Downhill skiing or snowboarding, in its simplest form, refers to the motion of a human sliding down an inclined plane on a porous medium. The extensive classic literature treating the science of skiing and snowboarding is summarized in (^{12}). A major contribution in this literature is the pioneering work of Colbeck (^{3-7}) on the reduced frictional force that results from the micron-thick fluid film that forms on the underside of the ski or snowboard due to frictional heating. This is of particular importance in cross-country skiing. More recently, Feng and Weinbaum (^{8}), hereafter referred to as F&W, have developed a theory for predicting the lift forces that are generated by planing surfaces in highly compressible porous media. This theory was initially developed to describe the motion of red blood cells on the endothelial surface layer (ESL) that lines our microvessels. F&W also showed that there was a remarkable dynamic similarity between this motion and the motion of a human skier or snowboarder skiing on soft snow powder even though their difference in mass is of order 10^{15}. F&W predict that the excess pore pressure generated by a planing surface moving on any compressible porous media scales as α^{2} = *h* ^{2}/*K*, where *h* is the layer thickness and *K* is the Darcy permeability, and that α is of order 10^{2} or larger for both red blood cells gliding on the ESL and humans skiing. Thus, the lift forces generated can be four or more orders of magnitude greater than the classical lubrication theory. This explains the enhanced lift phenomenon for a human skiing or snowboarding.

The feasibility of the enhanced lift concept is demonstrated in (^{19,20}), where the pore pressure generated during the rapid compression of snow was measured for the first time, and the F&W theory was verified experimentally. The enhancement in lift arises from the fact that, as the porous media compresses, there is a dramatic increase in the lubrication pressure, because of the marked increase in the hydraulic resistance of the air as it tries to escape from the confining boundaries of the planing surface through the compressed porous layer. The principal difference between the tightly fitting red cell in a capillary and a human skier or snowboarder is the leakage of the excess pressure at the lateral edges of the skis, as can be seen from Figure 1a. This leakage diminishes the maximum enhancement in the lift force by a factor *W* ^{2}/*L* ^{2}, where *W*/*L* is the ratio of the width to the length of the planing surface.

The fundamental insights gained from the F&W theory provide the basis and starting point for the present study to understand the lift mechanics of downhill skiing and snowboarding. However, the F&W theory has the following major limitations: (i) in the analysis by F&W, the local variations in the Darcy permeability in the compressed fiber layer are determined from solutions of the Stokes equations for the local average flow through a compressed two-dimensional fiber array. This approximate model was applied in F&W to both the ESL and fresh powder snow. In the case of snow, it was only intended to capture the essential physics of the compression process due to the wide variety of shapes and sizes of snow crystals from planar dendritic (hexagonal) to columnar within each snow classification (^{1}); (ii) when the theory in F&W was first developed, it was intended to apply to highly compressible porous media in the limit where the structure is so compressible that the normal forces generated by the compression of the solid phase were negligible compared with the pore-pressure forces generated within the porous media. Several previous experiments (^{17,19,20}) suggested that this might be true for snowboarding on fine-grained less permeable (wind-packed) snow at velocities greater than 15 m·s^{−1} for a 1.5-m snowboard, where the duration of the contact time of the snowboard with the snow is so short (less than 0.1 s) that the lift forces generated by the trapped air in the compressed snow layer might provide the major component of the lift force. However, for snowboarding on more permeable (fresh) snow where the medium is too porous, or for skiing where the dominant pressure relaxation length, the width of the ski, *W*, is too small, the air can not be efficiently trapped inside the porous media, and the compressed snow crystals should support much of the load; (iii) the F&W analysis emphasizes lift generation in porous media, but does not treat the other forces and moments acting on the skier or snowboarder, which are necessary for examining the overall lift mechanics involved in downhill skiing or snowboarding including control and stability; (iv) the analysis in F&W is based on a generalized Reynolds equation derived using effective medium theory (Brinkman equation). The predictions of F&W and later studies by Wu et al. (^{18,20}) suggest that it can be further simplified for the case of skiing or snowboarding using Darcy's law.

In the present study, we develop a new theoretical approach that combines the lift mechanisms from both the trapped air and the solid phase (ice crystals) and treat these limitations. The pore air pressure distribution beneath the planing surface is based on a consolidation theory where the local change of Darcy permeability due to the compression of the snow layer is obtained from Shimizu's (^{15}) classical empirical relationship. The local resistance force from the solid phase (ice crystals) is based on the static experiments of Wu et al. (^{19,20}). We then perform a force and moment balance acting on the skier or snowboarder to determine the corresponding slope of the planing surface and the depth of penetration at the leading edge as a function of velocity, friction coefficient, and location of the center of mass (CM) for fresh and wind-packed snow. This analysis is then used to develop a theory for control and stability in response to horizontal and vertical shift in the CM of the individual.

In the next section, we shall first employ Shimizu's (^{15}) equation to predict the local change of Darcy permeability *K* as a function of compression. Then, we will apply a consolidation theory to predict the pore air pressure distribution beneath a ski or snowboard surface. Next, we will consider the lift force from the solid phase, then the force and moment balance. The results of the theoretical model are presented in the Results section. In the final section, we conclude the paper with a discussion of the limitations of the present model.

## METHODS

### Darcy Permeability of Snow

Because of the wide variety of sizes and shapes of the solid phase of snow (ice crystals), theoretical models used to predict permeability of snow with snow density as a parameter have had only limited success, (see summary in (^{11})). However, there are semiempirical or empirical relationships to predict snow permeability. The most widely used formula for relating permeability to crystal size and snow density is the empirical expression of Shimizu (^{15}):

Here ρ*s* is the density of snow and *d* the mean diameter of the snow particles. After scaling the experimental data as a function of the crystal diameter *d*, Jordan et al. (^{11}) demonstrated that the porosity dependence of *K* follows Shimizu's equation given by equation 1 rather closely.

When a fresh snow layer of depth *h* _{0} and undeformed Darcy permeability *K* _{0} is compressed by a planar surface, the density ρ*s* of the snow layer changes due to the motion of the upper boundary. In the current application shown in Figure 1b, one assumes that the bottom surface of the ski or snowboard is planar and that there is no lateral tilt or edging. If one assumes that the snow layer is uniformly compacted in the vertical direction, whereas in the horizontal plane there is no crystal movement, the deformation-dependent Darcy permeability of snow based on equation 1 reduces to:

where *L* is the length of the ski or snowboard; *k* = *h* _{2}/*h* _{1} is the compression ratio, *h* _{2} and *h* _{1} are the local thickness of the snow layer at the leading (*x* = *L*) and trailing (*x* = 0) edges of the planing surface, respectively, and *K* _{2} is the Darcy permeability of the snow beneath the leading edge,

The detailed derivation of equations 2a and b can be found in Appendix A.

To illustrate how Darcy permeability (*K*) varies with position (*x*) and the compression ratio (*k* = *h* _{2}/*h* _{1}), we have plotted in Figure 2 the variation of *K* from leading to (*x′* = 1) trailing edge (*x′* = 0)for different compression ratios, *k*. In this calculation the snow layer at the leading edge is undeformed, *h* _{2} = *h* _{0} = 10 cm (typical value in the snow compaction measurements, (^{19,20})), the mean diameter of the ice crystal *d* is 1.0 mm, and the permeability of the snow beneath the leading edge of the planing surface is *K* _{2} = *K* _{0} = 1.7 × 10^{−8} m^{2} (typical value of fresh snow, (^{19,20})). One observes in Figure 2 a monotonic variation of Darcy permeability *K* because of compression. One expects that the large decrease in *K* from leading to trailing edge will produce a pronounced asymmetry with respect to *x* in the pressure loading and an increase in the maximum pressure and the resulting integrated lift force at the higher values of *k*.

### Pore Pressure

#### Governing equations.

In F&W, the authors considered a two-dimensional rigid boundary moving arbitrarily with velocity *U* = (*Ux*, *Uy*, *Uz*) over a fiber layer as shown in Figure 1a. Effective medium theory based on the Brinkman (^{2}) equation

was used to describe the flow in the fiber layer, where μ is the air viscosity. For large α = *h*/*√K*, F&W predict that a one-dimensional plug flow in the axial direction develops at all locations, that there is a thin fiber boundary layer at the upper and lower surfaces, and that there is little motion over most of the porous media except near boundaries and as one approaches the leading and trailing edges (see Figure 10 in F&W). For a 10-cm-thick snow layer, α is in the order of 10^{3}. It is clear that the *x* velocity component *u* satisfies Darcy's law and the profile approaches that of a uniform plug flow outside the thin fiber interaction layers adjacent to the bottom of the ski or the ground. Thus,

Because the length of a ski or snowboard is much larger than its width, air velocities will be much larger in the transverse *y*, *z* plane than in the *x* direction except at the upper boundary, where there is a very thin fiber interaction boundary layer due to the motion of the ski or snowboard. Wu et al. (^{18,20}) have proposed that the flow in the transverse plane can be viewed, to a first approximation, as a stagnation-point flow in a porous medium, where the vertical velocity at the upper boundary is equivalent to the incoming flow in the classical two-dimensional stagnation-point boundary layer. The key insight, that a similarity solution might exist for the flow in the transverse direction, was gleaned from the solutions for the pressure field beneath a snowboard shown in Figure 10 of F&W, where one observes that the transverse pressure profiles are parabolic at any axial location, the same as one finds for a classical two-dimensional stagnation-point flow. In the analysis of Wu et al. (^{18,20}) for a stagnation-point flow in a porous medium, a new fundamental dimensionless parameter β = υ*/KA* emerges, where υ is the kinematic viscosity and *A* is the characteristic velocity gradient imposed by the external flow. In the case of skiing or snowboarding, *A* = −*Uz*(*x*,*y*,*h*)/*h*. Here the vertical velocity of the upper boundary *Uz*(*x*, *y*, *h*) in the general case is given by

where *h*(*x, y*) is the local gap height and *Ux*, *Uy* the velocity of the ski or snowboard in the *x*, *y* directions. Thus,

In (^{18,20}), the structure of a new type of boundary layer is observed that evolves as β varies from zero, the classical limit of the Hiemenz (^{9}) and Homann (^{10}) solutions, to β >> 1, the classical Brinkman (^{2}) limit where inertial effects are negligible. These solutions show that for the case of skiing or snowboarding where β>> 1, the viscous boundary layer is vanishingly thin, and one is left with an outer flow for the velocity field, which is given by

The *v* velocity profile is a uniform plug plow in the *y* direction that increases linearly with distance *y* from the centerline (*y* = 0) of the ski or snowboard, while the downward velocity *w* decreases linearly with *z*.

Substituting equations 4, 7, and 8 into the continuity equation,

we obtain

Equation 10 is the new simplified governing equation for the pore pressure in skiing or snowboarding. The two terms on the righthand side have a simple interpretation. The first term proportional to *A* is the forcing due to the angle of attack and the lateral edging of the ski or snowboard (see definition of *A* in equation 6). The second term arises from the variation in permeability due to the compression of the snow. Both the first and the second term include lateral tilt or edging that is reflected in the partial derivative with respect to *y*. If both terms in equation 10 vanish, one obtains the potential equation for Darcy flow without any forcing.

#### No lateral tilt or edging.

In the present study, we shall only examine the simple case where there is no lateral tilt to the ski or snowboard, the local thickness of the snow layer, *h* = *h*(*x*), *K* = *K*(*x*), and *A* = (*U*/*h*)*dh*/*dx*, where *U* = *Ux*. For this case, equation 10 simplifies to

From the analysis in (^{18,20}), we know that the pressure distribution in the *y* direction is parabolic, which can be expressed as

where *W*(*x*) is the local width of the planing surface, *Pc*(*x*) is the centerline pressure corresponding to the cross section at the location *x*, and *P* _{0} is very close to the atmospheric pressure because the porosity of the compressed snow changes abruptly at the lateral edges where the compression ends and the hydrodynamic resistance falls precipitously. The detailed derivation of equation 12 can be found in Appendix A.

Substituting equation 12 into the generalized governing equation 11 and evaluating at *y* = 0, one obtains the governing equation for the centerline pressure distribution beneath a ski or snowboard,

In the current application, we will examine the case when the width of the planing surface is a constant along the axial direction, *W*(*x*) = *W*. However, one can readily apply equation 13 for more complex planar shapes, such as parabolic skis, where the width *W*(*x*) of the planing surface changes with the axial location *x*.

It is desirable to introduce the following dimensionless variables:

Substituting equations 14b, c, e, and f into equation 13, we obtain a dimensionless equation for *pc*′(*x*′):

where

At the leading and trailing edge of the planing surface, the pressure is close to the atmospheric pressure:

Equation 15 subject to the boundary conditions (^{17}) is a nonlinear, two-point boundary-value problem that contains two dimensionless parameters, ε and θ*L*. ε denotes the ratio of the square of two characteristic pressure relaxation lengths in the *y* and *x* direction. In general, due to the large leakage of air that occurs at the lateral edges, the primary pressure relaxation occurs at these lateral boundaries. One expects a decrease in ε to decrease the pore pressure beneath the planing surface because of the increased lateral air drainage. This is consistent with the predictions in F&W, where, for *L*/*W* >> 1, the solutions of the generalized Reynolds equation 2-23 in F&W for an elongated planform, such as a ski, differ greatly from the solutions of equation 2-25 in F&W when *L* ^{2}/*W* ^{2} >> 1, because the second term in (2-23) is O(*L* ^{2}/*W* ^{2}) larger than the first. θ*L* in equation 15 is a combination of two dimensionless parameters, μA|*x = L/P* _{0} and *L* ^{2}/*K* _{2}. *A* scales with the vertical compression velocity of the planing surface. For a given tilt angle, *γ*, either an increase in the forward velocity *U* or a decrease in the snow layer thickness at the leading edge *h* _{2} will increase the value of A. It is clear that this increase of *A* will increase the pore pressure because a faster compression generates a higher pore air pressure. The inverse of *K* is a measure of the resistance that the air encounters as it flows through the porous media. One expects that as *K* _{2} decreases, the maximum pore pressure will increase because the air will encounter an increased resistance when escaping.

#### ε << 1 (ski).

Equation 15 subject to the boundary conditions (^{17}) can be solved numerically. However, for ε << 1, a simple asymptotic solution can be obtained that is a good approximation for skiing. For a ski, *L* = 2 m, *W* = 0.1 m and ε = 2.5 × 10^{−3}. Inspection of equation 15 reveals that:

One can rewrite equation 15 as

where θ*W* =(μA*x = L*/*P* _{0}) · (*W* ^{2}/*K* _{2})

In equation 19 the small dimensionless parameter ε multiplies both the highest derivative term *d* ^{2} *p*′_{c}(*x*′)/*dx*′^{2} and the nonlinear term (1/*K*′(*x*′)) (*dK*′(*x*′)/*dx*′)(*dp*′*c*(*x*′)/*dx*′). Equation 19 is readily solved using singular perturbation methods,

The detailed derivation of equation 20 can be found in Appendix A.

Once *pc*′(*x*′) in equations 15 or 20 is determined, one can readily find the two dimensional pressure distribution beneath a snowboard or ski surface by applying equations 12 and 14:

The average dimensionless pressure generated by the trapped air inside the compressed snow layer is obtained by integrating equation 21 over the entire surface. This leads to:

### Lift Force from Solid Phase

The solid-phase (ice crystals) lift force is obtained by measuring the quasi-steady force generated when the snow is subject to incrementally increasing compressive forces using a porous cylinder-piston apparatus (^{19,20}). This solid-phase force is given approximately by an empirical relation of the form:

where *Fsolid* is the force exerted by the ice crystal phase, Δ*h* is the instantaneous deformation of the snow layer, Δ*h* _{max} is the final displacement of the piston when the full load *F* _{max} is supported solely by the ice crystals. In these static experiments, the loading area of the piston cancels out and equation 23 reduces to:

In the current study, we use equation 24 to predict the pressure from the solid phase during skiing or snowboarding. As shown in Figure 3, the skier descends directly down the fall line and the angle of the inclined slope is α*h*. We consider the case where there is no lateral tilt or edging. The centerline sketch of a ski or snowboard is shown in Figure 1b. At any location *x*, the snow is compressed uniformly and the local pressure from the ice grains is given by equation 24, where *P* _{max} = *Pmg* = *mg*cosα_{h}/*LW*, Δ*h* = *h* _{0} − *h*(*x*), Δ*hmax* = *h* _{0} − *hmin* and *hmin* corresponds to the thickness of the snow layer where maximum compression is achieved. *Pmg* is the ice crystal pressure when the pore air pressure vanishes. Note that for snowboarding, *m* is the mass of the snowboarder with his/her equipment, while for skiing it is half of this total mass if the two skis equally share the weight. Equation 24 can be written as

The maximum deformation that would be achieved for a given patch of snow corresponds to a density as high as ρ_{s,max} = 0.875 g·cm^{−3} (^{13,19,20}). Thus, the minimum snow porosity is φ_{min} = 1 − ρ_{s,max}/ρ*i* = 0.06, where the ice density is ρ*i* = 0.931 g·cm^{−3}. If the undeformed porosity of the snow layer is φ_{0}, *hmin*/*h* _{0} = (1 − φ_{0} + φ_{min})/1 = 1.06 − φ_{0}. If one defines *p*′*s*(*x*′) = *Psolid* (*x*)/*P* _{0}, equation 25 reduces to

where λ = *h* _{2}/*h* _{0}. If we define the average pressure generated by the solid phase (ice crystals) as

we have

### Force Balance

To begin discussion of the force balance, we first look at the representative forces acting on a skier gliding on an inclined snow slope. As shown in Figure 3, we wish to take notice of the skier's center of mass (CM) including his/her equipment. The weight *mg* may be resolved into two forces, *FS* parallel to the slope, and *FN* normal to the slope. The lift force, which refers to the total reaction force of the snow in the skiing community (^{14}), is in the present analysis the sum of the distributed forces due to the pore air pressure, *Na*, and the ice crystals' reaction force acting on the bottom of the ski, *Ns*. The skier gliding down the slope has a snow friction force *Ff* and a wind resistance or aerodynamic drag force *FD*, which are directed up the slope. An aerodynamic lift force *FL* is also present. When a skier has achieved terminal velocity, there is no acceleration and, thus, there is no inertial force, so all the forces as well as the torques shown must sum to zero. In this figure, the forces are shown at the points at which they act. The gravitational force acts at the center of mass. The aerodynamic drag *FD* acts at the effective center of the frontal area, in line with the center of mass in this case; the snow friction force *Ff* acts along the contact area where the ski meets the slope, which is greatly reduced by the presence of micrometer-thick fluid films that form beneath the ski due to frictional heating (^{5,6}). The small tilt angle ã between the bottom surface of the ski or snowboard with the inclined slope is not shown in this figure.

For the normal force balance we require that

where

and α*h* is the angle of the inclined slope. Here we neglect the aerodynamic lift force *FL* because it is small compared with *Na* and *Ns*, (^{14}).

Equation 28a can also be written in the form of an average pressure by dividing each term by *L*·*W*. If one defines the total dimensionless average pressure arising from both the trapped air and the solid phase as *P*′*av* = ((*Na* + *Na*/LW)/ *P* _{0} )= *P*′_{ava} + *P*′*avs*, and the total dimensionless loading pressure exerted by the skier or snowboarder as *P*′*avload* = *Pmg*/*P* _{0}, equation 28a reduces to

or equivalently 1= *fair* + *fsolid*, where *fair* = *Pava*′/*Pavload*′ and *fsolid* = *Pavs*′/*Pavload*′.

### Moment Balance

For the skier or snowboarder moving down the slope with a steady motion, the sum of all torques about the center of mass must be zero. In the case shown in Figure 3, the gravitational force *mg*, the aerodynamic drag *FD*, and the aerodynamic lift force *FL* act through the center of mass. Thus, we have

where *lc* is the normal distance of the center of mass from the ski surface, and *xc*, *xa*, and *xs* are the *x* coordinates of the center of mass, center of pore-pressure force, and center of solid lift force, respectively. Equation 30 can also be written as

where *xc*′= *xc*/*L*, *xa*′= *xa*/*L*, *xs*′= *xs*/*L*, *xc*′= *xc*/*L*, *lc*′= *lc*/*L* and *Ff*′= *Ff*/*mg*cosα_{h}. The dimensionless *x* coordinate of the center of pore-pressure force, *xa*′ can be obtained from a moment balance about the trailing edge (*x* = 0):

Similarly, the location of the center of lift force from the solid phase, *xs*′ is expressed as

The snow frictional force exerted on the ski or snowboard, *Ff*, is independent of velocity and may be expressed as *Ff* = η*Ns*, where η is the coefficient of friction.

In the current application, we have assumed that the aerodynamic drag *FD* acts at the effective center of the frontal area, in line with the center of mass. Thus, we do not take into account the contribution of *FD* in the moment balance about CM. This is a reasonable assumption because if *FD* deviates from the center of mass, this deviation is much smaller than *lc*; on the other hand, the friction coefficient, η < 0.05. *FD* is of the same order as *Ff* (^{12}), thus *FD*'s contribution in the moment balance is negligible.

When a skier or snowboarder glides down a slope at velocity *U*, over an undeformed snow layer of thickness *h* _{0} and Darcy permeability *K* _{0}, without changing the location of the skier's center of mass (CM), one has to adjust the tilt angle γ (or the compression ratio from the leading to trailing edge, *k* = *h* _{2}/*h* _{1}) as well as the compression ratio at the leading edge, λ = *h* _{2}/*h* _{0}, to satisfy the force and moment balance equations, equations 29 and 30, respectively.

## RESULTS

The lift distribution between the trapped air and the ice crystals strongly depends on the geometry of the planing surface, *W*/*L*, the speed of the skier or snowboarder, *U*, and the permeability and compression properties of the snow layer, *K* _{0} and φ_{0}. This is reflected in equation 15, where the pore air pressure generated as the planing surface glides over the compressed snow layer depends on two dimensionless parameters, ε and θ*L*. In this section we examine how the lift mechanism depends on these parameters. We first look at the case of snowboarding where *L*/*W* = 4.3 or ε = (1/4.3)^{2} = 0.054, a typical value for a 1.16-m-long snowboard (see Table 3.5 in (^{12})). In the case of skiing we use as representative values *L* = 1.7 m, *L*/*W* =17 or ε = (1/17)^{2} = 0.0035, (^{12}). We consider two typical snow types, wind-packed and fresh snow. These two snow types bracket the range of permeability for most skiing conditions, and they are also the snow conditions for which Wu et al. (^{17,19,20}) conducted their field experiments to determine the mechanical properties of snow under dynamic compression. We assume the thickness of the fresh undeformed snow layer, *h* _{0} = 10 cm. The mass of the skier with his or her equipment is 80 kg; the angle of the inclined slope α*h* = 15°. Unless specifically mentioned otherwise, we assume the location of the skier's center of mass (CM), *xc*′ = 0.45 for snowboarding and *xc*′ = 0.40 for skiing; the normal distance of the center of mass from the ski or snowboard surface, *lc* = 80 cm (^{14}); and the coefficient of sliding friction, ç = 0.04 (^{12}).

### Snowboarding

#### Wind-packed snow (*K* _{0} = 5.0 × 10^{−10} m^{2}, φ_{0} = 0.6, d = 0.42 mm). *U* = 10, 20, 30 m·s^{−1} for *fsolid* ≠ 0 compared with *U* = 20 m·s^{−1} for *fsolid* = 0.

For snowboarding on wind-packed snow with undeformed Darcy permeability *K* _{0} = 5.0 × 10^{−10} m^{2} and undeformed porosity φ_{0} = 0.6 (^{19,20}), we have numerically solved equation 15 subject to boundary conditions (^{17}) to obtain the centerline pore-pressure distribution beneath a snowboard surface. The results are shown in Figure 4a and the values for *U*, λ, *k*, *xa*′, *xs*′, *fair* and *fsolid* are summarized in Table 1A. Here the tilt angle γ (or the compression ratio, *k* = *h* _{2}/*h* _{1}) as well as the precompression ratio at the leading edge, λ *= h* _{2}/*h* _{0} are determined to satisfy the force and moment balance equations, 29 and 30, respectively. The dash-dotted line is the prediction of the model when (λ, *k*) is obtained by requiring that the entire load be supported by the pore air pressure only (*fair* = 1). It is compared with the case where the lift forces arise from both the trapped air and the solid phase (*fsolid* = 46%) for *U* = 20 m·s^{−1}. As can be seen from Table 1A the compressions at the leading and trailing edge are significantly reduced and the slope of the snowboard *k* increased when the solid phase is considered (*h* _{2}/*h* _{0} = λ*f* *air* = 0 = 0.58, *h* _{1}/*h* _{0} = λ/*k* *f* *air* = 0 = 0.55 while *h* _{1}/*h* _{0} *f* *air* = 46% = 0.65, *h* _{2}/*h* _{0} *f* *air* = 46% = 0.70). It is evident from Figure 4a that approximately 50% of the total lift force is generated by the trapped air when one snowboards on a fine-grained, wind-packed snow layer at relatively high speed (*U* > 10 m·s^{−1}); an increase in velocity leads to an increase in the trapped air's contribution to the total lift and a decrease in the compression of the snow layer (for *U* = 10 m·s^{−1}, *fair* = 43%, *h* _{2}/*h* _{0} = 0.65, *h* _{1}/*h* _{0} = 0.60; while for *U* = 20 m·s^{−1}, *fair* = 54%, *h* _{2}/*h* _{0} = 0.70, *h* _{1}/*h* _{0} = 0.65). We attribute this behavior to the fact that as one increases his/her velocity, the contact time of the planing surface with the snow layer decreases, the air that is trapped inside snow has less time to escape before the pore pressure decays and thus, one needs a smaller compression to generate the required lift force.

The dimensionless pressure from the solid phase (ice crystals) defined in equation 26, is shown in Figure 4b. It is clear from this figure that the solid-phase pressure increases nearly linearly from the leading (*x*′ = 1) to the trailing edge (*x*′ = 0) where maximum compression is achieved; as the velocity increases, *fair* increases, and, thus the ice crystal's contribution to the total lift decreases.

The predictions shown in Figure 4 qualitatively agree with the experimental measurements in (^{17,19,20}) for the compression of wind-packed snow, in their porous walled piston-cylinder apparatus. In these experiments, one measured the time for the air to escape through the porous side walls of the porous cylinder-piston apparatus and also measured the dynamic forces on the piston from the trapped air and the snow crystals as the air escaped. Although this is obviously not a realistic model for a ski or snowboard, it provides some guide for the forces that one can anticipate. Furthermore, the experiments suggest that for snowboarding on fine-grained less permeable (wind-packed) snow at velocities greater than 10 m·s^{−1}, the lift forces generated by the trapped air in the compressed snow layer should provide a significant fraction of the total lift.

#### Friction (wind-packed snow, *U* = 20 m·s^{−1}, η = 0.02, 0.04, 0.06 and 0.08).

The study of snowboard or skis as sliders on snow has a long history related to the more general study of friction. Skiing is unique among these studies. For alpine skiing or snowboarding, the skier seeks minimal sliding friction to increase their downhill speed, while for track and cross-country skiing one requires his or her skis to combine minimum friction on the forward glide and maximum friction or grab when the skier pushes off against the snow for acceleration or for travel uphill. To our knowledge, most of the literature on ski or snowboard friction is focused on the presence of a micron thick fluid film that forms beneath the running surface as a melt water lubrication layer (^{3-7}). No one has ever investigated the associated lift mechanism due to the trapped air and the solid phase (ice crystals) for different values of friction coefficient.

In Figure 5a, we plot the centerline pore-pressure distribution beneath a snowboard surface for various values of friction coefficient, η, as one glides over a 10-cm wind-packed snow layer at a velocity, *U* = 20 m·s^{−1}. The corresponding λ, *k*, *xa*′, *xs*′, *fair* and *fsolid* are listed in Table 1B. The values of *η*, 0.02 to 0.08, are chosen to span a broad range of thermal, bottom snowboard surface and ice crystal conditions (^{12}). It is clear from this figure that a decrease in the sliding friction leads to an increase in the trapped air's contribution to the total lift and compression ratios at the leading and trailing edges, ë = *h* _{2}/*h* _{0} and ë/*k* = *h* _{1}/*h* _{0} increase. As η decreases from 0.08 to 0.02, *fair* increases from 24 to 63%, the compression ratio at the leading edge ë = *h* _{2}/*h* _{0} increases from 0.55 to 0.77, and the compression ratio at the trailing edge, *h* _{1}/*h* _{0} = ë/*k* increases from 0.54 to 0.68 providing for a much larger tilt angle. This behavior is further confirmed in Figure 5b, where one can readily see the pronounced increase in solid-phase lift force as η increases. These results indicate that for downhill skiing, a decrease in the sliding friction not only reduces the snow frictional drag force that the skier encounters as he or she glides down the slope at a given speed, but also increases the trapped air's contribution to the total lift. These results provide a new insight into the role of friction in ski and snowboard performance. For a smooth air cushioned glide one wants a low value of η. This can be obtained by waxing the bottom surface of the snowboard or ski.

#### Snowboard control and stability.

**Shift of x**_{c}**′.** Figure 6 provides the critical insights for snowboard control and stability. In contrast to an airplane, where the center of mass is fixed and one controls the angle of attack and moments about the center of mass by flap and rudder control, a snowboarder can alter his/her center of mass by shifting their weight from the front to the rear foot. Under most conditions this shift in *xc*′ is small and varies between roughly 0.37 and 0.47. However as shown in Figures 6a and b, this has a dramatic impact on the distribution of the pressure loading from the air and solid (ice crystal) phases. When one places more of their weight on the rear foot *xc*′ decreases and approaches the lower limit of 0.37, whereas when one shifts their weight to the front foot *xc*′ increases and approaches the upper limit of 0.47. When *xc*′ = 0.37 the contribution of the trapped air pressure is both large and asymmetrically distributed with large pressures near the rear of the snowboard. In contrast, when *xc*′ = 0.47 the contribution of the trapped air is much smaller and the pressure distribution is symmetric. This change is accompanied by a transfer of lift forces from the air to the solid phase and a change in angle of attack of the snowboard, which is plotted in Figure 6c. The dashed lines in Figure 6a and b crossing the pressure profiles show this shift in *xc*′.

The curves in Figure 6 apply to a neutral stability condition in which the sum of moments about the CM vanishes (see equation 30). For an airplane one examines stability by changing the angle of attack from the neutral stability position and sees if the moment about the CM is restoring (stable) or produces a nonrestoring moment (unstable). A similar analysis can be performed for a ski or snowboard except that one wants to examine the moment created by a shift in *xc*′.

One can rewrite equation 30 as

Introducing dimensionless variables, equation 34a reduces to

where *M*′ = *M*/*mg*cosα_{h} *L*. In Figure 6d we have plotted the equivalent moment (*M′*) or the pitching moment curves for a snowboard. For each set of curves in Figure 6a, b (same value of *xc*′) we ask how the moment would increase or decrease about the neutral equilibrium position if the angle of attack of the snowboard was fixed but *xc*′ shifted. This calculation leads to the moment profiles in Figure 6d. Each curve crosses the neutral position at *M′* = 0 and all curves have a positive slope indicating that a forward shift of *xc*′ produces a clockwise moment and a backward shift a counterclockwise moment. If one shifts their weight (changes *xc*′) and does not change their angle of attack one must apply the moment indicated in this figure to maintain the same snowboard position. This requires a muscular input and for minimum effort one will want to go to a new neutral stability condition where no effort is required. This is shown in Figure 6d as the trajectory a-b-c. This requires that one change their angle of attack and move along the curve shown in Figure 6d where *M′* = 0.

In the same way that one can shift their weight from the front to rear foot or the reverse, one can tilt the snowboard from one lateral edge to the other in a maneuver called edging. This maneuver requires a more difficult analysis since the lateral tilt changes the simple parabolic pressure distribution described by equation 12 in the transverse plane of the snowboard. Edging maneuvers, which are required for turning, are a subject for future study.

*Shift of lc.* The vertical displacement of the center of mass (CM) will also affect the snowboarder's stability, see equation 34. Similar to the shift of *xc*′ in Figure 6, if one raises or lowers his or her center of mass vertically through up-turn-down or down-turn-up body motion without changing their compression ratios at the leading and trailing edges, the initial neutral moment balance is broken and an unbalanced pitching moment *M*′ defined in equation 34b is generated. To maintain stability, one has to input a muscular moment or change the compression ratios of the snow layer to get back to a new neutral moment balance position. The latter requires no muscular input, and is accompanied by a transfer of lift forces between the trapped air and the solid ice crystals as well as changes of snow compression at the leading and trailing edges. We have plotted in Figure 7a the centerline pore-pressure distribution beneath a snowboard surface as one snowboards over a 10-cm-think wind-packed snow layer at *U* = 20 m·s^{−1}. The air-pressure profiles for *M*′ = 0 for these different values of *lc, lc* = 0.8, 1.0, and 1.2 m are shown. The corresponding solid-phase pressure is plotted in Figure 7b and the results for *U*, λ, *k*, *xa*′, *xs*′, *fair*, and *fsolid* are summarized in the Table 1D. Table 1D shows that as one vertically lowers his or her CM from *lc* = 1.2 to 0.8 m, the trapped air's contribution to the total lift increases from 40 to 63% and the solid-phase lift contribution decreases from 60 to 37%, accordingly. Meanwhile, the required compression ratio at the leading edge ë = *h* _{2}/*h* _{0} changes from 0.62 to 0.77, and the compression ratio at the trailing edge, ë/*k* = *h* _{1}/*h* _{0} changes from 0.60 to 0.69. In the current study we choose the vertical displacement of the CM from crouching to standing position to be approximately 0.4 m (^{12}).

**Fresh snow** (*K* _{0}** = 1.7 × 10****−8**** m****2****, φ**_{0}** = 0.8, d = 1.0 mm).** Because the Darcy permeability of fresh snow is roughly 34 times larger than the wind-packed snow, air can not be trapped efficiently. At a given speed, one needs a much larger compression of the fresh snow layer and a larger contribution from the solid phase to generate the required lift, see Figure 8. Figure 8a shows the centerline pore-pressure distribution beneath a snowboard surface for the two different snow types. It is clear from this figure that, for a given speed, *U* = 20 m·s^{−1}, the trapped air's contribution to the total lift is much larger in less permeable wind-packed snow than in highly permeable fresh snow (*fair* *wind-packed snow* = 54%, *fair* *fresh snow* = 18%). The values of *h*/*h* _{0} at both the leading and trailing edge decrease by roughly a factor of two (for fresh snow, *h* _{2}/*h* _{0} = 0.38, *h* _{1}/*h* _{0}=0.31; while for wind-packed snow, *h* _{2}/*h* _{0} = 0.70, *h* _{1}/*h* _{0} = 0.65). The maximum compression ratio that can be achieved for a given type of snow is obtained from the expression: *h* _{1}/*h* _{0} *min* = 1 − φ_{0} + φ_{min}. For fresh snow *h* _{1}/*h* _{0} _{min} = 0.26, for wind-packed snow *h* _{1}/*h* _{0} _{min} = 0.46. Figure 8a shows that for fresh powder snow, an increase in velocity *U* leads to an increase in *fair* and a decrease in *k*, the same behavior observed in Figure 4 for wind-packed snow.

Figure 8b shows the corresponding solid-phase pressure beneath the snowboard surface. Because the trapped air inside fresh snow is not nearly sufficient to support the weight of the snowboarder with his/her equipment, the lift force from the ice crystals carries much of the load (*fsolid* *fresh snow* = 82%). This is in sharp contrast to the case of wind-packed snow, where *fsolid* *wind-packed snow* = 46% at the same velocity. This behavior was previously predicted in (^{19,20}) where the relaxation of the pore pressure in the piston-cylinder apparatus was too rapid to support the full weight of the piston when it was dropped from rest.

### Skiing

The primary difference in the lift mechanics of snowboarding and skiing is due to the dimensionless parameter ε. The width of a ski is one third that of a snowboard, and ε for a ski is roughly 1/16 that for a snowboard. Thus, solutions of equation 15 for a long slender planing surface (ski) differ greatly from those for snowboarding. In general, due to the large increase in the pore-pressure relaxation at the lateral edges, the required snow compression is larger in skiing than in snowboarding. In Figure 9a, we have compared the centerline pore-pressure distribution beneath a ski and snowboard for wind-packed snow for a velocity of 20 m·s^{−1}. As shown in this figure, although the required compression at the leading edge is almost the same, the tilt angle γ of the planing surface (or the compression ratio from the leading to trailing edge, *k* = *h* _{2}/*h* _{1}) is much larger in skiing than in snowboarding (*k* *skiing* = 1.315, *k* *snowboarding* = 1.072), and thus the snow layer at the trailing edge of the ski is compressed more (*h* _{1}/*h* _{0} *skiing* = 0.54, *h* _{1}/*h* _{0} *snowboarding* = 0.65). For *U* = 20 m·s^{−1}, 42% of the total lift force is generated by the trapped air for skiing on wind-packed snow, which is less than that for snowboarding, where *fair* = 54%. The difference in *fair* is not great, considering the large difference in ε and the pore-pressure profiles shown in Figure 9a., The skier's center of mass is closer to the trailing edge (*x*′*c* *skiing* = 0.4, *x*′*c* *snowboarding* = 0.45), and one needs to increase the tilt angle γ to satisfy the moment balance equation 30. By doing so, the center of pore air pressure shifts towards the trailing edge enhancing the pore pressure's contribution to the total lift and compensates in part for the pressure leakage at the lateral edges. The dependence of the pore-pressure distribution on the velocity *U* of the skier is similar to that for snowboarding. Here *U* is chosen as 10, 20, and 30 m·s^{−1} for skiing.

In Figure 9b we plot the corresponding solid-phase pressure, which clearly demonstrates the significant contribution of the solid phase to the total lift in the case of skiing. Furthermore, due to the larger value of *k* = *h* _{2}/*h* _{1}, one observes a steeper increase in the local solid-phase pressure from the leading to trailing edge for skiing.

Having obtained the centerline pore-pressure distribution, we can plot two dimensional pore-pressure distributions beneath a snowboard or ski surface using equation 21. The results are shown in Figures 10a for snowboarding and 10b for skiing, where the snow is wind-packed, *h* _{0} = 10 cm, and *U* = 20 m·s^{−1}. This figure shows the effect of the dimensionless parameter, ε. One also notes the parabolic pore-pressure distribution in the transverse plane of a ski or snowboard. For skiing, due to the large tilt angle and prevailing continuous drainage in the lateral direction, the pore air pressure increases sharply to its peak value near the trailing edge and then decays rapidly to atmospheric pressure, in contrast to that observed for snowboarding, where the axial pressure relaxation at the trailing edge is much more gradual.

## CONCLUSIONS

In this paper, we have developed a new theoretical analysis of the lift forces generated during downhill skiing or snowboarding, which incorporates the lift contribution from both the transiently trapped air and the compressed ice crystals. This study is an important practical application and extension of the lubrication theory for highly compressible porous media developed in F&W. The results presented herein agree with the more qualitative predictions in (^{17,19,20}), where the pore pressures generated in snow were measured for the first time using a porous cylinder-piston apparatus.

The pore air pressure distribution was obtained using a consolidation theory based on Darcy's law, where the local change of Darcy permeability of snow was estimated using Shimizu's (^{15}) classical empirical relationship. In our analysis we have neglected the motion of solid particles in the horizontal (*x*, *y*) plane and have assumed that the snow is compressed uniformly in the vertical direction. This assumption is appropriate for straight downhill skiing because one leaves a relatively clean track of compressed snow and the motion of the escaping snow at the lateral edges of the skis or snowboard is small compared with that of the escaping air. However, during edging and rapid turning maneuvers one has to consider the motion of the snow powder, and in particular its momentum, in determining the lateral forces, because under these conditions the snowboard or skis also act as a snowplow. The assumption of uniform vertical compression implies that the Darcy permeability *K* is primarily a function of *x*, or *K* does not change with the depth of the snow layer. This is reasonable because most skiing conditions involve a fresh snow layer on a packed base, which consists of recrystallized snow having a permeability that is 20-30 times smaller than fresh powder snow (^{11}). Thus, the packed snow with much lower permeability acts as an impermeable base.

The lift force generated by the solid phase (ice crystals) was obtained from the experiments in (^{19,20}). These experiments neglect the shear forces produced in the ice crystal structure at the lateral edges of the skis or snowboard. However, studies examining the deformation of the snow at the edge of a compression surface suggest that this force is small compared with the main compression force exerted by the snow directly beneath the planing surface (^{16}).

The great enhancement in lift generated by a planing surface as it glides over a soft porous media is a new concept. The application to human skiing or snowboarding was first suggested in F&W. In a more recent study, Wu et al. (^{17,20}) have applied this idea in the design of a future generation train that can glide on a soft porous track whose mechanical properties are similar to goose down pillows. The key insight in the latter application is that one could greatly enhance the lift and reduce the drag due to friction in the solid phase if the lateral loss of pore pressure at the side walls of the track could be eliminated. If this could be achieved, the authors predicted that it would be possible to support a 50-ton train car moving on a porous material with permeability properties that do not differ greatly from fresh snow at velocities greater than approximately 10 m·s^{−1}.

This research was performed in partial fulfillment of the requirements for the Ph.D. degree from the City University of New York by Qianhong Wu. Dr. Wu was supported by NIH grant 19544 and the Mario Capelloni Dissertation Fellowship.

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**Appendix A**

## Darcy Permeability of Snow beneath a Ski or Snowboard

The empirical relationship of Shimizu (^{15}) for the Darcy permeability of snow is expressed as:

where ρ*S* is the density of snow and *d* the mean diameter of the snow particles.

In general, as a fresh snow layer is compressed by a planar surface, the density of the snow layer changes due to the motion of the upper boundary. In the present analysis we assume that a fresh snow layer of the depth *h* _{0} either rests on the ground or over a previously packed snow base whose permeability is much smaller than the new snow layer. If one assumes that the new snow layer is uniformly compacted in the vertical direction, whereas in the horizontal plane there is no crystal movement, the deformation-dependent average snow density, ρ*S*(*h*), where *h* is the local instantaneous height of the new snow layer, is expressed as

if the snow layer has an initial uniform density ρ*S0* at its initial height *h* _{0}. If the mean diameter of the ice crystal, *d*, remains the same, and the initial Darcy permeability of snow is *K* _{0}, one obtains from equations A1 and A2,

In the current application shown in Figure 1b, one assumes that the bottom surface of the ski or snowboard is planar and that there is no lateral tilt or edging. Thus, the local thickness, *h*, of the fresh snow layer beneath the planing surface is a linear function of *x*:

where *h* _{2} is the thickness of the snow layer at the leading edge and γ is tilt angle of the ski or snowboard. The slope of the ski relative to the ground or snow base is given by

where *k* = *h* _{2}/*h* _{1} is the compression ratio. Substituting equations A4 and A5 into equation 3, we have

where *K* _{2} is the Darcy permeability of the snow beneath the leading edge of the planing surface, which can be determined from equation A3 for given values of undeformed Darcy permeability, *K* _{0}, undeformed snow layer thickness, *h* _{0}, and the value of *h* _{2}, as shown in Figure 1b.

Equations A1 and A6 are described in the main text as equations 1 and 2a, respectively.

## Pore-Pressure Distribution beneath a Ski or Snowboard without Lateral Tilt or Edging

From the analysis in (^{18,20}) and Figure 13 in F&W, we know that the pressure distribution in the *y* direction is parabolic. Thus, one assumes a parabolic pressure profile in the cross-sectional plane of a ski or snowboard,

where *E*(*x*), *F*(*x*) are unknown functions and *Pc*(*x*) is the centerline pressure corresponding to the cross section at the location *x*. From symmetry *P*(*x*, *y*) = *P*(*x*, *−y*), it is evident from equation A7 that *F*(*x*) = 0. Thus, equation A7 can be written as:

E(x) is determined by applying a simplified boundary condition *P*∣*y* = ±*W*/2 = *P* _{0} at the lateral edges of the ski or snowboard,

*P* _{0} is very close to the atmospheric pressure because the porosity of the compressed snow changes abruptly at the lateral edges where the compression ends and the hydrodynamic resistance falls precipitously. Substituting equation A9 into A8, one obtains

Equation A10 is described in the main text as equation 12.

## Centerline Pressure Distribution beneath a Ski Surface (ε << 1)

The governing equation for the centerline pressure distribution beneath a ski surface is given by:

where θ*W* = (μ*A*∣*x* = *L*/*P* _{0}) · (*W* ^{2}/*K* ^{2}). In equation A11 the small dimensionless parameter ε multiplies both the highest derivative term *d* ^{2} *p*′*c*(*x*′)/*dx*′^{2} and the nonlinear term (1/*K*′(*x*′))(*dK*′(*x*′)/*dx*′)(*dp*′*c*(*x*′)/*dx*′). Equation A11 is readily solved using singular perturbation methods.

The outer solution of equation A11 is obtained by neglecting the first two terms on the lefthand side,

Near the trailing edge, *x*′ = 0, one expects a sharp increase of pressure in a narrow region of thickness O(ε^{1/2} *L*). Both *K*′(*x*′) and *dK*′(*x*′)/*dx*′ remain nearly constant (see Fig. 2) within this layer and can be approximated by their values at *x*′ = 0. Introducing σ = *x*′/√ε, we obtain the inner solution at *x*′ = 0 by requiring

Similarly, at the leading edge, one introduces ζ = (1 − *x*′)/√ε to stretch the pressure boundary layer, and obtains the inner solution at *x*′ = 1

Thus the composite solution for the centerline pressure distribution beneath a ski surface is given by

Equations A11 and A15 are described in the main text as equations 19 and 20, respectively.