3.3.1. Equilibrium Scour Depth
 One of the objectives of this study is to predict the maximum scour hole depth SD because this parameter rather than the pool shape determines the rate of soil erosion from head cuts (equation 3). The approach taken here is based on the conceptual model of equilibrium depth presented by Stein et al. . In this model the equilibrium scour depth is that obtained when the scour hole is so large that jet diffusion has established a balance between the eroding and resistive stresses and a steady state scour depth SD is reached.
 It is seen from Figure 6 that the following geometric function relates SD, the distance Ji between the jet entry point and the point of impingement, the jet entry angle θe, and the distance from the brink to the tail water surface, h [Stein et al., 1993]:
where the left-hand side characterizes the flow depth in the pool. Equation (29) implies that the maximum pool depth and the point of impingement coincide, and that the submerged jet trajectory does not exhibit any eccentricity. In actuality, when a jet plunges into the pool, the flow reattaches at a location further downstream, and a confined eddy is formed between the entry and reattachment points. When the presence of lateral walls result in nonventilated conditions, as is the case in the Bennett data set, the jet is pulled upstream by the reduction of pressure inside the captive eddy. This phenomenon is known as the Coanda effect and has been studied elsewhere [Sawyer, 1960; Newman, 1961; Rajaratnam and Subramanya, 1968]. Beltaos [1976a] shows that jet eccentricity is significant in the range 30° < θe < 60°, and this finding is confirmed by the streamline pattern shown in Figure 1b. However, it was decided in favor of neglecting jet eccentricity to retain the simple functional structure of equation (29).
 It is also important to note that equation (29) is based on the concept that when the scour hole becomes sufficiently large for a balance to exist between the eroding stresses induced by the diffused jet and the resistive soil stresses, both the scour depth and impingement distance reach their steady equilibrium limits [Blaisdell et al., 1981; Stein et al, 1993]. In actuality, the head cut is migrating continuously and the jet does not impinge on the bed at the same location for an extended period of time, thus preventing Ji from reaching its true equilibrium limit. In order to retain the steady state framework of the present analysis, we introduce the following ratio:
where Ji* represents a virtual equilibrium limit assumed to exist under migration conditions, SD and SD*are the corresponding true and virtual equilibrium scour depths, respectively, and β is treated as a calibrating coefficient.
 Points of maximum shear occur on both wall jet zones outside the impingement region, the latter being dominated by pressure gradients that are centered at the stagnation point and vanish rapidly outside this region [Beltaos, 1976a; Robinson, 1989]. Once the particles are detached, the flow carries them into the wall jet regions where shear stresses are dominant. However, little is known about soil entrainment by turbulent normal stresses and thus current measuring techniques focus exclusively on shear-stress characterizations [Hanson, 1990, 1991]. In order to take advantage of available data, and as proposed by Stein et al. , scour by jet impingement is herein attributed to an effective maximum shear stress, τm, that is in turn related to the impinging velocity, Vi (Figure 6). It should be noted that this velocity is interpreted as the centerline velocity reached at the distance Ji in a jet diffusing in an unbounded fluid. This conceptualization results in:
where Cf is a bed shear coefficient. Beltaos [1976a] found that this coefficient can be approximated in the wall region as
for the jet Reynolds-number range 3 × 103 < Rj < 5 × 105 within the range 30° ≤ θe ≤ 90°. The coefficient δf is of order 1/20, Rj = qw/ν, and ν is the fluid kinematic viscosity. In the Bennett data set Rj varied between 2,400 and 8,300, and thus Cf can be expected to be of order 0.1 for small head cuts.
 The jet centerline velocity is given by [Albertson et al., 1950; Rajaratnam, 1976]:
where cd is a jet diffusion coefficient of order 5/2, and J is the distance from the entry point along the jet trajectory assumed greater than the length of the jet's potential core. When scour reaches the true-equilibrium limit, the condition τm = τc is assumed to hold, where τc is the soil's true critical-shear stress. Hence combining equations (31), (32), and (33) yields
Eliminating Ji between equations (30) and (34) and recalling equation (32) results in the virtual-equilibrium scour depth:
and τ*c = τc/β is the critical shear-stress at virtual equilibrium conditions. Equations (30) and (36) show that the true values of Ji and τc are modified simultaneously by adjusting the single coefficient β, and in this fashion, the observed scour depths are reached instantaneously in any given soil regardless of the migration rate. With this modification, the equilibrium model can be extended to head cuts migrating in a virtual soil whose erodibility properties are identified through empirical calibration of the coefficient β.
 It should be noted that the product cd2δf is of order 5/16 and independent of whether or not the jet is ventilated. Therefore the experimental coefficient σ depends essentially on fluid and soil properties. It is also noteworthy to observe that (1) SD* vanishes as H → 0 because θe → 0, and (2) SD* grows indefinitely as H → ∞ because θ → π/2 regardless of the degree of jet ventilation. However, the latter limit is physically impossible because eventually the jet becomes unstable and breaks down into droplets falling at terminal velocity that do not have the same erosive power of the jet [Goedde and Yuen, 1970].
3.3.2. Rate of Head Cut Migration
 As the head cut propagates upstream, scouring soil as it does so, energy must be consumed to overcome the resistance presented by the soil. Obviously, only the impinging jet that drives the flow in the plunge pool can supply that energy. A simple way to relate the supply and consumption of energy within the pool is to recall that the pool system must satisfy the first principle of thermodynamics, namely, that the heat added plus the total work done must equal the change in total energy of the system. To apply this law to the plunge pool domain, let us consider a control volume enclosing the flow in direct contact with the soil boundary (Figure 6). Noting that the control volume boundary does not change, the inflow and outflow control boundaries are positioned normal to the flow, and the process is steady, the control-volume form of the energy conservation law can be expressed as
where e = u + gζ + V2/2 is the stored specific energy of the fluid, u is the internal energy, gζ is the potential energy defined with respect to an arbitrary datum passing through the entry point, p denotes pressure, υ is the fluid specific volume, pυ is the work of the normal stresses, external heat sources are ignored, are area and velocity vectors, and the subscripts e and d identify the inflow and outflow sections, respectively. The work term Wτ includes the work on the control volume boundaries by normal (pressure) and tangential (shear) stresses, which is largely expended in eroding the soil bed. Introducing the definition and ignoring heat generated by viscous friction, equation (37) yields
 By further neglecting density variations (i.e., ρe = ρd ≈ ρw, ρdVdAd ≈ ρeVeAe = ρwqw) and recognizing that the pressure distribution outside the impingement region is not far from hydrostatic, the pressure at the outflow section can be approximated by pd = gρw(ζe − ζd). Beltaos [1976a] found that the maximum velocity in the wall jet region, Vd,max, varies as
where α is the distance along the wall measured from the point of impingement. Moreover, studies of the classical wall jet [Rajaratnam, 1976] have established the similarity of velocity profiles along the jet cross-sections, and this similarity has been verified for both normal [Beltaos and Rajaratnam, 1973] and oblique [Beltaos, 1976b] impinging jets. Thus it is to be expected that the average velocity of the wall jet decays at the same rate as Vd,max, and by further assuming that Ad is far enough from the stagnation point such that α/be = 10 and rounding off the coefficient 5.5 to 5.0, equation (38) can be expressed as
The right-hand-side of equation (40) has the units (FLT−1)L−1 and therefore can be interpreted as the jet power consumed by head cut scour per unit width of soil bed. Obviously, any given amount of available jet power will result in varying amounts of scour depending upon the erodibility of the soil. It was shown in Section 3.1 that the rate of steady state head cut erosion depends only on the rate of migration and the height of the eroding face. The rate of migration can be interpreted as a rate of soil detachment per unit area uniformly distributed over the eroding face. Thus we can assume that
where kd represents the soil erodibility, τ is the point shear per unit area applied by the flow on the eroding face, and τc* = τc/β is the virtual critical stress introduced in equation (36). Equation (41) is commonly used to characterize the erosion of soil materials encountered in engineering applications [Stein et al, 1993; Stein and Nett, 1997]. Multiplying both sides of this equation by M and integrating over the portion of the head cut height directly exposed to erosive flow action, SD* − h, results in:
where r is a variable of integration. Using the units [kd] = F−1T−1L3, [τ] = FL−2, we note that the integral term on the R.H.S. of this equation has the dimensions (F LT−1L−1), i.e., jet power applied per unit width. Hence equation (42) can be expressed as
The energy rates (power) computed from equations (40) and (43) are expected to be equal under virtual-equilibrium conditions. Thus equating these expressions and solving for M we arrive at the final equation for the migration rate:
It should be noted that the parameter μ is a function of material properties and jet entry angle, and hence it will depend on jet ventilation conditions even when all other parameters remain constant.