In small and intermediate sized streams, the interaction between wood and bed material transport often determines the nature of the physical habitat, which in turn influences the health of the stream's ecosystem. We present a stochastic model that can be used to simulate the effects on physical habitat of forest fires, climate change, and other environmental disturbances that alter wood recruitment. The model predicts large wood (LW) loads in a stream as well as the volume of sediment stored by the wood; while it is parameterized to describe gravel bed streams similar to a well-studied field prototype, Fishtrap Creek, British Columbia, it can be calibrated to other systems as well. In the model, LW pieces are produced and modified over time as a result of random tree-fall, LW breakage, LW movement, and piece interaction to form LW jams. Each LW piece traps a portion of the annual bed material transport entering the reach and releases the stored sediment when the LW piece is entrained and moved. The equations governing sediment storage are based on a set of flume experiments also scaled to the field prototype. The model predicts wood loads ranging from 70 m3/ha to more than 300 m3/ha, with a mean value of 178 m3/ha: both the range and the mean value are consistent with field data from streams with similar riparian forest types and climate. The model also predicts an LW jam spacing that is consistent with field data. Furthermore, our modeling results demonstrate that the high spatial and temporal variability in sediment storage, sediment transport, and channel morphology associated with LW-dominated streams occurs only when LW pieces interact and form jams. Model runs that do not include jam formation are much less variable. These results suggest that river restoration efforts using engineered LW pieces that are fixed in place and not permitted to interact will be less successful at restoring the geomorphic processes responsible for producing diverse, productive physical habitats than efforts using LW pieces that are free to move, interact, and form LW jams.
 Developing an improved understanding of the processes that control channel morphology remains an important goal of fluvial geomorphology, particularly for streams that are strongly influenced by inputs of large wood (LW) from the riparian zone and/or adjacent hillslopes. LW has a profound impact on the storage and transport of bed material in small and intermediate streams, especially in the forested watersheds of the Pacific Northwest. In such streams, the volume of LW also strongly influences the quality and availability of the physical habitat for salmonid species. While the processes influencing the wood load of a stream are relatively well studied and modeled, less attention has been paid to the quantification of the interactions between LW pieces that are responsible for jam formation or to the interactions between LW and bed material sediment transport. In this paper, we present a physically based, reach-scale channel simulator that models LW jam formation in a gravel bed stream as a function of LW input, movement, decay, and breakage.
 A number of stochastic [e.g., Lancaster et al., 2003; Marcus et al., 2011], physical [e.g., Braudrick and Grant, 2000, 2001; Wallerstein et al., 2001], theoretical [e.g., Braudrick et al., 1997], and computer-simulated models have been developed in order to estimate LW recruitment, storage, and movement in channels.Gregory et al.  compared 14 models: 11 were developed in the Pacific Northwest, 2 in the United States Midwest, and one in the Rocky Mountain region. Most of the models Gregory et al.  reviewed are deterministic, while three are stochastic based on probabilities of wood processes and rates: the reviewed models were either compared with limited numbers of field cases, or no attempt was made to test the model outcomes against field cases. In order to develop a more reliable model, a better understanding of the relation between flow, sediment supply, and LW dynamics is needed.
 The objective of this work is to construct and test a reach-scale channel simulator relating LW input, movement, and decay to the storage and release of sediment using a Monte Carlo modeling approach. As far as we know, other numerical models have not attempted to explicitly link LW dynamics to the storage and release of sediment at the reach scale using a stochastic modeling approach.Gregory et al. noted that one short-coming of the majority of the models they reviewed was the absence of stochastic forcing and the lack of information about the variance of the model predictions. We proceed by describing the simulator structure and rules and then presenting the distributions from a series of several hundred runs, each of which spans several centuries. The Monte Carlo modeling approach allows us to estimate the typical steady state functional LW load and sediment storage volume associated with a set of governing parameters, as well as the expected distribution of LW loads and sediment storage volumes. We then proceed to perform a series of sensitivity analyses on the governing parameters to determine the degree to which they influence the modeled distributions.
2. Reach-Scale Channel Simulator
2.1. Prototype System
 The reach-scale channel simulator (RSCS) was constructed using Fishtrap Creek as a prototype: this stream flows into North Thompson River about 50 km north of Kamloops, British Columbia. Like many other streams in the interior region of British Columbia, Fishtrap Creek flows across a high plateau of relatively low relief for most of its length, but becomes steeper and incised as it enters the main, structurally controlled valley system. Near the lower end of the incised reach, Fishtrap Creek flows over a confined floodplain. This part of the stream has been studied extensively since a forest fire occurred there in 2003 [Phillips, 2007; Leach and Moore, 2008; Phillips and Eaton, 2009; Andrews, 2010; Eaton et al., 2010a, 2010b], and a physical model has been used to study the effect of LW load on bed material sediment storage and transport at the study site [Davidson, 2011]. As a result, the characteristics of large wood in that system are relatively well known, as are the bed material transport dynamics.
 Fishtrap Creek is relatively steep (about 0.02 m/m), with a bankfull width of about 10 m and a bankfull depth of 0.5 m [Eaton et al., 2010a, 2010b]. The annual peak flow varies from just under 4 m3/s to a maximum value of 15 m3/s [Eaton et al., 2010b], with the formative discharge estimated to be 7.5 m3/s [Phillips, 2007]. The median bed surface texture (D50) at Fishtrap Creek varies from about 30 mm to 60 mm throughout the study reach, with a coarse tail that often extends into the 128–181 mm size range [Phillips, 2007; Andrews, 2010]. Based on an analysis of tracer movement data and documented morphologic change, we have estimated the annual bed material sediment load (Qbm) to typically vary between 200 and 400 m3/yr for a reach that is about 400 m in length, with peak transport rates on the order of 1000 m3/yr occurring within the reach following the failure of a large LW jam [Eaton et al., 2010a].
 In order to parameterize the RSCS, we have specified channel dimensions that are comparable to Fishtrap Creek: we have chosen a bankfull channel width (Wch) of 10 m, and mean bankfull depth (dch) of 0.5 m (see Figure 1 and Table 1). The reach length is arbitrarily set at 15 times the channel width (i.e., Lch = 150 m). Based on our field estimates of the typical bed material transport rate, we have set the rate of bed material transport entering the reach (Qbm) to a value of 300 m3/yr.
Table 1. Variables Used in the Reach-Scale Simulator
Number of tree mortalities per year adjacent to the reach
Distance of each tree mortality from the channel bankb
Orientation of an individual LW piece
Diameter of an individual LW piece
LW decay coefficient for calculating DLW
Reference LW breakage rate for calculating Pb–LW
Length of an individual LW piece
Distance a mobilized LW piece travels per year
Total number of LW pieces in the reach
Number of jams in the reach
Probability of a tree breaking as it falls into the reach
Probability of an LW piece breaking in any given year
Probability of an LW piece moving in any given year
Conditional probability of LW movement based on LLW
Conditional probability of LW movement based on ΘLW
Probability that mobilized LW will be trapped by a jam
LW key piece definition ratio
Total volume of LW stored in the reach
Bed Material Storage
Bed material trapping efficiency for and LW piece
Surface area of an LW piece, projected across the stream
Bed material sediment transport into the reach
Volume of sediment stored by an LW piece
2.2. Overview of the Simulator
 Unlike most previous models that estimate LW load deterministically by specifying average rates for various input terms and using a large time step [see Gregory et al., 2003, Table 1], the RSCS has a time step of one year and tracks individual LW pieces that enter the system by stochastically driven processes. All of the ‘events’ that pertain to LW input and behavior—such as a tree falling, an LW piece breaking, a jam forming—are triggered by first specifying event probabilities and then using random number generators to determine whether the event happens or not. As a result, the actual LW inputs for a single time step arise stochastically, rather than being determined by a specified rate parameter. Thus, while the RSCS structure and LW input relations are conceptually based on the model presented by Benda and Sias , the structure of the RSCS (and thus the variables included in the RSCS, which are listed in Table 1) are fundamentally different from those in their model.
 The RSCS tracks the diameter, length, and orientation of each piece, as well as whether or not the piece is trapped in a jam; it also tracks the volume of sediment stored in association with each piece, since the main purpose of the RSCS is to investigate the interactions between in-stream wood and stream morphodynamics. Therefore, the RSCS only considers the functional LW load (i.e., LW that interacts with the flow and sediment transport processes), not the total wood load of the stream. It should be noted that the position of each LW piece within the reach is not considered: interactions between pieces are dealt with in a probabilistic fashion instead of a spatially explicit one. The RSCS calculates and saves the total functional LW load at the end of each year, as well as the volume of sediment stored in the channel; it records the number of LW jams, the volume of LW associated with jams and the volume of sediment stored in association with jams; and it records the jam age at failure, as well as the total volume of sediment that was stored behind the jam just prior to failure.
 Each RSCS run produces a single time series of data. In order to estimate the typical characteristics of the reach, the RSCS is run multiple times, and the results are aggregated to estimate the typical reach characteristics. The Monte Carlo modeling approach also provides information about the range of possible channel characteristics.
 The governing equations have been kept as simple as possible so that we can assess the degree to which complex behavior arises from a simple set of behavioral rules. As a result, some of the processes and interactions that affect LW dynamics have been ignored: for example, we have not attempted to model the interactions between LW pieces not included in a jam, nor have we attempted to model LW accumulations that do not span a significant portion of the channel; we have not considered the interactions between LW pieces or jams and the channel banks; and we have not considered the exchange of LW between the stream and adjacent floodplain. Some of these issues can be accounted for by modifying the model, while others cannot due to the limitations of this modeling approach.
2.3. Riparian Forest Inputs
 Large volumes of LW can be recruited to stream channels by episodic events, such as mass movements (including debris flows and snow avalanches, in particular), windstorms, and wildfire- or disease-related tree mortality [e.g.,Bragg, 2000; Benda and Sias, 2003; Czarnomski et al., 2008]. LW can also be recruited at lower but more persistent rates determined by natural tree mortality, lateral migration of the stream, and slow mass movements [Bisson et al., 1987; Fetherston et al., 1995; Hassan et al., 2005]. While LW inputs to mountainous headwater streams are dominated by episodic mass movements from the hillslopes, inputs to alluvial channels with moderately extensive floodplains come primarily from the surrounding riparian forests and from the channel network upstream.
Benda and Sias identified a relatively complete suite of LW input processes that are relevant at the reach scale, including influx of LW from upstream via fluvial transport, chronic forest mortality, fire or wind storm-related tree toppling, episodic bank erosion, landslide/avalanche occurrence, and exhumation of wood buried in the floodplain. We have followedBenda and Sias  closely in developing the LW input component of the RSCS, and we therefore use an annual mortality rate (assumed to represent the rate at which trees topple to the ground) as the dominant input term. While the RSCS is ultimately intended to be used to investigate the effects of disturbances such as forest fires on channel dynamics [cf. Bragg, 2000; Benda and Sias, 2003], and can incorporate episodic inputs due to forest fire-related mortality and due to bank erosion, both of these input terms are set to zero for all of the runs presented herein. As a result, the characteristics of the riparian forest are held constant over time. The forest stand is described by an average tree height (Htr), tree diameter (Dtr), and forest density (ρtr). We set Htr = 30 m, Dtr = 0.4 m, ρtr = 500 stems/ha, values that are consistent with the lodgepole pine and Douglas fir forest adjacent to Fishtrap Creek [cf. Farnden, 1996]. As a point of comparison, Benda and Sias  and Czarnomski et al.  used a similar approach assuming a mean tree height of 40 m to describe the coastal forests of the Pacific Northwest. By using a single mean tree height and diameter, we are effectively assuming that the average tree dimensions are reasonably representative of the LW that enters the stream, and that the rate at which trees fall into the channel is equal to the rate at which they are replaced by regrowth (i.e., the forest stand is operating at steady state).
 In the RSCS, chronic mortality (M) varies randomly from year-to-year, with rates chosen from a uniform distribution between 0.2% and 0.3%, which is similar to the rates used byBenda and Sias  for an undisturbed forest. The number of trees that die and fall to the ground every year (NM), and could thus potentially contribute to the in-stream wood load of the reach, is
The term 2HtrLch represents the total floodplain area adjacent to the reach from which trees can enter stream (Figure 1), and ρtr ∙ 2HtrLch represents the total number of trees adjacent to the reach.
 The tree fall direction for each tree, Θtr, is chosen from a uniform random distribution from 0 to 360°. Trees assigned orientations between 0° and 180° fall toward the stream channel and can potentially input LW to the stream, while the rest fall away from it. The distance of each fallen tree from the stream bank (Xtr; Figure 1) is assigned using a uniformly distributed random number between 0 and Htr. It is assumed that, for LW to actually enter the stream and become functioning LW capable of interacting with the sediment transport processes in the stream, the tree must break at the intersection of the stream bank nearest the falling tree and the tree stem (Figure 1): this simplifying assumption ignores the role of bank retreat, which introduces LW pieces that are attached to a root wad, and which are therefore less mobile than similarly sized LW pieces without one. Considering the fall direction and the tree position, the length of the tree that enters the stream channel (LLW) is calculated as follows:
If the top of the tree reaches the other side of the channel (i.e., LLWsinΘtr > Wch where Wch is the bankfull width of the channel), then the tree is assumed to break at the intersection with the far bank as well, such that LLW = Wch/sinΘtr.
2.4. LW Piece Decay and In-Stream Breakage
 LW leaves a reach when it is transported downstream, decays in situor moves out of the channel onto the floodplain. While such over-bank losses may be important in larger systems, they are generally assumed to be negligible in small to intermediate streams [Martin and Benda, 2001; Benda et al., 2002]. Decay involves a number of different processes, and occurs at rates that depend on the climate, species of tree, cause of tree mortality, and LW recruitment mechanism [Harmon et al., 1986; Webster and Benfield, 1986]. The effect of wood submergence on decay processes is also important, but relatively poorly studied [Keller and Swanson, 1979; Sedell et al., 1988; Cederholm et al., 1997; Benda and Sias, 2003]. Because decay rates are so variable, it is difficult to make any generalizations about the typical wood decay rates in streams and riparian zones.
 We again follow Benda and Sias , and assume that over-bank losses of LW to the floodplain are approximately balanced by inputs due to exhumation of buried wood, and that the fluvial input of LW to the reach is balanced by the LW export out of the reach, leaving decay as the dominant mechanism by which LW is lost from a reach. In their model,Benda and Sias  attributed losses of wood at the reach scale to in situ decay, as well as the physical disaggregation of LW into pieces that are too small to be retained within the reach: both of these processes are included in their decay function. In the RSCS, we consider these two processes separately, specifying a decay rate based on observations of wood decay on the forest floor, and by simulating the advection of small wood out of the system by incorporating breakage in the model.
 In the RSCS, several things can happen once an LW piece enters the stream. Immediately upon entering the channel, the LW piece may break. The probability of the piece breaking during the input process (Pb−tr) is assumed to be proportional to its length relative to the stream channel width, and the position at which the break occurs is assumed to occur randomly at a location between 25% and 75% along the length of the tree. We use a scaled error function curve to estimate the breakage probability, as shown in Figure 2a. The probability of a small piece breaking is very low, and increases to about 40% for a piece as long as the channel is wide. Regardless of whether or not a piece breaks when it is input, the orientation of the piece (ΘLW) is inherited from the direction in which the tree falls (Θtr). At this point, the total age of the piece (t) and the time that the piece has been at one location (tx) are set to zero, and both will increase every year. If a piece breaks again or is moved downstream, tx is reset to zero, while t continues to increase. The variable tx is used to estimate the sediment trapping efficiency, as described below, while t is used to estimate the rate of decay.
 We assume that the LW piece diameter (DLW) declines exponentially at a rate determined by a decay coefficient, Kdecay [cf. Harmon et al., 1986]:
In our simulations we set Kdecay= 0.01, which corresponds to an exponential decay curve with a half-life of 69 yrs. Decay is typically expressed in terms of volumetric loss, and since volume is proportional toD2, the equivalent volumetric decay rate is 2Kdecay. The implicit volumetric decay rate of 0.02 is consistent with the range of decay rates for wood on the forest floor reported in the literature [e.g., Bilby, 2003].
 Advection losses of small wood are calculated based on the size of the wood piece: any LW piece that has a diameter of less that 0.1 m (which is reached at t = 138 yrs in the RSCS) is removed from the system, assuming that the piece is likely too old and too small to function effectively. Similarly, any piece for which LLW < Wch/5 is removed from the system, assuming that such short LW pieces would be easily transported out of the reach. In reality, pieces shorter than Wch/5 may be trapped in jams, thereby increasing the sediment storage capacity of the jams, but the residence time of this wood is probably much shorter than for larger pieces, so this interaction has not been included in the model. Thus, in our system LW loads may decline due to decay at a rate determined by Kdecay or due to advection once the piece becomes too small to be retained within the channel.
 During every year of a run, each LW piece in the channel is evaluated with respect to an in-stream LW breakage probability,Pb−LW. If a piece does break, then it is assumed to break at a location randomly located between 25% and 75% of the way along the piece. The first part of the piece is assumed to remain in place and to retain a linear proportion of any sediment the unbroken piece may have stored. The second part is assumed to release any sediment it may have stored and to re-orient itself so that it is either slightly skewed with respect to the flow direction (i.e., ΘLW is between 120° and 150°, where the streamflow direction is assumed to be 180°), or nearly parallel to the flow direction (i.e., ΘLW is between 150° and 180°). Based on the observations made by Davidson  in a series of flume studies of wood movement, we specify in the RSCS that one third of the broken pieces adopt a skewed orientation, while the rest take up an orientation nearly parallel to the flow.
 The probability of an LW piece breaking once it is in the channel is much lower than the probability of breakage when it falls into the channel. For the sake of simplicity, we estimate this breakage probability using the shape of the LW piece (LLW/DLW), based on the assumption that long, narrow pieces are more susceptible to breakage than short, wide ones. We use a reference breakage probability (Kbreak, assumed to be 0.10 for our base case) that applies to an LW piece for which LLW/DLW = 100, and then assume a linear relation between piece shape and breakage probability.
The annual probability of breakage for a piece that has a length of 10 · DLW is 0.01, and for a piece with a length of 1.0 · DLW is 0.001, based on our assumed value for Kbreak. Since the probability of breakage depends on the diameter of the piece, which declines over time due to decay, breakage is also driven primarily by LW decay, with longer LW pieces being more susceptible than short ones for the same level of decay (and hence the same piece diameter).
 These rules, taken together, produce an exponential distribution of LW piece lengths (see Figure 3), which is similar to the LW distribution assumed in the model presented by Benda and Sias . The median LW length for the distribution is about 4 m, which is close to the median LW length observed in Fishtrap Creek by Andrews .
 The rules governing the movement of individual LW pieces are relatively simple, and are based on the observations made by Davidson during a series of experiments using a physical model of Fishtrap Creek. If a piece moves, it is assumed to release any sediment it may have stored and to re-locate within the reach at an orientation that is either slightly skewed with respect to the direction of flow or parallel to it, according to the same rules as for LW breakage. The probability of movement (Pmove) is estimated as the product of two conditional probabilities. The first (PmoveL), estimates the probability of movement based on the relative size of the LW piece (LLW / Wch), assuming the piece is oriented perpendicular to the flow direction. Since the flow depth at bankfull conditions is large (i.e., 0.5 m) relative to the median diameter of the wood in the RSCS (0.2 m), we ignore the effect of LW diameter on movement. In streams where the flow depths are closer to half of the LW piece diameter, the effect of diameter would also need to be considered [Braudrick et al., 1997; Braudrick and Grant, 2001; Bocchiola et al., 2006; Manners et al., 2007]. The second conditional probability ( ) adjusts the probability of movement by considering the effect of orientation on the likelihood of movement. Thus:
 LW often accumulates into jams once it is entrained, especially in intermediate size streams [Gurnell et al., 2002; Wohl and Jaeger, 2009]. The organization of wood into jams complicates the processes of wood export and storage, alters bank and bed stability, and changes the geomorphological and sedimentological character of the reach.
 When LW pieces in the RSCS are allowed to interact and form jams, the rules related to piece movement are modified, and the orientation of the pieces that interact with a jam are changed. We first identify key pieces that are of a size and orientation that they block enough of the channel to become the nuclei for potential jam formation. The threshold ratio for defining a key piece, TRkey, is set to a value of 0.8 for the base cases, and is compared to the width of the key piece projected across the channel. The criteria for a key piece can be written as follows:
 Once a key piece is identified, it is assigned a unique identifying code that represents a potential jam in the reach. Any individual LW pieces that move in a given year are assessed to determine whether or not they are likely to encounter (and be trapped by) a jam as they move downstream. The distance of movement is estimated by fitting an exponential curve to the upper envelope of the data on LW piece movement from Mack Creek, Oregon, presented by Gurnell et al. , as shown in Figure 2d. Those data represent the cumulative distance of movement in Mack Creek over a 10-year period, which obviously includes pieces that moved in some years but not in others. We assume that a piece that is known to move in a given year will therefore move a distance similar to the largest observed movements in the Mack Creek data set. We have reduced the data to travel distances in m/yr (Ltravel), and then we have normalized both the travel distances the LW piece lengths by the bankfull width of Mack Creek to facilitate the application of this equation across a range of scales. The empirically fitted equation has the form:
 The probability of an LW piece being trapped by a jam depends on distance moved by the piece, the number of jams in the reach, and the length of the reach (Lch). Consider a reach with a single key piece located somewhere within it. The probability of a moving LW piece interacting with that key piece is Ltravel / Lch. The probability of the piece not interacting with the jam is therefore 1 − Ltravel / Lch. If there are Nj jams in the reach, then the probability of an LW piece not interacting with any of them is (1 − Ltravel / Lch) . The probability of a piece being trapped by a jam (Ptrap) in a reach of length Lch that contains Nj key pieces is given by:
Any piece that is trapped in a jam (based on a comparison between Ptrap and a random number) is assigned to one of the key pieces identified in the reach.
 If a piece is trapped in a jam, two things happen. The first is that the piece is forced into an orientation that is nearly perpendicular to flow (ΘLWfalls between 75° and 105°), rather than being slightly skewed or parallel to flow as are the individual pieces. While, in reality, some pieces that are trapped by a jam take up skewed or parallel orientations, most of them have larger blockage ratios than do LW pieces that are not part of a jam, due either to a more perpendicular orientation or because a larger proportion of each piece is submerged at high flow. The result is an increased trapping efficiency for jam-related LW [Davidson, 2011]. In order to replicate this increase in sediment trapping efficiency, we focus on the orientation of LW rather than trying to account for changes in LW piece submergence.
 The second thing that happens when an LW piece interacts with a jam is that the piece is prevented from either moving or breaking until the key piece that initiated the jam breaks. When the key piece eventually breaks, all of the sediment stored in association with the jam is released, and all of the pieces move into orientations that are skewed or nearly parallel to flow. While the process of jam decay is more complicated than the RSCS rule admits, we know from experimental studies that most of sediment stored behind a jam will be transported out of a reach within a single flood event when the LW is removed [Davidson, 2011], though some will remain stored in channel bars and along the channel margins [e.g., Lisle, 1995].
2.7. Sediment Storage
 The governing equations for sediment storage are based on results from laboratory experiments conducted by Davidson . We simulate sediment retention using a reach-average reference bed material trapping efficiency (ζbm) that is related to the total LW load in the stream channel (VLW). The reference trapping efficiency is simulated using a scaled error function that was fit to the maximum observed trapping efficiencies from the experiments conducted by Davidson , and which asymptotically approaches 1 for high wood loads (see Figure 2e). We also assume that the trapping efficiency drops off exponentially over time as the available storage space is filled with sediment: the reduction in trapping efficiency has a half-life of 0.70 yrs, which produces trapping rates that are less than 5% of the reference efficiency by 3 yrs after the piece was last moved (seeFigure 2f).
 The trapping rate for each LW piece is calculated by distributing to the total trapping potential (Qbm · ζbm, where Qbm is the annual bed material transport rate entering the reach; see Figure 1) using a weighted-average approach. We use the ratio ofBi, the area of the ith LW piece that is projected across the channel, to the sum of all the projected areas for all pieces (∑B), and then adjust the trapping rate based on the length of time that the LW piece has been in its current location, tx. B is calculated as follows:
 The change in sediment storage for a given year is:
wherein Vsed is the total volume of sediment stored by an individual LW piece.
3. Monte Carlo Modeling Results
3.1. Base Case Characteristics
 One set of Monte Carlo runs is presented in which the RSCS only invokes the rules for LW input and breakage (see Figure 1). In this base case, LW pieces are not permitted to move downstream, to form jams, or to change their orientation unless they break: this is referred to as base case 1 (BC 1). There are two other base cases that use the same set of parameter values but in which the LW pieces are permitted to:
 1. move downstream, changing their orientation as they do so (BC 2); and
 2. move downstream, change orientation, and interact with LW jams (BC 3).
 The average wood load and stored sediment volume for all three base cases are plotted against time in Figure 4. All of the base cases use the same values of the governing parameters (see Table 1). The data from the individual runs for BC 3 are also presented in Figure 4 to provide a visual representation of the typical range of data.
 At t= 0 yrs, the channel contains no in-stream LW, and is conceptually equivalent to a forest channel that has had all of the in-stream LW removed from it. LW removal was a common practice in managed streams in the Pacific Northwest in the 1960's and 1970's, so this initial condition is not a historically uncommon one. All three base cases show a similar response time, reaching a steady state wood load after about 150 yrs, and a steady state sediment storage by between 150 and 200 yrs. The choice of rules governing LW dynamics does not have a detectable influence on the time required to reach a steady state wood load.
 The LW dynamics do influence the steady state wood load and sediment storage dynamics. The distribution of wood load and sediment storage volume at steady state (which is conservatively defined as the period between 200 and 300 yrs) is represented using a boxplot for each base case in Figure 5. The distributions of year-to-year changes in sediment storage at steady state for each base case are also presented inFigure 5. The wood loads for BC 1 and BC 2 are essentially identical, since mobilized LW that leaves the bottom end of the reach in BC 2 is assumed to be compensated by LW entering the reach from upstream. In both cases, wood volume is only ever lost from the reach via decay (according to equation (3)) and advection of small wood pieces out of the system (i.e., once LLW < 0.20 Wch or DLW < 0.10 m), and these processes are not linked to LW movement in the model.
 The rules governing LW movement introduced in BC 2 do have an effect on the volume of sediment stored in the reach. Every time an LW piece is entrained, it releases all of the stored sediment and moves into an orientation that is either slightly skewed with respect to the flow direction or parallel to it, which reduces the sediment trapping efficiency for the piece. As a result, the reach in BC 2 stores about 25% less sediment on average than it does in BC 1. The range of the storage distributions for BC 1 and BC 2 are similar. The distributions of the year-to-year changes in sediment storage are also similar for BC 1 and BC 2, though the range for BC 2 is greater than for BC 1.
 The relation between the wood load and the volume of sediment stored in the channel is effectively linear for both base cases (see Figure 6a and 6b). For BC 1, the slope of a linear relation between the total volume of wood in the channel and the total volume of sediment stored in the channel is 8.7 m3/m3 (solid dark grey line on Figure 6), while it is 6.4 m3/m3 for BC 2 (solid light grey line on Figure 6). These values compare well with the volumes of sediment stored in the flume experiments reported by Davidson , who estimated storage values of 3.3 to 8.0 m3/m3, and are of the same order as the estimate of 3.5 m3/m3 reported by Brooks et al. .
 BC 3 is different from the other two base cases in all respects. The steady state wood load has about the same range, but the mean wood load for BC 3 is slightly higher than the others. LW that is trapped in a jam is prevented from breaking or moving until the jam itself breaks, so the jams act to preserve the wood stored in it and reduce the LW losses due to piece breakage and advection; the effect is noticeable but small compared to the range of the data (Figure 5). The volume of sediment stored in the reach during BC 3 has a mean that is nearly identical to that for BC 1, but the range of the data is greater. The relation between the wood volume and the sediment storage volume is again linear, and similar to BC 1, though the data are much more scattered (Figure 6c): for BC 3, the volume of sediment stored per unit of functional LW is 7.9 m3/m3 (for BC 1 it is 8.7 m3/m3). The biggest difference is in the year-to-year changes in sediment storage (Figure 5c). Due to the increased trapping efficiency of LW in a jam, the maximum year-to-year sediment storage increases are elevated by about 50% relative to BC 1. The maximum year-to-year sediment losses are even greater, and approach the annual sediment load,Qbm, since jams release all of their sediment once the key piece breaks.
 For BC 3, the characteristics of the LW jams that form can be examined. At the time that each jam breaks, the jam age and volume of sediment stored in association with that jam is recorded. To remove any distortions associated with the rapid increase in wood load that typically occurs from t = 0 to t = 100 yrs, we only analyze jams that failed between 100 and 300 yrs. The results are presented as a bivariate probability density plot based on the age of the jam at failure and the volume of sediment stored in the jam (Figure 7). Most jams fail after about 20 yrs and store about 20 m3 of sediment, though they may store as much as about 150 m3 or as little as 1 m3. It is also relatively common for jams to persist for about 60 yrs, but these jams store more sediment (50 m3) on average, and exhibit a narrower range of storage values (from about 20 m3 to 150 m3). Relatively few jams persist for 100 yrs, but if they do, they invariably store large volumes of sediment (i.e., between 50 m3 and 150 m3). As a point of reference, the largest jam that formed at Fishtrap Creek stored on the order of 176 m3 [Eaton et al., 2010a], which indicates that the upper size limit of the jams formed in theRSCS are reasonable.
 BC 3 appears to be the most realistic configuration, and differs from the other two base cases in a predictable (but not extreme) fashion. The remaining analysis of the RSCS behavior will focus on BC 3.
3.2. Sensitivity Analysis of LW Inputs
 To determine the effect that the governing parameter values have on the output, we conducted a series of sensitivity analyses. While many of the parameters were assessed, three of them had significant effects on the model behavior, and are discussed herein. The first analysis considers the forest stand density, ρtr: it was varied by ±20%, and ±40%, which is within the range of stand densities for our prototype stream [see Farnden, 1996]. This parameter controls the rate of LW input to the stream channel.
 The value of ρtr has no discernible effect on the time required to reach steady state, though it does influence the typical steady state wood load (Figure 8). As the stand density goes up, the mean wood load goes up linearly, but the range of the distribution gets wider (Figure 9a). The mean wood load for the lowest stand density (300 stems/ha) is about 0.01 m3/m2, while the wood load for the highest density (700 stems/ha) is about 0.024 m3/m2.
 The distributions of stored sediment volumes mirrors those for wood load: the mean sediment volume increases with stand density from about 125 m3 to about 275 m3 (Figure 9b). The proportion of the total sediment that is stored behind LW jams is also presented (Figure 9c). For all values of ρtr there are instances when jam storage represents only a small fraction of the total stored sediment. More often, jam storage dominates sediment storage. The median proportion of the sediment stored in association with jams changes relatively little: it is about 0.80 for the lowest stand density and about 0.86 for the highest. The most noticeable change is that, as the stand density increases, the upper quartile remains virtually constant at about 0.90 but the lower quartile increases from about 0.68 to 0.82. The result is that the distributions in Figure 9c become more tightly clustered around the median value, indicating that the effects of jams on sediment storage become more persistent as stand density (and wood load) increase, even if their median effect does not change by much.
 The average number of jams is also tracked over time (see Figure 10a). The number of jams increases until about 200 yrs, after which the number of jams varies about a mean value that is positively correlated with the stand density. For the lowest density, the average number of jams in the reach plateaus at about 1.9 jams, which corresponds to a jam spacing of about 7.5 Wch (or 75 m). The number of jams at steady state varies from 0 to 9 for this stand density (see Figure 10b), indicating a wide range of possible jam numbers and associated jam spacings.
 The highest stand density (700 stems/ha) produces about 5.5 jams at steady state, corresponding to a mean spacing of 2.7 Wch (27 m), while the standard density used in the base case (500 stems/ha) is associated with a jam spacing of about 3.8 Wch (38 m). The range of jam numbers at steady state for both 500 stems/ha and 700 stems/ha varies from 0 to 16, indicating that in rare circumstances, jam spacing can be about 1 Wch (10 m), according to the Monte Carlo modeling results, so long as the assumed stand density is ≥500 stems/ha.
3.3. Sensitivity Analysis of LW Breakage Rate
 The reference coefficient that determines whether or not an LW piece will break (Kbreak) influences the steady state wood load as well as the dynamics of the jams that form in the reach. As the coefficient increases, jam failure becomes more likely at every time step. As a consequence, jams are shorter lived for high values of Kbreak than they are for low values (see Figure 11a). For the range of Kbreak values that we have tested (i.e., the base value ±40%), the mean age of the jams at failure ranges from 16 yrs for the highest value of Kbreak to 30 yrs for the lowest value, while the maximum age attained or exceeded only 20% of the time ranges from about 20 yrs to 60 yrs.
 The value of Kbreak also affects the number of jams present in the reach at steady state (see Figure 11b). For the lowest value of Kbreak, the median value is about 6 jams per reach, while the upper quartile value is about 7 jams per reach and the lower quartile is about 4. For the highest value of Kbreak, the lower quartile value is about 1 jam per reach, the median is 3 and the upper quartile is 4. The overall relation between the number of jams present and the value of Kbreak is a linear one for which a 40% decrease/increase in Kbreak produces about a 40% increase/decrease in the median number of jams.
 Since the influence of Kbreak on the age distribution and the number of jams in the reach is linear, the model is not particularly sensitive to the value of Kbreak. Furthermore, the chosen base value (Kbreak = 0.10) seems reasonable (although admittedly untested against empirical data) based on the results in Figures 11a and 11b, which agree reasonably well with the conceptual model for jam formation and decay proposed by Hogan et al. .
3.4. Sensitivity Analysis of Key Piece Threshold
 The final sensitivity test investigates the effect of the parameter TRkey (used to identify key pieces that can form LW jams) on the model behavior. This parameter represents the ratio of the LW length projected across the channel to the channel width. Monte Carlo runs are presented for TRkey values of 0.70, 0.75, 0.80, 0.85 and 0.90%. The results are shown in Figure 12. By definition, lower values of TRkey produce more key pieces and thus more jams (Figure 12a). The lowest value of TRkey produces between 4 and 8 jams per reach (based on the upper and lower quartile values), while the highest value of TRkey produces between 2 and 4 jams per reach. Despite this relatively large effect on the number of jams present in the reach, the volume of stored sediment remains about the same for all runs (Figure 12b), though slightly more of that sediment is typically stored in association with LW jams for lower values of TRkey (Figure 12b); however, the difference is relatively small, and the sediment storage dynamics of the model are not sensitive to the value of TRkey used in the Monte Carlo run.
 As far as we know, the RSCS is the first attempt to link LW input processes to both jam formation and bed material storage and transport using a physically based set of rules. The RSCS accounts for the stochastic processes of breakage, decay, movement, and bed material sediment trapping for each LW piece that enters the stream channel. It employs equations that calculate the volume of sediment each LW piece will trap, and it allows individual LW pieces to interact with one another, thereby forming LW jams which have an effect on sediment trapping that is disproportionate to the volume of LW within the jam. In order to determine the typical characteristics of a reach, a Monte Carlo modeling approach is used, in which the results of numerous RSCS runs are aggregated to determine the mean values of the wood load, the volume of sediment stored in the channel, and the number of jams within the reach. This approach also provides estimates of the ranges of values that can be expected; most of the models reviewed by Gregory et al.  do not include stochastic forcing and cannot provide information about the range.
 In some respects, our model is much simpler than others. For example, the models presented by Benda and Sias  and by Bragg  explicitly represent a wider range of LW input processes. The RSCS uses a single mortality rate, M, which is treated as a probability that any one of the riparian trees will die and fall over in any particular year. The actual LW input rates in the RSCS are stochastic, since trees die randomly, are located at random distances from the channel banks, and fall in random directions; however, the average LW input rate is effectively constant and is determined by M.
 A recent field study of streams on the Interior Plateau of British Columbia by Johnston et al. identified five distinct LW input processes: landslides; wind throw; stem breakage; bank erosion; and standing dead fall. The streams they studied are physiographically and climatologically similar to our prototype stream, Fishtrap Creek. They concluded that, for streams of a size and morphology similar to our prototype, the dominant input mechanism is standing dead fall, which accounts for nearly 60% of all in-stream LW, followed by bank erosion, which accounts for about 20%. Their results also indicate that landslide-related inputs are negligible for streams such as Fishtrap Creek, though landslide-related inputs are certainly important (perhaps dominant) in the more mountainous areas of British Columbia. Our chronic forest mortality rate corresponds most directly to the process of standing dead fall but can reasonably be extended to include wind throw and stem breakage, which taken together account for nearly 80% of the LW inputs for the streams studied byJohnston et al. . Our simplified approach to LW input can be conceptually extended to include the inputs due to bank erosion, but only provided that the rates of bank erosion—averaged over the length of the reach—are relatively constant over time, and that the channel morphology for the reach does not change significantly over time. This simplified approach to the LW input parameterization is therefore not applicable to systems subject to disturbances such as wildfire or large floods, or to reaches where jams becomes large enough and numerous enough to force a net change in the reach-average channel pattern [Collins et al., 2012; Wohl, 2011]. However, so long as the channel pattern remains statistically stable, the LW input rate will remain constant, regardless of the process by which it enters the stream. From this point-of-view, our use of a single input parameter (M) is justified for stable channels that are not subject to external disturbances like forest fires: in order to consider disturbances such as forest fires, additional input terms must be considered [e.g., Benda and Sias, 2003], as do any potential changes in the channel characteristics [Eaton et al., 2010b].
 An obvious and important question is how the wood loads modeled using the RSCS compare to wood loads observed in the field. In Table 2, the wood load observed in Fishtrap Creek is compared with the mean wood loads for our most realistic model, BC 3. In order to facilitate comparison with other field data, we have transformed the model output to the more commonly used m3/ha. The wood load observed in Fishtrap Creek corresponds relatively well with the simulation results associated with the default forest density of 500 stems/ha, and is consistent with the ranges for simulations assuming a density of 300 stems/ha and 700 stems/ha.
Table 2. Mean Wood Loads Predicted by the Model and Observed in the Fielda
 The calculated wood load is generally consistent with the mean wood loads reported by various other researchers in mountainous environments (Table 2), with the exception of the loads reported by Lienkaemper and Swanson  in coastal Oregon and by Andreoli et al.  in the Chilean Andes. The forest type that we have modeled is similar to the forests found in the Colorado Front Range, so the data reported by Richmond and Fausch  and Wohl and Cadol  are particularly relevant. Our simulated, steady state mean wood load of 178 m3/ha compares well with the mean of 182 m3/ha reported by Richmond and Fausch , and the range of the simulated data (72–327 m3/ha) also compares well to their observed range (92–254 m3/ha). The mean load of 161 m3/ha reported by Wohl and Cadol  compares favorably, although their reported range of observations is significantly wider than the simulated range. However, the analysis conducted by Wohl and Cadol  indicates the reason for this discrepancy: they analyzed the range of wood loads as a function of channel size, and found that, for channels having a width similar to our prototype stream, the range of data is closer to 0 to 400 m3/ha (i.e., all of their higher wood loads are associated with smaller channels), which agrees nicely with our simulated range, especially considering that our wood load estimates are for functional LW only, not the total wood load.
 The typical LW piece sizes in the RSCS also compare well with the sizes reported by Wohl and Cadol : they report average LW piece lengths ranging from 2.1 to 4.1 m for their study reaches in the Colorado Front Range, and average LW piece diameters ranging from 0.16 to 0.23 m. In the RSCS, LW piece lengths are exponentially distributed with a mean length of 4.7 m and a median of 4.1 m (see Figure 3). LW piece diameters are also exponentially distributed, with a mean of 0.21 m and a median of 0.19 m. These comparisons between the RSCS and various data sets from the field imply that the rules that we have invoked for LW input, decay, breakage and advection out of the study reach are reasonable, since the product of these rules (i.e., the simulated wood volumes and the piece dimension distributions) is comparable to observations made in the field.
 In order to assess the performance of our rules related to LW piece interaction and jam formation processes, we compare the jam spacing for similar channels in the field to the simulated jam spacing. Generally, the range of jam spacing that occurs in the RSCS (i.e., 3.8 Wch for BC 3; 2.7 Wch for a forest stand density of 700 stems/ha; and 7.5 Wch for 300 stems/ha) is consistent with estimates made in the field. For example, Petts and Foster  reported jam spacings between 2 and 3 Wch, Kreutzweiser et al.  estimated the jam spacing in boreal streams to be between 6 and 16 Wch, and the channels in the Colorado Front Range with stream gradients less than 0.06 m/m studied by Wohl and Cadol  have a mean jam spacing of 3.9 Wch. The latter are probably most similar to the prototype for the RSCS, and the mean jam spacing corresponds almost exactly with the BC 3 results. For the prototype stream, Fishtrap Creek, Andrews  measured a jam spacing of 6.7 Wch, which is well within the range of values for the various runs. Therefore, our simplified rules for LW piece interaction and jam formation appear to capture at least some of the key aspects of jam formation.
 The general model for jam development and decay in the Pacific Northwest presented by Hogan et al.  fits well with the Monte Carlo modeling using the RSCS. Hogan et al. posited that most of the bed material trapping occurring during the first decade after a jam forms. In their model, jams will begin releasing any stored sediment between 20 to 30 yrs after jam formation, and the channel will return to the pre-jam state by between 30 and 50 yrs. According to our Monte Carlo modeling, jam decay (which is driven by randomly occurring LW piece breakage events) typically occurs between 16 and 30 yrs after jam formation (seeFigures 5 and 11), at which point all of the stored sediment in the jam is released. Typically for jams in the Pacific Northwest, the release of sediment is a more gradual process, involving the failure of multiple key pieces over a period of several decades. The timing and nature of jam failure likely varies from place to place, depending on the typical species of trees found in the riparian area and the local decay rate, as well as the characteristics of the flow regime that determine the forces acting on the jam.
 Temporal changes in the sediment output, volume of stored sediment, and number of jams that occur during individual model runs are also informative. The outputs for the second half of two BC 3 runs—chosen to represent a run with relatively high wood loads and one with low wood loads—are presented in Figure 13. For the high wood load scenario, the number of jams within the reach fluctuates between 2 and 9 for most of the run, falling to 1 jam near the end of the simulation (Figure 13a). The output clearly shows that the RSCS is creating jams and destroying them over timescales of one to three decades, and that the random nature of the jam creation and destruction process can produce a wide range of wood loads and associated jam numbers.
 We can also use the RSCS output to determine the potential morphologic consequences of the jams. Figure 13b depicts the net storage of sediment associated with the wood in the channel, expressed relative to the volume of the channel for bank full flow conditions. The dashed line at 0.33 indicates a somewhat arbitrary threshold above which the bed morphology is likely to be substantially different from that associated with lower volumes of sediment storage, and to be associated with a very diverse physical habitat [e.g., Montgomery et al., 2003]. In contrast, when the volume of stored sediment drops below 0.33, we speculate that the channel is likely to enter into a down-cutting phase, during which the channel morphology becomes much simpler, morphologically, representing a physical habitat that is much less variable. In fact, aggradation levels above about 0.33 may produce dynamics not considered within this model, such as shifting of the channel away from the jam structure or avulsion of the channel around it, forming a multiple-thread channel pattern, as described byCollins et al.  and Wohl . The interaction between jams and the local channel morphology is an avenue for future research using -the RSCS.
 For this high wood load run, the channel switches back and forth across this morphologic complexity threshold, with about five periods of major aggradation lasting for about two or three decades, and four shorter periods of degradation. In contrast, the low wood load BC 3 run stores much less sediment for most of the run (Figure 13e). Nevertheless, there are still two brief periods of aggradation, separated by a long period of degradation. The implication of these two contrasting runs is that the same morphodynamic rules can yield very different channel types and morphologic histories when the stochastic forcing is used to drive the RSCS. The implication is that the natural range of channel types may be very wide indeed, even when the governing conditions are held constant [cf. Trainor and Church, 2003].
 The effect of episodic sediment storage and release by jams also has a clear effect on the sediment routing through the system. The simulated changes in sediment storage over a year were used to estimate the sediment output from the RSCS for both the high and low wood load BC 3 runs (Figures 13c and 13f). Both runs show large departures from the mean bed material transport rate of 300 m3/yr imposed at the upstream end of the RSCS reach; they also both show large negative departures associated with jam formation and sediment trapping.
 As a point of comparison, the reconstructed relative storage volumes and sediment transport rates for randomly chosen BC 1 and BC 2 runs are shown in Figures 13b, 13c, 13e, and 13f. The BC 1 simulations, which do not allow wood to move or to form jams, trap about the same volume of sediment as the BC 3 simulations do, on average (refer to Figure 4); however, the year-to-year variations in the BC 1 runs shown inFigures 13b and 13e are relatively small, and the channel does not cross the relative storage threshold of 0.33 at any time during either run. The BC 2 runs (which permit LW movement but not jam formation) trap less sediment and also remain below the threshold in the two examples plotted. As a result, those simulated channels are presumably less morphologically diverse and, more importantly, less like the kinds of streams we find in the field. Furthermore, while the storage and release of sediment by individual LW pieces does generate some variation in the sediment output (Figures 13c and 13f), it is much lower than for BC 3, and much lower than one would expect, based on field studies of LW jams [cf. Hogan et al., 1998]. In short, the morphodynamics predicted by BC 1 and BC 2 are quite different from those for BC 3, and the formation and decay of LW jams in the RSCS dominates the temporal changes in sediment storage/channel morphology, thereby generating significant year-to-year variation in the bed material sediment transport rate.
 The placement of engineered in-stream LW, wherein LW pieces are effectively prevented from moving, is conceptually similar to BC 1: this implies that approaches to restoration in which LW is anchored in place cannot be expected to reproduce the dynamics that create and maintain diverse physical habitats, since it is the cyclic accumulation and release of sediment behind jams that imparts the variability in the sediment transport field and the channel morphology that is responsible for creating diverse physical habitats. Furthermore, the placement of un-achored LW in a stream is likely to retain on average about as much sediment as anchored LW, but has the added effect of creating substantial temporal and spatial complexity in the channel morphology. As a restoration strategy the placement of un-anchored LW appears to be preferable, when the restoration project is not constrained by private land-owners and/or public infrastructure adjacent to the stream channel.
 One important limitation of the RSCS has not been discussed: all of the runs assume that the same volume of bed material enters the reach each year, and that the channel geometry does not change. In reality, year-to-year variations in the flow regime have an impact on the volume of bed material supplied to the reach, the mobility of the in-stream wood, the stability of the jams, and the channel morphology. In intermediate channels, changes in channel morphology can be episodic and are often associated with disturbances such as extreme floods or forest fires [e.g.,Eaton et al., 2010b]. An important avenue for future research is to investigate the degree to which the flow regime and/or the riparian disturbance regime influence the interaction between LW and the bed material flux.
 By using a Monte Carlo modeling approach, we have come to the following conclusions. Based on the results of the three base case simulations (BC 1, BC 2 and BC 3), it is clear that the ability of LW pieces to interact and form jams is an essential part of any model relating wood load to stream channel morphodynamics. While both the steady state wood load and mean sediment storage volume for the three base cases are similar (Figures 5a and 5b), the year-to-year changes in sediment storage become much greater once jams are allowed to form in BC 3 (Figure 5c), resulting in larger temporal variations in sediment storage and sediment output from the reach (Figure 13); this presumably results in greater spatial and temporal diversity of the channel morphology and physical habitat. The significance of LW jams with respect to sediment storage depends only slightly on the density of the riparian forest (Figure 9), and on the definition of potentially jam-forming key pieces used in the RSCS (Figure 12).
 The mean wood load of 178 m3/ha and mean jam spacing of 3.8 Wch predicted by the BC 3 model are very similar to the mean wood load and jam spacing observed in streams that are similar to our prototype stream, Fishtrap Creek. Furthermore, the range of predicted wood loads corresponds to range of wood loads observed in the field. This implies that the rules which govern the RSCS are reasonable, despite the fact that most of them have not been calibrated to any particular field case. These results also demonstrate that there is a wide range of wood loads (72–327 m3/ha) that can be expected even under steady state conditions.
 The volume of sediment trapped by the in-stream wood (7.9 m3/m3for BC 3) is comparable to the upper range values from laboratory experiments and field studies. Since the RSCS assumes that all in-stream wood is submerged and functioning, we expect this positive bias. If we consider the size of individual jams in the RSCS, they compare well to the jams observed in the prototype stream: the largest jams in Fishtrap Creek store about 176 m3, which corresponds to the largest jams produced by the RSCS, regardless of how long the jams persist (see Figure 7). Therefore, our empirical functions estimating the sediment stored behind each LW piece are reasonable for streams similar to the prototype, provided we realize that the model considers only functional LW.
 The behavior of the LW jams in the model is also reasonable: the timescale for sediment accumulation, jam building and jam decay correspond well to existing, field-based conceptual models of jam behavior [Hogan et al., 1998]. Furthermore, the RSCS exhibits temporal and spatial variations in sediment storage (and by extension, channel morphology) that closely match the typical pattern described for LW dominated streams. This suggests that the RSCS produces temporal variations that are similar to those documented in the field.
 Since the RSCS behaves reasonably for steady governing conditions, it can be used in future to investigate the likely effects of changes such as forest fire and climate change. Furthermore, the model can be adapted for use in restoration site selection and prioritization. Trainor and Church  argue that restoration efforts should be focused on channels that are highly dissimilar from the population of undisturbed channels, and our Monte Carlo modeling approach using the RSCS could be used to determine if a potential restoration site falls out of the natural range for a given set of governing parameters.
 This work was supported by a Discovery Grant (held by B. Eaton) and by a Post Graduate Scholarship (held by S. Davidson) from the Natural Science and Engineering Research Council of Canada. The paper was much improved by reviews from T. Lisle and two anonymous reviewers.