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A linear basal drag law has resisting traction $-\mu \beta u_b$, with $\beta >0$ of dimension inverse length. It idealizes lubrication by subglacial till and is equivalent to a Navier slip boundary condition of slip length $1/\beta$ when the ice is a Newtonian fluid. It is a constitutive idealization, not a universal sliding law.

A Navier slip boundary condition relates fluid velocity relative to a stationary flat wall to the wall-normal tangential velocity gradient, $u_b={\ell }_s{u}_z$, with positive slip length ${\ell }_s$. For a Newtonian fluid, it is equivalent to linear resisting wall traction of magnitude $\mu u_b/{\ell }_s$. The normal points from the wall into the fluid; this fixes the sign convention.

A transversely waving Taylor swimming sheet can have a Navier slip boundary condition relating surface tangential velocity to shear and a slip length $\gamma$. The small-amplitude parameters are $ϵ=k{y}_0$ and $\delta =k\gamma$. The first-order velocity is unchanged by slip, but evaluation of shear at the displaced surface changes the second-order mean velocity.

Use the Cartesian streamfunction convention $u_x={\partial }_y\Psi$, $u_y=-{\partial }_x\Psi$. In the creeping-flow regime implicit in the question's biharmonic equation hint, the Stokes equations are $\mu {\nabla }^2\mathbf{u}=\nabla p$, $\nabla \cdot \mathbf{u}=0$. Taking their curl eliminates pressure and givesThe fluid occupies the region above the actual sheet; the perturbation calculation expands its boundary about $y=0$. In the swimming frame the material velocity is $(0,-\omega y_0\cos (kx-\omega t))$. The vertical kinematic condition and the supplied Navier slip boundary condition therefore give, at $y=y_0\sin (kx-\omega t)$,At infinity, ${\Psi }_y\to U$ and ${\Psi }_x\to 0$, with bounded velocity and no imposed shear; all fields are periodic in $x$ with period $2\pi /k$. The sign is important: a sheet translating at $-U{\mathbf{e}}_x$ sees the remote fluid translating at $+U{\mathbf{e}}_x$.

Set $X=kx$, $Y=ky$, $\tau =\omega t$, $\psi =k^2\Psi /\omega$, and $U^{\ast }=kU/\omega$. The two dimensionless parameters are the small-slope approximation parameter $ϵ=k{y}_0$ and dimensionless slip length $\delta =k\gamma$. With $\theta =X-\tau$, the dimensionless biharmonic stream function for planar Stokes flow satisfiesExpand $\psi =ϵ{\psi }_1+{ϵ}^2{\psi }_2+\cdots$ and $U^{\ast }=ϵU_1+{ϵ}^2{U}_2+\cdots$. At first order,The decaying first harmonic of the biharmonic equation has the form $(A+BY)e^{-Y}\sin \theta$. The vertical condition fixes $A=1$ and excludes a cosine component. The horizontal condition givesThus $(1+2\delta )(B-1)=0$ and $B=1$ for every $\delta \ge 0$. The spatially averaged first-order streamfunction is affine in $Y$; its Navier slip boundary condition forces $U_1=0$. ThereforeThis first-order slip independence of a transverse sheet is independent of slip length. Its surface tangential velocity and surface shear are both zero: $({\psi }_1{)}_Y(0)=0$ and $({\psi }_1{)}_{YY}(0)-({\psi }_1{)}_{XX}(0)=0$. This explains why the first-order streamfunction agrees with the no-slip boundary condition.

To find the slip-enhanced swimming speed of a transverse sheet, expand only the Navier slip boundary condition through second order. Put $Q_j=({\psi }_j{)}_{YY}-({\psi }_j{)}_{XX}$. Taylor expansion at the displaced boundary yieldsDirect differentiation givesHenceAverage over $X$. The zero Fourier series mode of a biharmonic streamfunction is a cubic polynomial in $Y$; bounded velocity at infinity removes the quadratic and cubic terms. Thus its velocity is the constant $U_2$, and its averaged shear $⟨Q_2{⟩}_X$ is zero. The mean boundary velocity determines Taylor-sheet swimming speed principle then givesThe even remainder follows because changing the sign of the amplitude is equivalent to a half-period phase shift. In dimensional form,For the Navier-slip Taylor swimming sheet, the ratio of speed to the no-slip value is $1+2\delta >1$ whenever $\delta >0$: slip increases the swimming speed. The expansion is for fixed $\delta$ as $ϵ\to 0$; it is not a uniform prediction of arbitrarily large speed if the slip length is allowed to diverge with the inverse amplitude.

Take $z$ upward from the horizontal bed and measure $x$ away from an ice divide. In the shallow-ice approximation, the hydrostatic pressure is $p=p_a+\rho g(h-z)$ and the horizontal balance for a Newtonian fluid isThe upper stress-free boundary condition gives $u_z(h)=0$. The resisting basal traction is $-\mu \beta u_b$, so the fluid-side shear stress satisfies $\mu u_z(0)=\mu \beta u_b$. This is a Navier slip boundary condition with slip length ${\ell }_s=1/\beta$. Integrating twice givesThe first term is the uniform basal sliding contribution, while the quadratic term is internal viscous deformation. Both are positive where the thickness decreases downstream.

The shallow-ice flux with linear basal drag isNo ice accumulation or ice ablation appears. For a symmetric glacier, work on one half, $0\le x\le x_N(t)$, with $q(0,t)=0$, $h(x_N,t)=0$ and no outgoing volume flux. Its conserved half-volume per unit width is $V_g={\int }_0^{x_N}hdx$; the full glacier has volume $2V_g$. This specifies the volume convention rather than silently supplying an unspecified value. For a general finite release on the whole line, the late similarity solution is centred on its conserved centre of mass and uses half its total volume for $V_g$.

The internal-shear and linear basal drag law mobilities are equal at $H_{\ast }=3/\beta$. Suitable vertical, horizontal and temporal scales areWrite $h=H_{\ast }\mathcal{H}$, $x=L_{\ast }X$, $t=T_{\ast }\tau$. ThenThe initial typical thickness is $\mathcal{H}\simeq 10$, the initial extent is of order $L_{\ast }/10$, and its shear-controlled spreading time is of order $T_{\ast }/10^5$. The shallow-ice approximation also requires small aspect ratio: in particular the initial thickness and extent must satisfy approximately $100H_{\ast }/L_{\ast }\ll 1$. The equations model the broad shallow bulk; the very tip can require physics beyond that approximation.

The two limiting equations are porous medium equations of the form ${\mathcal{H}}_{\tau }=({\mathcal{H}}^n{\mathcal{H}}_X{)}_X$, with $n=3$ for a thick shear-dominated glacier and $n=2$ for a thin sliding-dominated glacier. Volume conservation and balance of the time derivative give exponent $\alpha =1/(n+2)$. Set $\mathcal{H}={\tau }^{-\alpha }f(\eta )$, $\eta =X{\tau }^{-\alpha }$. One integration, using zero volume flux at the ice divide, gives $f^n{f}^{\prime }=-\alpha \eta f$, henceThe planar volume-conserving nonlinear-diffusion similarity is thereforeThe Beta function fixes its constants:Consequently the explicit early shear and late sliding limits areTheir centre depths are $A_3{\tau }^{-1/5}$ with $A_3\simeq 0.842252$, and $A_2{\tau }^{-1/4}$ with $A_2=\sqrt{2/\pi }$. Thus the early extent grows as $t^{1/5}$ and the late extent as $t^{1/4}$. These are asymptotic regime profiles; they solve the respective limiting equations, not the full sum of both mobilities. For arbitrary finite-width initial data they are appropriate after the corresponding spatial relaxation, with a virtual time origin depending on that data.

The shear-to-slip transition in a shallow ice current occurs at thickness of order $H_{\ast }$. Using equality at the centre of the early similarity solution givesIf the initial profile is already approximated by that early similarity solution and its centre depth is 10, the virtual age is ${\tau }_0=A_3^5/10^5\simeq 4.2385\times 10^{-6}$, and the elapsed estimate is ${\tau }_{\mathrm{tr}}-{\tau }_0$. Other reasonable definitions of typical thickness change the order-one coefficient; the initial thickness alone does not uniquely determine an exact transition time or an exact initial profile. The robust nondimensional estimate is a transition time of order one in the $T_{\ast }$ scale.

Finally, let a local nose translate steadily at positive speed $v$ and write $s=x_N-x\ge 0$. This is a local traveling nose of a glacier with linear basal drag; the speed of the globally spreading glacier changes slowly with time. The local mass conservation equation integrates to $q=vh$, since both $q$ and $h$ vanish at the front. Therefore

The right side is strictly increasing, giving a unique positive thickness at every $s>0$. With $S=X_N-X$ and $c=d{X}_N/d\tau$, the same relation reads $cS={\mathcal{H}}^3/3+{\mathcal{H}}^2/2$. Its limiting profiles areThe square-root tip reflects basal sliding; the cube-root thicker region reflects internal viscous deformation. Even during the early thick regime, the very nose is thin and has the square-root inner form. At late times that sliding balance governs nearly the whole profile. The dimensionless crossover thickness is $\mathcal{H}=1$ and its distance is $S=5/(6c)$; the early thick cube-root outer profile matches this smaller sliding region. The diverging geometric slope at the idealized tip also marks the local limit of the long-wave shallow-ice approximation.

The slip length is the distance obtained by extrapolating a locally linear tangential velocity profile back to zero velocity beneath a wall. A Navier slip boundary condition has $u_b={\ell }_s{\partial }_{n}u$. Zero slip length gives a no-slip boundary condition; large slip permits approximately plug-like motion when the surrounding geometry and forces allow it.