaged with respect to distance increases for the outer boundary and decreases for the inner boundary (section 3c). This sensitivity is considered undesirable and is largely eliminated by combining the inner and outer boundary results either as an arithmetic mean (

## 4. Application to satellite data

The preceding sections 2 and 3 motivate use of the streamline definition of MIZ width, with a spatial average width defined by averaging with respect to distance along both MIZ boundaries (

### a. MIZ spatial average width from streamline definition

The bold curve in Fig. 8a is the recommended

(a) MIZ average width for July–September based on analysis of satellite data. For the streamline definition,

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

(a) MIZ average width for July–September based on analysis of satellite data. For the streamline definition,

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

(a) MIZ average width for July–September based on analysis of satellite data. For the streamline definition,

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

The widening trend is consistent with the decline in the inner pack ice area outpacing the decline in total ice area. To visualize this, we plot the effective radius of the total ice area (^{−1} for ^{−1} for

### b. MIZ spatial average width from alternative definitions

We now consider three alternative definitions of MIZ spatial average width that are less computationally expensive than

For the third and final computationally efficient alternative, we consider here the MIZ spatial average width calculation that was applied to the Antarctic satellite record in Stroeve et al. (2016). This is the meridian definition discussed in section 2 implemented with a time-averaging operation. For this definition of the MIZ average width (

To complement the seasonal mean results presented above, we present a daily comparison of

As in Fig. 8a, but shown daily for July–September 2010.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

As in Fig. 8a, but shown daily for July–September 2010.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

As in Fig. 8a, but shown daily for July–September 2010.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

## 5. Summary and discussion

The width of the MIZ is a fundamental length scale for polar physical and biological dynamics, and a variety of width definitions have emerged in the literature over the past three decades. The streamline definition recommended here defines width as the arc length of streamlines through the solution of Laplace’s equation within the MIZ, and it features invariance with respect to translation and rotation, uniqueness at every point on the MIZ, and generality to nonconvex shapes. Our recommended definition for MIZ spatial average width, tested by sensitivity analyses using an annulus as an analytically tractable idealized MIZ geometry, is the average taken with respect to distance along the MIZ boundaries. We applied the recommended definitions to the warm-season (July–September) Arctic satellite record (1979–2015), extending the previously reported widening trend analysis by 3 yr and updating the total widening to 40%. The three most recent years in the record featured notable MIZ narrowing during two sequential years (2013–14), consistent with observed 25%–33% increases in sea ice volume in autumn 2013 and 2014 relative to the 2010–12 mean (Tilling et al. 2015).

Three computationally efficient alternatives to the streamline-based MIZ spatial average width were presented for comparison over the satellite record. While some approximations such as the difference in effective radii (

All three computationally efficient alternatives indicated statistically significant widening trends, suggesting that a broad range of reasonable definitions will agree on the presence or absence of trends, and in some cases the associated percent change (widening amounted to 33% for

In summary, we recommend using the streamline definition of MIZ width because of its mathematical properties (invariance, uniqueness, and generality), and we recommend defining MIZ spatial average as an average with respect to distance along the MIZ boundaries based on results of our sensitivity analyses. The basis for these recommendations is objective and mathematical. The context for these recommendations spans both MIZ physical dynamics and climate science research. When multidecadal time series are constructed, the focus is on spatiotemporal changes, including trends indicating the response of the ice field to natural variability and anthropogenic forcing. At the smaller spatial and temporal scales of MIZ physical dynamics, Laplace’s equation can be thought of as a steady-state heat equation, in which case its solution would define a smooth and monotonic decrease in temperature from the outer to inner edge of the MIZ that could in principle thermodynamically force a concentration increase in the same direction. Laplace’s equation is also the steady-state solution to Fick’s second law and thus defines a concentration field that would arise from a constant source (constant concentration) diffusing inward from the MIZ inner edge with a constant melting rate at the MIZ outer edge. In this way, while sea ice concentration does not formally obey it, Laplace’s equation is linked to potentially relevant thermodynamic and dynamic processes as a steady-state solution in the context of heat and diffusion. Advection is important in the actual MIZ, and this is not explicitly considered in Laplace’s equation. Ultimately, the goal of using Laplace’s equation is not to represent the sea ice physics, but to provide an idealized sea ice concentration field whose streamlines provide an objective definition of width satisfying invariance, uniqueness, and generality.

We now offer some remarks on the fact that the streamline definition uses the arc length of streamlines to define MIZ width as opposed to, say, great circles or straight lines on a stereographic projection. It is the use of streamlines that enables the streamline definition to simultaneously satisfy the properties of invariance, uniqueness, and generality. For simple planar shapes like rectangles, the width measurement path is a straight-line orthogonal to the opposing edges, yet the curvilinear streamline and planar straight-line definitions of width are not entirely unrelated. First, observe that the planar straight-line width for a rectangle is the shortest distance side to side across the shape, meaning a line orthogonal to two opposing edges. The streamlines through the solution to Laplace’s equation are also (by definition) orthogonal to the boundaries, and they thus take on curvature where the MIZ departs from simple rectangular (or spherical quadrangle) geometry (e.g., consider Fig. 3b). Second, we found that *minimal geodesic* defining the shortest path between the MIZ edges if distance were to be measured along the three-dimensional concentration surface. This three-dimensional mathematical concentration surface does not exist in physical space, and we emphasize that MIZ width is the arc length of the streamline along the surface of the earth rather than along the mathematical surface, but it is worth noting that each streamline does define a form of shortest distance. Recall also that measuring width along the shortest distance on the spherical earth is the shortest-path definition of width, which lacks the uniqueness property featured by the streamline definition as illustrated in section 2.

It should be recognized that other definitions of MIZ width may be more useful in operational settings where the context may be pragmatic rather than scientific. For example, the captain of a vessel is not likely to follow a streamline through the MIZ unless there is reason to traverse along the gradient of the (idealized) concentration field. The width in such an operational context might instead be more usefully defined as the shortest distance across the MIZ. In the absence of extreme curvature, though, note that the shortest-distance and streamline definitions are not markedly different (e.g., consider the streamlines in Figs. 2c,d). Local streamline-based MIZ widths may thus provide a useful overlay for operational resources such as the National Ice Center’s MIZ product (NIC 2016), especially for reasonably behaved MIZ geometries.

The MIZ continues to be the focus of innovative modeling and intensive field campaigns such as the Marginal Ice Zone Program (Lee et al. 2012), and much has been learned about, for example, the role of melt ponds in the summer breakup of Arctic sea ice cover (Arntsen et al. 2015), the role of lateral melting in the seasonal evolution of sea ice floe size distribution (Perovich and Jones 2014; Zhang et al. 2015), and the potential for sea ice prediction on seasonal time scales (Lindsay et al. 2012; Steele et al. 2015). Considering directions for continued research on MIZ width, Stroeve et al. (2016) analyzed MIZ trends for the Antarctic in all months except summer, when signal-to-noise ratios decline, and Strong and Rigor (2013) found a 39% widening of the warm-season Arctic MIZ with much smaller narrowing of the cold-season Arctic MIZ. Comparatively little has been reported on long-term change and variability in MIZ width during the Arctic transition seasons. Also, the approximately 40% widening of the Arctic warm-season MIZ may motivate modeling–observational analyses to understand the roles of thermodynamic versus dynamic, and oceanic versus atmospheric drivers of these changes.

## Acknowledgments

We gratefully acknowledge the support from the Division of Mathematical Sciences at the U.S. National Science Foundation (NSF) through Grants DMS-1413454 and DMS-0940249. We are also grateful for the support from the Arctic and Global Prediction Program at the Office of Naval Research (ONR) through Grant N00014-13-10291, and we thank the NSF Mathematics and Climate Research Network (MCRN) for its support of this work. Finally, we thank two anonymous reviewers for comments that helped to improve an earlier draft of the manuscript.

## APPENDIX A

### Solution to Laplace’s Equation

*z*plane. Its outer circle centered at the origin is scaled to have a unit radius

*w*plane (Fig. A1) via the linear fractional transformation (e.g., Brown and Churchill 2009)

*a*is given by

*z*plane is mapped to a circle on the

*w*plane with the center at the origin and radius

*w*plane, Laplace’s equation

*ϕ*and

*ψ*on the

*z*plane are given by

*ψ*through that point on the

*z*plane. On the

*w*plane, the level set

*z*plane as

Conformal mapping of the eccentric annulus from the *z* plane in Fig. 4a to the *w* plane.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

Conformal mapping of the eccentric annulus from the *z* plane in Fig. 4a to the *w* plane.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

Conformal mapping of the eccentric annulus from the *z* plane in Fig. 4a to the *w* plane.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

## APPENDIX B

### Existence and Uniqueness of

*ψ*exists and is unique, we will show that the function

*τ*goes to zero and to

*π*using L’Hôpital’s rule, as both limits are indeterminate. Using expressions for

*a*and

Now we show that the function *τ* and only on the domain mentioned earlier. Let us split this domain into two halves. First, consider the behavior of the function on the interval

From these analytic results, we glean several useful pieces of information. First, since the function (B1) is continuous, it has the intermediate value property. This translates to the fact that every length between *ψ*. The fact that Eq. (B1) is monotonic also means that the each unique length has a unique value of *ψ*. There exists no two streamfunctions, *ψ*, that have the same length on

Departure of *h*).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

Departure of *h*).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

Departure of *h*).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0171.1

## REFERENCES

Arntsen, A. E., A. J. Song, D. K. Perovich, and J. A. Richter-Menge, 2015: Observations of the summer breakup of an Arctic sea ice cover.

,*Geophys. Res. Lett.***42**, 8057–8063, doi:10.1002/2015GL065224.Barber, D. G., and Coauthors, 2015: Selected physical, biological and biogeochemical implications of a rapidly changing Arctic Marginal Ice Zone.

,*Prog. Oceanogr.***139**, 122–150, doi:10.5194/tc-6-881-2012.Brown, J. W., and R. V. Churchill, 2009:

*Complex Variables and Applications.*McGraw-Hill, 468 pp.Cavalieri, D. J., and C. L. Parkinson, 2012: Arctic sea ice variability and trends, 1979–2010.

,*Cryosphere***6**, 881–889, doi:10.5194/tcd-6-957-2012.Comiso, J. C., 2006: Abrupt decline in Arctic winter sea ice cover.

,*Geophys. Res. Lett.***33**, L18504, doi:10.1029/2006GL027341.Comiso, J. C., 2012: Large decadal decline of the Arctic multiyear ice cover.

,*J. Climate***25**, 1176–1193, doi:10.1175/JCLI-D-11-00113.1.Comiso, J. C., and H. J. Zwally, 1984: Concentration gradients and growth/decay characteristics of the seasonal sea ice cover.

,*J. Geophys. Res.***89**, 8081–8103, doi:10.1029/JC089iC05p08081.Glendening, J. W., 1994: Dependence of boundary layer structure near an ice-edge coastal front upon geostrophic wind direction.

,*J. Geophys. Res.***99**, 5569–5581, doi:10.1029/93JD02925.Jones, S. E., B. R. Buchbinder, and I. Aharon, 2000: Three-dimensional mapping of cortical thickness using Laplace’s equation.

,*Hum. Brain Mapp.***11**, 12–32, doi:10.1002/1097-0193(200009)11:1<12::AID-HBM20>3.0.CO;2-K.Lagarias, J., J. A. Reeds, M. H. Wright, and P. E. Wright, 1998: Convergence properties of the Nelder–Mead simplex method in low dimensions.

,*SIAM J. Optim.***9**, 112–147, doi:10.1137/S1052623496303470.Lee, C. M., and Coauthors, 2012: Marginal Ice Zone (MIZ) Program: Science and experiment plan. Applied Physics Laboratory, University of Washington, Tech. Rep. APL-UW 1201, 48 pp.

Lindsay, R., and Coauthors, 2012: Seasonal forecasts of Arctic sea ice initialized with observations of ice thickness.

,*Geophys. Res. Lett.***39**, L21502, doi:10.1029/2012GL053576.Livina, V. N., and T. M. Lenton, 2013: A recent tipping point in the Arctic sea-ice cover: Abrupt and persistent increase in the seasonal cycle since 2007.

,*Cryosphere***7**, 275–286, doi:10.5194/tc-7-275-2013.Meier, W., F. Fetterer, M. Savoie, S. Mallory, R. Duerr, and J. Stroeve, 2012: NOAA/NSIDC Climate Data Record of Passive Microwave Sea Ice Concentration, version 2 (updated 2016). National Snow and Ice Data Center, accessed 11 November 2016, doi:10.7265/N55M63M1.

NIC, 2016: Products. Naval Ice Center. [Available online at http://www.natice.noaa.gov/Main_Products.htm.]

Perovich, D. K., and K. F. Jones, 2014: The seasonal evolution of sea ice floe size distribution.

,*J. Geophys. Res. Oceans***119**, 8767–8777, doi:10.1002/2014JC010136.Perrette, M., A. Yool, G. D. Quartly, and E. E. Popova, 2010: Near-ubiquity of ice-edge blooms in the Arctic.

,*Biogeosciences***8**, 515–524, doi:10.5194/bg-8-515-2011.Polyakov, I. V., J. E. Walsh, and R. Kwok, 2012: Recent changes of Arctic multiyear sea ice coverage and the likely causes.

,*Bull. Amer. Meteor. Soc.***93**, 145–151, doi:10.1175/BAMS-D-11-00070.1.Post, E., and Coauthors, 2013: Ecological consequences of sea-ice decline.

,*Science***341**, 519–524, doi:10.1126/science.1235225.Ribic, C. A., D. G. Ainley, and W. Fraser, 1991: Habitat selection by marine mammals in the marginal ice zone.

,*Antarct. Sci.***3**, 181–186, doi:10.1017/S0954102091000214.Rogers, T. S., J. E. Walsh, T. S. Rupp, L. W. Brigham, and M. Sfraga, 2013: Future Arctic marine access: Analysis and evaluation of observations, models, and projections of sea ice.

,*Cryosphere***7**, 321–332, doi:10.5194/tc-7-321-2013.Schmale, J., M. Lisowska, and M. Smieszek, 2013: Future Arctic research: Integrative approaches to scientific and methodological challenges.

,*Eos, Trans. Amer. Geophys. Union***94**, 292–292, doi:10.1002/2013EO330004.Shaw, W. J., R. L. Pauley, T. M. Gobel, and L. F. Radke, 1991: A case study of atmospheric boundary layer mean structure for flow parallel to the ice edge: Aircraft observations from CEAREX.

,*J. Geophys. Res.***96**, 4691–4708, doi:10.1029/90JC01953.Squire, V. A., 1998: The marginal ice zone.

*Physics of Ice-covered Seas*, M. Lepparanta, Ed., Vol. 1, Helsinki University Printing House, 381–446.Squire, V. A., 2007: Of ocean waves and sea-ice revisited.

,*Cold Reg. Sci. Technol.***49**, 110–133, doi:10.1016/j.coldregions.2007.04.007.Stammerjohn, S., R. Massom, D. Rind, and D. Martinson, 2012: Regions of rapid sea ice change: An inter-hemispheric seasonal comparison.

,*Geophys. Res. Lett.***39**, L06501, doi:10.1029/2012GL050874.Steele, M., S. Dickinson, J. Zhang, and R. W. Lindsay, 2015: Seasonal ice loss in the Beaufort Sea: Toward synchrony and prediction.

,*J. Geophys. Res. Oceans***120**, 1118–1132, doi:10.1002/2014JC010247.Stephenson, S. R., L. C. Smith, and J. A. Agnew, 2011: Divergent long-term trajectories of human access to the Arctic.

,*Nat. Climate Change***1**, 156–160, doi:10.1038/nclimate1120.Stroeve, J. C., S. Jenouvrier, G. G. Campbell, C. Barbraud, and K. Delord, 2016: Mapping and assessing variability in the Antarctic marginal ice zone, pack ice and coastal polynyas in two sea ice algorithms with implications on breeding success of snow petrels.

,*Cryosphere***10**, 1823–1843, doi:10.5194/tc-10-1823-2016.Strong, C., 2012: Atmospheric influence on Arctic marginal ice zone position and width in the Atlantic sector, February–April 1979–2010.

,*Climate Dyn.***39**, 3091–3102, doi:10.1007/s00382-012-1356-6.Strong, C., and I. G. Rigor, 2013: Arctic marginal ice zone trending wider in summer and narrower in winter.

,*Geophys. Res. Lett.***40**, 4864–4868, doi:10.1002/grl.50928.Thomson, J., and W. E. Rogers, 2014: Swell and sea in the emerging Arctic Ocean.

,*Geophys. Res. Lett.***41**, 3136–3140, doi:10.1002/2014GL059983.Tilling, R. L., A. Ridout, A. Shepherd, and D. J. Wingham, 2015: Increased Arctic sea ice volume after anomalously low melting in 2013.

,*Nat. Geosci.***8**, 643–646, doi:10.1038/ngeo2489.Wadhams, P., 2000:

*Ice in the Ocean*. Gordon and Breach Science Publishers, 351 pp.Weeks, W. F., 2010:

*On Sea Ice*. University of Alaska Press, 664 pp.Williams, R., and Coauthors, 2014: Counting whales in a challenging, changing environment.

,*Sci. Rep.***4**, 4170, doi:10.1038/srep04170.Williams, T. D., L. G. Bennetts, V. A. Squire, D. Dumont, and L. Bertino, 2013: Wave–ice interactions in the marginal ice zone. Part 2: Numerical implementation and sensitivity studies along 1D transects of the ocean surfaces.

,*Ocean Modell.***71**, 92–101, doi:10.1016/j.ocemod.2013.05.011.WMO, 2009: WMO sea-ice nomenclature. WMO/OMM/BMO 259, Suppl. 5, 23 pp. [Available online at http://www.jcomm.info/index.php?option=com_oe&task=viewDocumentRecord&docID=4438.]

Zang, J., A. Schweiger, M. Steele, and H. Stern, 2015: Sea ice floe size distribution in the marginal ice zone: Theory and numerical experiments.

,*J. Geophys. Res. Oceans***120**, 3484–3498, doi:10.1002/2015JC010770.