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Define $g(\mathbf{x},h,t)dh$ as the fraction of horizontal ocean area covered by sea ice of thickness in $[h,h+dh]$. Allow a Dirac delta mass at $h=0$ for open water, so the whole-area distribution has $\int gdh=1$; its first moment $H=\int hgdh$ is ice volume per ocean area, rather than mean thickness conditional on ice being present. Let $f=dh/dt$ be thermodynamic growth, $\mathbf{v}$ the horizontal ice velocity and $\psi$ the mechanical redistribution of sea ice.

Conservation in a horizontal cell and a thickness interval gives horizontal flux $\mathbf{v}g$ and thickness-space flux $fg$. For $h_1<h_2$, their contribution is $-\nabla \cdot {\int }_{h_1}^{h_2}\mathbf{v}gdh-[fg{]}_{h_1}^{h_2}$; adding ${\int }_{h_1}^{h_2}\psi dh$ and shrinking the interval yieldsThis is the sea-ice thickness distribution equation. Its distributional formulation must include creation and removal of open water when characteristics reach or leave $h=0$. On a positive-thickness smooth portion it is equivalentlyIf a nonconservative form uses a redistribution function $\tilde{\psi }$, then $\tilde{\psi }=\psi -g\nabla \cdot \mathbf{v}$; the two definitions must not be interchanged.

With the open-water boundary flux accounted for consistently, whole-area normalization requires $\int \psi dh=\nabla \cdot \mathbf{v}$. Mechanical rearrangement preserves ice volume locally, so $\int h\psi dh=0$. The volume moment consequently obeyswhere formation of new sea ice from the open-water category is included in the thermal term. Compression raises local volume per area through convergence, even though ridging does not create ice mass.

The three processes have distinct effects on the curve. Pure advection imports the distributions found upstream, so a fixed location may acquire a different mode or thick tail. Along an incompressible material trajectory, advection alone does not alter its distribution. Convergence and divergence also change area fractions through the horizontal flux, with the redistribution and open-water terms ensuring normalization.

Thermodynamic growth moves probability along thickness characteristics $\dot{h}=f(h,t)$, with $g(h,t)dh=g(h_0,0)d{h}_0$ when mechanics and horizontal divergence vanish. For example, winter conductive growth $f=\kappa /h$ gives $h^2=h_0^2+2\kappa t$ andThin ice grows faster, so the mode moves to larger thickness and neighboring thin-ice characteristics converge, sharpening the level-ice peak. Summer ablation moves classes left and creates open water when ice disappears; if melt rate depends on thickness, the shape changes as well.

Mechanical redistribution of sea ice removes area from thin or level ice and populates thick rafted or classes containing sea-ice pressure ridges, broadening the upper tail. It can also create sea-ice leads. A simple volume-conserving example, stacking a fraction $r$ of an initial distribution into double-thickness ice, isThe stacked component has area $r/2$ and unchanged volume $r\int h{g}_0(h)dh$; the freed area $r/2$ becomes open water. Thus mechanical thickening is not merely a translation of the entire curve.

The figure uses illustrative distributions, not measured data. The advection panel compares local ice with incoming thicker ice; the growth panel applies the thickness-characteristic formula; the mechanical panel uses $r=0.4$ and shows its open-water mass separately from the continuous density.

For the thick tail, take the common-shape sea-ice pressure ridge idealization to mean triangular profile sections with a common effective side slope $s=\tan \delta$, while peak thicknesses vary. Let $n(H)dH$ count sea-ice pressure ridge peaks per track length with maximum thickness in $[H,H+dH]$. Above a level-ice baseline $h_0$, a sea-ice pressure ridge with $H\ge h$ contributes two flank lengths $dh/s$ at thickness $h$; it contributes nothing at $h>H$. For a representative transect, summing these lengths gives the triangular ridge occupation identitySuppose $n(H)=B{e}^{-\lambda H}$ for $H\ge h_0$. ThenThus an exponential ridge-draft model, expressed in the same vertical coordinate as the profile, generates an exponential occupation tail with the same decay parameter. The peak-count distribution and the area-weighted thickness distribution are different distributions, so their amplitudes carry essential information.

If $\mu ={\int }_{h_0}^{\infty }n(H)dH$ is ridge line density and $p={\int }_{h_0}^{\infty }{g}_{\mathrm{tail}}(h)dh$ is the tail's area fraction, the same result becomesThe decay rate alone does not fix slope: sea-ice pressure ridge frequency or tail amplitude is also needed. If side slopes vary independently of peak height, replace $1/s$ by $⟨1/s⟩$, giving an effective harmonic-mean slope. Measured along-track slopes differ from slopes normal to a ridge if the transect is oblique. Overlap, irregular shape and porosity limit this simple geometric model. Literally identical complete ridge cross-sections would have one height, so “common shape” here necessarily permits different ridge sizes.

Sea-ice pressure ridge spacings are positive and broadly distributed. A spatial Poisson process gives an exponential distribution of gaps, a useful simple approximation. Observed sea-ice pressure ridge spacings, however, often fit a shifted lognormal distribution better, with few very small spacings because of ridge geometry, identification rules and finite beam width. This differs from the successful exponential model for peak drafts. Observed ridge-spacing and draft distributions establish this distinction.

Sea-ice lead-width distributions contain very many narrow openings and a broad tail. A power law often describes widths over a resolved range better than one negative exponential; its exponents and apparent cutoffs depend on the lead definition, resolution and count-versus-area weighting. Sea-ice lead spacings may be approximately exponential over an intermediate range, but clustering produces excess close pairs and large gaps, so there is no universal independent-spacing law. Both behaviors were found in Greenland Sea and Eurasian Basin lead statistics. Advection imports thickness classes, thermodynamics transports them in $h$, and mechanics exchanges thin-ice area for ridged ice and leads; common-slope exponential ridge populations reproduce the exponential thick tail.

Let $H$ denote a ridge's peak sea-ice draft, reserving $h$ for the draft at a randomly sampled position. In the exponential ridge-draft model, normalization by the line density $\mu$ givesThe mean peak sea-ice draft isConsequentlywith $h_m>h_0$, $b$ having dimensions inverse length and $B$ inverse length squared. The normalized peak probability density function is a shifted exponential distribution.

For the triangular argument, interpret the common ridge shape as geometrically similar triangles with common along-track slope $\tan \delta$ and variable peak height. Literal congruence would require identical sizes and could not coexist with an exponential peak-draft distribution. Each side of a triangle has $dx=|dh|/\tan \delta$. A ridge reaching draft $H\ge h$ therefore contributes $2\cot \delta dh$ of horizontal track in the interval $[h,h+dh]$. Summing this occupation length over all qualifying peaks proves the triangular ridge occupation identity:ThusThis is a tail relation for sampled draft occupation, not an instruction to normalize $n$ and $g$ identically. Below $h_0$, the ideal triangles contribute $2\mu \cot \delta$ rather than the same exponential; level ice and gaps contribute their own draft distributions. If triangular keels are referenced to a level-ice base, the vertical coordinate must be shifted consistently. We also require nonoverlapping occupation: arbitrary choices of $\mu$, mean draft and slope can otherwise demand more than the available track length.

Observed mean keel slopes are typically of order $20^{\circ }$–$30^{\circ }$, with broad individual variation rather than a single universal angle. Orientation matters: if a track crosses a straight ridge at angle $\psi$ to the crest, $\tan {\delta }_{\mathrm{track}}=\tan {\delta }_{\perp }|\sin \psi |$. The track slope can therefore approach zero at a grazing crossing. A sonar morphology study found location-dependent mean slopes about $22^{\circ }$–$27^{\circ }$ after correcting for ridge orientation.

Young sea-ice pressure ridges often have recognizably triangular sections with angular, porous rubble and comparatively continuous crests. Melting, refreezing and repeated cracking modify older ridges: their blocks can become rounded and consolidated, and their keel or crest can fragment into separated hummocks rather than retain one triangular shape. A pre-exam multibeam study found first-year sea ice ridge slopes averaging roughly $27^{\circ }$, while multi-year sea ice ridges often consisted of irregular separated smooth blocks. Multi-year sections can be broader or locally shallower, but age alone does not determine one slope angle. The constant-slope triangle is a useful statistical idealization, not a faithful shape for every old ridge.