Interaction of waves and currents with kelp forests ......Interaction of waves and currents with...

13
Interaction of waves and currents with kelp forests (Macrocystis pyrifera): Insights from a dynamically scaled laboratory model Johanna H. Rosman, a,* Mark W. Denny, b Robert B. Zeller, c Stephen G. Monismith, c and Jeffrey R. Koseff c a Institute of Marine Sciences, University of North Carolina at Chapel Hill, Morehead City, North Carolina b Hopkins Marine Station, Stanford University, Pacific Grove, California c Environmental Fluid Mechanics Laboratory, Stanford University, Stanford, California Abstract The interaction of surface waves and currents with kelp forests was examined under controlled conditions using a dynamically matched 1/25-scale physical model in a laboratory flume. In experiments with kelp mimics, waves increased the time-averaged drag by a factor of 2 and altered the shape of current profiles. Relative motion between model kelp and water under waves increased wake generation of turbulence, resulting in turbulent kinetic energies 2–5 times larger, and eddy viscosities 20–50% larger, than for experiments without waves. Because mixing lengths were reduced to wake-scales in the model kelp forest, eddy viscosities were 25–50% smaller than when kelp was absent. In the model kelp-forest surface canopy where solid obstacles were most densely spaced, wave orbital velocities were reduced by , 10% from linear wave theory predictions. This decrease in wave orbital velocities is thought to result primarily from inertial forces exerted on water by model kelp. Stokes drift was reduced by , 20% as a result of the change in wave orbital velocities. Although hydrodynamics within kelp forests are more complex than in the laboratory experiments and a wide range of flow conditions can occur, laboratory results suggest that (1) kelp forest drag is increased by waves; (2) wave properties can be altered by drag and inertial forces; and (3) wake production of turbulence caused by waves may be the main source of turbulence in dense kelp stands. Interactions between kelp and waves must therefore be taken into account when developing models for drag and mixing in these systems. Along temperate coasts, the giant kelp Macrocystis pyrifera (Linneaus) Agardh forms dense forests that are highly productive and rich in species diversity. Mature M. pyrifera grow on rocky substrates in the shallow subtidal and extend from the bottom to the surface, often forming a dense surface canopy. As in many marine ecosystems, water motion and biological processes in kelp forests are linked. Currents deliver resources such as nutrients and organic matter to kelp forests and affect the paths of larvae and spores, contributing to both growth rates and recruitment patterns of new individuals (Graham 2003; Fram et al. 2008). At smaller scales, flow past blade surfaces increases nutrients transfer and dissolved gas exchange (Stevens and Hurd 1997; Hepburn et al. 2007). Turbulent eddies bring larvae and spores closer to substrates on which they can settle (Gaylord et al. 2002). Moving water also exerts forces on kelp forest organisms, leading to dislodgement, breakage, and mortality (Seymour et al. 1989; Utter and Denny 1996; Graham 1997). Water motion thus plays an important role in structuring kelp forests and their evolution at seasonal and inter-annual time scales (Dayton et al. 1992). In addition to being strongly affected by water motion, kelp forests modify water motion, altering resource availability, larval dispersal patterns, and hydrodynamic forces for organisms living within their boundaries. Within M. pyrifera forests, current speeds may be as small as 20% of those in surrounding kelp-free areas (Jackson 1998; Gaylord et al. 2007) and internal waves are damped (Jackson 1984; Rosman et al. 2007). The effects of kelp forests on currents and internal waves are typically modeled by including a linear or quadratic drag term in the depth-averaged momentum budget to represent kelp drag (Jackson and Winant 1983; Jackson 1984; Rosman et al. 2010). Previous studies have estimated kelp-forest drag coefficients by approximating the kelp forest as an array of cylinders (Jackson and Winant 1983). In recent work that used a small-scale model of a kelp forest in a laboratory flume, we illustrated that a dense surface canopy could increase kelp-forest drag coefficients by a factor of 1.5–3 (Rosman et al. 2010). To date, studies of drag on currents within kelp forests have ignored the effects of surface waves; however, M. pyrifera grow in shallow coastal waters and are typically exposed to ocean swell. Field observations and modeling efforts indicate that wave energy loss during propagation across kelp forests is typically small—only a few percent of the incident wave energy flux (Elwany et al. 1995; Gaylord et al. 2003). Therefore, wave heights within kelp forests are similar to those offshore. It is well-known that the interaction of waves and currents with rough bottoms increases drag on currents because drag is a nonlinear function of water velocity (Grant and Madsen 1979). In a similar way, relative motion between water and kelp fronds due to waves may increase the drag on currents within kelp forests, although this has never been investigated. Relative motion between water and kelp fronds due to waves varies with time and therefore, in addition to * Corresponding author: [email protected] Limnol. Oceanogr., 58(3), 2013, 790–802 E 2013, by the Association for the Sciences of Limnology and Oceanography, Inc. doi:10.4319/lo.2013.58.3.0790 790

Transcript of Interaction of waves and currents with kelp forests ......Interaction of waves and currents with...

Page 1: Interaction of waves and currents with kelp forests ......Interaction of waves and currents with kelp forests (Macrocystis pyrifera): Insights from a dynamically scaled laboratory

Interaction of waves and currents with kelp forests (Macrocystis pyrifera): Insights from

a dynamically scaled laboratory model

Johanna H. Rosman,a,* Mark W. Denny,b Robert B. Zeller,c Stephen G. Monismith,c

and Jeffrey R. Koseff c

a Institute of Marine Sciences, University of North Carolina at Chapel Hill, Morehead City, North CarolinabHopkins Marine Station, Stanford University, Pacific Grove, Californiac Environmental Fluid Mechanics Laboratory, Stanford University, Stanford, California

Abstract

The interaction of surface waves and currents with kelp forests was examined under controlled conditionsusing a dynamically matched 1/25-scale physical model in a laboratory flume. In experiments with kelp mimics,waves increased the time-averaged drag by a factor of 2 and altered the shape of current profiles. Relative motionbetween model kelp and water under waves increased wake generation of turbulence, resulting in turbulent kineticenergies 2–5 times larger, and eddy viscosities 20–50% larger, than for experiments without waves. Because mixinglengths were reduced to wake-scales in the model kelp forest, eddy viscosities were 25–50% smaller than when kelpwas absent. In the model kelp-forest surface canopy where solid obstacles were most densely spaced, wave orbitalvelocities were reduced by , 10% from linear wave theory predictions. This decrease in wave orbital velocities isthought to result primarily from inertial forces exerted on water by model kelp. Stokes drift was reduced by ,20% as a result of the change in wave orbital velocities. Although hydrodynamics within kelp forests are morecomplex than in the laboratory experiments and a wide range of flow conditions can occur, laboratory resultssuggest that (1) kelp forest drag is increased by waves; (2) wave properties can be altered by drag and inertialforces; and (3) wake production of turbulence caused by waves may be the main source of turbulence in densekelp stands. Interactions between kelp and waves must therefore be taken into account when developing modelsfor drag and mixing in these systems.

Along temperate coasts, the giant kelp Macrocystispyrifera (Linneaus) Agardh forms dense forests that arehighly productive and rich in species diversity. Mature M.pyrifera grow on rocky substrates in the shallow subtidaland extend from the bottom to the surface, often forminga dense surface canopy. As in many marine ecosystems,water motion and biological processes in kelp forests arelinked. Currents deliver resources such as nutrients andorganic matter to kelp forests and affect the paths of larvaeand spores, contributing to both growth rates andrecruitment patterns of new individuals (Graham 2003;Fram et al. 2008). At smaller scales, flow past bladesurfaces increases nutrients transfer and dissolved gasexchange (Stevens and Hurd 1997; Hepburn et al. 2007).Turbulent eddies bring larvae and spores closer tosubstrates on which they can settle (Gaylord et al. 2002).Moving water also exerts forces on kelp forest organisms,leading to dislodgement, breakage, and mortality (Seymouret al. 1989; Utter and Denny 1996; Graham 1997). Watermotion thus plays an important role in structuring kelpforests and their evolution at seasonal and inter-annualtime scales (Dayton et al. 1992).

In addition to being strongly affected by water motion,kelp forests modify water motion, altering resourceavailability, larval dispersal patterns, and hydrodynamicforces for organisms living within their boundaries. WithinM. pyrifera forests, current speeds may be as small as 20%of those in surrounding kelp-free areas (Jackson 1998;

Gaylord et al. 2007) and internal waves are damped(Jackson 1984; Rosman et al. 2007). The effects of kelpforests on currents and internal waves are typicallymodeled by including a linear or quadratic drag term inthe depth-averaged momentum budget to represent kelpdrag (Jackson and Winant 1983; Jackson 1984; Rosman etal. 2010). Previous studies have estimated kelp-forest dragcoefficients by approximating the kelp forest as an array ofcylinders (Jackson and Winant 1983). In recent work thatused a small-scale model of a kelp forest in a laboratoryflume, we illustrated that a dense surface canopy couldincrease kelp-forest drag coefficients by a factor of 1.5–3(Rosman et al. 2010).

To date, studies of drag on currents within kelp forestshave ignored the effects of surface waves; however, M.pyrifera grow in shallow coastal waters and are typicallyexposed to ocean swell. Field observations and modelingefforts indicate that wave energy loss during propagationacross kelp forests is typically small—only a few percent ofthe incident wave energy flux (Elwany et al. 1995; Gaylordet al. 2003). Therefore, wave heights within kelp forests aresimilar to those offshore. It is well-known that theinteraction of waves and currents with rough bottomsincreases drag on currents because drag is a nonlinearfunction of water velocity (Grant and Madsen 1979). In asimilar way, relative motion between water and kelp frondsdue to waves may increase the drag on currents within kelpforests, although this has never been investigated.

Relative motion between water and kelp fronds due towaves varies with time and therefore, in addition to* Corresponding author: [email protected]

Limnol. Oceanogr., 58(3), 2013, 790–802

E 2013, by the Association for the Sciences of Limnology and Oceanography, Inc.doi:10.4319/lo.2013.58.3.0790

790

Page 2: Interaction of waves and currents with kelp forests ......Interaction of waves and currents with kelp forests (Macrocystis pyrifera): Insights from a dynamically scaled laboratory

modifying time-averaged drag, it causes periodic drag andinertial forces (Utter and Denny 1996). These periodic forcesmay alter the properties of waves as they propagate throughkelp forests. Previous studies that have measured waves withbottom-mounted pressure sensors have not observed mea-surable changes in wave periods or amplitudes within kelpforests (Elwany et al. 1995; Rosman et al. 2007). However,more detailed measurements of wave properties, such aswave orbital velocities and propagation speeds, within kelpforests are currently lacking. As water parcels move underthe influence of waves, they drift slowly in the direction ofwave propagation because their trajectories are not closedellipses; this is a phenomena known as Stokes drift. Previousfield studies suggest that Stokes drift could contributesignificantly to cross-shore transport and hence exchangebetween kelp forests and offshore waters (Gaylord et al.2003; Rosman et al. 2007). However, these studies have usedexpressions from linear wave theory and have neglected anychange to wave orbital motion caused by kelp.

The drag forces described above result in removal ofenergy from currents and waves, and this energy isconverted to turbulent energy in plant wakes (Nepf 1999).Therefore turbulence mixing within kelp forests could bequite different from that in the surrounding coastal ocean.There are no previous field studies of turbulence withinkelp forests because of challenges associated with turbu-lence measurement in the coastal ocean, and within kelpforests in particular. Previous results from our small-scalelaboratory model without waves suggest that eddy viscos-ities within kelp forests are , 25–50% smaller than for thesame current in the absence of kelp due to the combinedeffects of increased turbulent energy and reduced turbulentlength scales (Rosman et al. 2010). However, within densekelp forests, currents are strongly damped, and thereforeturbulence generation by this mechanism could be limited.Waves propagate through kelp forests with little dampingand periodic wake production of turbulence due to relativemotion between kelp and water under waves could be asignificant mechanism for vertical mixing.

Our understanding of currents, wave properties, andmixing within kelp forests would benefit from measure-ments of wave orbital motion and turbulence in thesesystems. Detailed investigations of wave and turbulenceproperties are difficult in the field because water velocitymeasurements can be noisy and this is exacerbated byinterruption of acoustic beams by kelp fronds. Because kelpforests are heterogeneous, measurements at a single pointare strongly influenced by the positions of nearbyindividuals. To deal with this issue, the equations ofmotion are typically averaged in space (Raupach and Shaw1982; Rosman et al. 2010); however, it is not feasible tomeasure spatially averaged quantities in real kelp forests.Conditions in the coastal ocean are highly variable in bothspace and time, and it may be difficult to isolate the effectsof kelp on flow properties. These issues can be overcome inlaboratory settings, where accurate measurements can bemade at high spatial and temporal resolution and flowconditions can be controlled and are repeatable.

Here, we describe the results of laboratory experimentsdesigned to investigate interactions of currents and surface

waves with kelp forests using a dynamically matched 1/25-scale model in a laboratory flume. This paper extends workdescribed in Rosman et al. (2010) that used the samelaboratory model to investigate currents and turbulencewithin kelp forests. This study addressed the followingquestions: (1) How does the interaction between waves andkelp modify drag on currents?; (2) How are the propertiesof waves modified by their interaction with kelp?; and (3)How do waves affect turbulence and vertical mixing withinkelp forests? We present laboratory results and introduceanalytical models that can be used to understand labora-tory observations and predict patterns in real kelp forests.

Methods

Scaling the kelp forest—The experiments used kelpmimics in a laboratory flume, as described in Rosman etal. (2010). Laboratory experiments were designed using theprincipal of dynamic similarity (Kundu 1990); both theratios of different forces acting on kelp and the relativesizes of terms in the momentum budget for the fluid werematched with values typical of real kelp forests. Rosman etal. (2010) described the dynamical scaling of the kelp forestfor currents alone. Here we describe the dynamical scalingof the kelp forest in a more general way that includes theeffects of waves.

First we discretize a kelp frond into segments andconsider the forces acting on each segment. The effectivebuoyancy force (FB,i) acting on the ith kelp segment is equalto its buoyancy minus its weight. In the absence of waves, theonly other significant external force is the drag force due towater moving past the kelp (FD,i; Rosman et al. 2010). Whensurface waves are present, there is an additional oscillatingdrag force due to relative motion between water and kelpunder waves. In the presence of waves, there is also aninertial force (FI,i) that is the sum of a component due to theunsteady pressure gradient in the accelerating fluid (FI1,i)and a component that results from an additional pressuregradient generated by fluid accelerating around the kelpstructure (FI2,i; Dean and Dalrymple 1991). These compo-nents are also termed the virtual buoyancy and added massforces, respectively (Denny et al. 1997). Additionally, tensionforces (Ti+1,Ti) act at either end of each kelp segment (i). M.pyrifera stipes are very flexible; therefore, we assume thatbending moments and shear stresses between adjacent kelpsegments are negligible.

The position, velocity, and acceleration of kelp segment iare Xi, X

.i, and Xi, respectively. The acceleration of the ith

kelp segment (Xi) with mass mi can be expressed in terms ofthe forces acting on it as (Denny et al. 1997)

mi€XXi~FB,izFD,izFI1,izFI2,izTiz1{Ti ð1Þ

Vector quantities are bold and scalar quantities areitalicized. Expressions for the external forces acting oneach kelp segment are

Waves and currents in a kelp forest 791

Page 3: Interaction of waves and currents with kelp forests ......Interaction of waves and currents with kelp forests (Macrocystis pyrifera): Insights from a dynamically scaled laboratory

FB,i~(r{rk,i) Vk,ig

FD,i~1

2rcdAk,i ui{ _XXi

� �ui{ _XXi

�� ��~

1

2rcdAk,if

2v2 �uui

fvz

~uui

fv{

_XXi

fv

!�uui

fvz

~uui

fv{

_XXi

fv

��������

FI1,i~rVk,iLLt

uið Þ~rVk,ifv2 LvLt

~uui

fv

� �

FI2,i~rVk,ikm

LLt

uið Þ{€XXi

� �

~rVk,ikmfv2 LvLt

~uui

fv

� �{

€XXi

fv2

" #

ð2Þ

Total water velocity is u, u is current speed, and u is waveorbital velocity. The water density and kelp density are rand rk, respectively; kelp volume is Vk; g is acceleration dueto gravity; cd is drag coefficient; Ak is kelp frontal area; f iswave amplitude; and v is wave frequency. The added masscoefficient km represents the additional local pressuregradient established as water adjacent to the kelp acceler-ates around it.

If the expressions in Eq. 2 are substituted into Eq. 1 andall terms are divided by rVk,ifv2, the result is

rk,i

rzkm

� � €XXi

fv2~ 1{

rk,i

r

� �g

fv2

z1

2cd

Ak,if

Vk,i

�uui

fvz

~uui

fv{

_XXi

fv

!�uui

fvz

~uui

fv{

_XXi

fv

��������

z 1zkmð Þ LvLt

~uui

fv

� �z

Tiz1{Ti

rVk,ifv2

ð3Þ

To ensure that the relative sizes of terms in Eq. 3 were

matched between the field situation and the laboratorymodel, we matched ratios of all length scales, the ratio ofkelp density to water density (rk,i/r), the acceleration offluid under waves relative to gravity (fv2/g), and theFroude number (u2/gH). The distributions of blade areaand density along the length of model kelp fronds werebased on measurements of real kelp fronds obtained from8 m to 10 m of water in Monterey Bay. A comparisonbetween properties of real and model kelp fronds isprovided in Rosman et al. (2010).

Parameters for the laboratory experiments and theirequivalent field values are given in Table 1. These valueswere selected to be within the range of conditions typicallyfound in kelp forests (Jackson 1998; Gaylord et al. 2007;Rosman et al. 2007). In the laboratory model, all lengthscales were 25 times smaller, time scales were 5 times

shorter, and velocities were 5 times smaller than theirequivalent field values. Although realistic for kelp forestperipheries, U 5 0.2 m s21 is quite a large for currentswithin kelp forests. We were limited to currents . 0.04 m s21

in laboratory experiments because this was the minimumvalue for which the water height over the weir at thedownstream end of the flume was sufficient to avoidproblems with wave reflections.

Although we were able to match other dimensionlessnumbers reasonably well, Reynolds numbers for thelaboratory experiments were a factor of , 20–100 smallerthan field values. Reynolds numbers based on currentspeed and cylinder diameter were , 500 in the laboratorymodel and could be as large as 5 3 104 in real kelp forests.Although these values suggest that wakes may not havebeen fully turbulent in the laboratory experiments,Reynolds numbers are within the range where wakes areirregular and unstable, and drag coefficients for flowaround cylinders are relatively constant (Rosman et al.2010). Reynolds numbers can seldom be replicated insmall-scale laboratory models. The implications of themismatch in Reynolds number are considered further in theDiscussion.

Laboratory model—Experiments were conducted in a12 m long, 1.2 m wide recirculating flume (Fig. 1a). Apump at the downstream end of the flume drove a returnflow through a pipe beneath the flume that entered themain flume at the upstream end, forcing a steady currentalong the test section. Waves were generated at theupstream end of the flume using a plunger-type wavemakerbased on the design of Yao (1992). At the downstream end,waves passed over a shallow weir designed to limitreflections. The nominal water depth, depth-averagedcurrent, wave amplitude, and wave period were H 50.4 m, U 5 0.04 m s21, f 5 0.02 m, and T 5 2 s,respectively. About 0.5% of the wave energy was reflectedfrom the downstream end of the flume, resulting in a 3 31023 m amplitude standing wave superimposed on theprogressive wave. A typical wave form generated in theflume is shown in Fig. 2.

Model kelp was constructed from plastics havingdensities similar to kelp stipes, blades, and pneumatocysts(Fig. 1b). Stipes were constructed from 4 mm diameterplastic tubing, pneumatocysts were 10 mm fishing floats,and blades were cut from 0.1 mm vinyl sheeting. Each kelp

Table 1. Ranges of parameters observed in real kelp forestsand values selected for the laboratory model. Numbers inparentheses are field values that correspond to the laboratoryvalues used.

Parameter Field Laboratory

Water depth h (m) 7–15(10) 0.4Frond length L (m) 5–20(15) 0.6Stipe bundle diameter d (m) 0.2–0.3(0.3) 0.012Kelp spacing S (m) 3–10(6) 0.15Current U (m s21) 0–0.3(0.2) 0.04Wave height H (m) 0–1.5(1) 0.04Wave period T (s) 5–15(10) 2

792 Rosman et al.

Page 4: Interaction of waves and currents with kelp forests ......Interaction of waves and currents with kelp forests (Macrocystis pyrifera): Insights from a dynamically scaled laboratory

mimic had three fronds that were intertwined; each frondrepresented the blade area and volume of 10 real fronds.For all experiments described in this paper, model kelpwere 60 cm in length, 50% longer than the water depth, andspread out in the upper 5–10 cm of the water columnforming a surface canopy. The model kelp were arranged inlines across the flume but were staggered in the flowdirection. The model kelp spacing was 0.15 m, correspond-ing to one individual per 14 m2 in the field (Table 1). This

corresponds to the ‘dense’ configuration experiments ofRosman et al. (2010). The along-channel length of themodel kelp forest was 2 m, corresponding to a 50 m longreal kelp forest.

Measurements and data processing—Measurements weremade inside the model kelp forest, 1.7 m from its upstreamend. Velocity profiles were measured using an acousticDoppler velocimeter (ADV, Nortek Vectrino+) mountedon a traverse capable of sliding vertically and laterallyacross the flume. The elevation of the water surface wasmeasured with a capacitance wave gauge (RBR WG-50)aligned at the same along-flume position as the ADVsample volumes. The velocity measurements, surface-levelmeasurements, and the wavemaker were synchronizedusing LabVIEW. Velocities were measured at 12 points inthe vertical and profiles were repeated at 6 positions acrossthe flume. Velocities and surface elevations were measuredat 25 Hz for 10 min at each ADV position.

Water velocities u 5 (u, v, w) were decomposed intocurrents u 5 (u, v , w ), wave orbital velocities u 5 (u, v, w),and turbulent fluctuations u9 5 (u9, v9, w9), such thatu 5 (u + u + u9) . For each measurement position, u wascalculated as the time-averaged velocity over the 10 minrecord. The current was then subtracted from the velocity timeseries and the record was divided into 2 s segments, the lengthof a wave cycle. The wave orbital velocity, u, was computed byensemble-averaging the 300 2 s segments, yielding 50 valueswithin a wave cycle. The time series of turbulent fluctuationswas then computed from u9 5 u 2 u 2 u.

Within vegetation canopies and other arrays of solidobstacles, flow conditions depend strongly on the posi-tions of nearby obstacles and vary significantly in space. Aformal spatial averaging approach is typically used toanalyze measurements from this type of system (Raupachand Shaw 1982). We denote spatially averaged quantitiesby angular brackets, <u>, and deviations from spatiallyaveraged quantities by double primes, u0 5 u 2 <u>. In themodel kelp forest, spatially averaged currents, wave orbitalvelocities, and turbulence statistics were computed at eachheight above bottom as the mean of their values at the six

Fig. 1. Experimental setup: (a) schematic diagram of flume, and (b) photograph of model kelp forest.

Fig. 2. Phase-averaged (a) surface elevations, (b) stream-wisevelocities, and (c) vertical velocities without kelp in the flume. Thetime average has been removed in each case and quantities havebeen normalized by the wave amplitude f, wave frequency v, andwave period T. Solid lines are measured quantities and colorsindicate velocities at different heights above bottom. Dashed linesare linear wave theory predictions for a progressive wave.

Waves and currents in a kelp forest 793

Page 5: Interaction of waves and currents with kelp forests ......Interaction of waves and currents with kelp forests (Macrocystis pyrifera): Insights from a dynamically scaled laboratory

across-flume positions. Similarly, spatial standard devia-tions were the standard deviation of the six values acrossthe flume; these are used as an indicator of the spatialvariability of the flow and to estimate uncertainties inspatially averaged currents and turbulence statistics.

Analysis of the effects of model kelp and waves oncurrents—The effects of the model kelp forest on currentswere investigated using a simplified form of the time- andspace-averaged stream-wise (x) momentum budget:

{1

r

LS�ppTLx

pressure gradient term

zLS{u0w0T

Lzshear stress gradient term

{ �ff Dxdrag term

~0 ð4Þ

This form of the momentum equation assumes steady, fullydeveloped flow; that is, the unsteady and advectiveacceleration terms are zero. Terms in Eq. 4 representspatially averaged forces per unit fluid mass acting onwater parcels. The first and second terms are the along-channel time-and spatially averaged pressure gradient forceand the vertical gradient in Reynolds stresses. Dispersivestresses, <u0w 0>, arising due to spatial correlationsbetween u and w were noisy and small compared withother terms and have therefore been neglected (Rosman etal. 2010). We also did not include wave stresses in thisanalysis because slight wave reflections from the down-

stream end of the flume resulted in non-zero S~uu~wwT and thisprevented us from measuring more subtle effects of kelp onwave stresses. The time-averaged drag force on the currentdue to the kelp is f Dx. The net drag is typically dominatedby form drag, the integrated result of pressure gradientsestablished in the fluid as it moves past solid obstacles inthe flow.

The instantaneous drag force fDx was modeled using aquadratic drag law (Nepf 1999),

fDx~1

2cdaurel u2

relzw2rel

� �1=2 ð5Þ

where cd 5 O(1) is the sectional drag coefficient, a is thefrontal area of solid obstacles at height z per unit fluidvolume, and urel 5 u + u 2 X

.and wrel 5 w 2 Z

.are the

horizontal and vertical components of the relative velocitybetween water and kelp. For experiments without waves,the time-averaged drag term is therefore given by

�ffDx~1

2cda �uu2 ð6Þ

By analogy, for experiments with waves, an effective dragparameter (cda)eff was defined as

�ffDx~1

2cdað Þeff�uu

2 ð7Þ

Any increase in drag on currents due to kelp–water relativemotion associated with waves appears as an elevated valueof (cda)eff.

The time-averaged drag force acting on the water due tokelp was calculated in the following way. The drag at z/H 50.1 was calculated from the quadratic drag law (Eq. 5)assuming that, at this height above bottom, kelp fronds

behaved like rigid cylinders with cd 5 1 and a 5 d/S2, where dis stipe bundle diameter and S is kelp spacing. Thisapproximation should be reasonable because there were noblades or floats near or below this height and the motion ofmodel kelp is small close to the base. The along-channeltime-averaged pressure gradient was computed at z/H 5 0.1using Eq. 4 and assumed to be uniform over the water depth,a reasonable assumption for fully developed flow (Rosmanet al. 2010). The vertical profile of f Dx was then computedfrom the momentum budget (Eq. 4) using cubic spline fits tomeasured currents and Reynolds stresses. The effective dragparameter (cda)eff was then calculated from Eq. 7.

For illustration purposes, values of (cda)eff from thetime-averaged momentum budget with waves were com-pared with values calculated from Eqs. 5 and 7 using twosimple descriptions for relative motion between model kelpand water. These models were a rigid (stationary) canopy inwhich relative motion was equal to water motion and apartially flexible canopy in which the model kelp was fixedat the base, moved exactly with wave motion at the surface,and varied linearly between the two. Profiles of cda derivedfrom the unidirectional momentum budget were used inEq. 5 for this analysis.

In the depth-averaged momentum equations, which havepreviously been used to describe flow patterns in andaround kelp forests (Jackson and Winant 1983; Rosman etal. 2010), the depth-averaged drag on currents is modeledusing a quadratic drag law with a bulk drag coefficient(CD), defined as

FD~CDU2

H~

1

H

ðH0

�ff Dx dz ð8Þ

CD values were calculated from laboratory measurementsby extrapolating f Dx to the bottom and water surface andintegrating numerically over the water depth.

Analysis of the effects of model kelp on wave orbitalmotion and Stokes drift—Because the wave form in theflume was not exactly sinusoidal, an equivalent waveorbital velocity amplitude was defined as Uw~

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2var ~uuð Þ

p,

the relationship between amplitude and variance for asinusoidal signal. This expression was used to comparewave orbital velocity profiles with and without model kelpto predictions from linear wave theory.

Measured wave orbital velocities were used to estimateStokes drift in the laboratory experiments. Stokes drift canbe computed from wave orbital velocities (u, w) andexcursions (Dx, and Dz) using the expression (Dean andDalrymple 1991)

�uuS~~uuzL~uu

LxDxz

L~uu

LzDz ð9Þ

We assumed that wave orbital velocities were well-approxi-mated by sinusoids with wave frequency v, wavenumber k,and amplitudes Uw and Ww. Substituting u 5 Uw cos(kx 2vt),Dx 5 2Uw/v sin(kx 2 vt), Dz 5 Ww/v cos(kx 2 vt), and(hu/hz) 5 (dUw/dz) cos(kx 2 vt) and time-averaging, yields

794 Rosman et al.

Page 6: Interaction of waves and currents with kelp forests ......Interaction of waves and currents with kelp forests (Macrocystis pyrifera): Insights from a dynamically scaled laboratory

�uuS~1

2vkUw

2zLUw

LzWw

� �ð10Þ

This expression was evaluated using Uw and Ww profiles fromexperiments with and without model kelp.

Analysis of the effects of model kelp and waves onturbulence and mixing—Turbulent length scales are usuallycalculated either directly from spatial measurement arrays orfrom de-correlation time-scales by assuming that turbulenteddies are ‘frozen’ and advected past fixed sensors by steadycurrents. For our laboratory experiments with waves,turbulent length scales could not be computed directly fromintegral time scales because turbulent eddies were advectedpast the sensor by unsteady wave orbital motion. To overcomethis issue, each time series was first converted to an effectivespatial series. The effective initial position of a water parcelwhose velocity was measured at time t was calculated from

xeff~{

ðt0

�uuz~uuð Þdt ð11Þ

Measurement times were replaced by xeff to generate a spatialseries of turbulent velocity fluctuations. The irregularly spacedseries was then interpolated onto a set of regularly spaced xeff

points. The integral length scale was then calculated from theautocorrelation of vertical turbulent fluctuations as

lxz~2

w’2

ðj50

0

w0xw0xzj dj ð12Þ

where Rxz(j) 5 w’xw’xzj is the autocorrelation function valueat lag j, and j50 corresponds to the minimum time lag at whichthe autocorrelation function drops below 0.5.

Eddy viscosities were calculated using the definition

nt~{Su’w’T

LS�uuT=Lzð13Þ

Cubic splined fits to spatially averaged Reynolds stressesand currents were used for this calculation. This methodwas problematic where velocity gradients were small andwas found to be very sensitive to details of the cubic splinefit. Therefore, eddy viscosities were also calculated from thescaling estimate (Pope 2000)

nt!k1=2‘m ð14Þ

where k is turbulent kinetic energy and ,m is mixing length,which we approximated by lxz.

Results

Effects of model kelp and waves on currents—For experi-ments with model kelp, currents were reduced in the upperwater column due to the large drag of the surface canopy(Fig. 3c). The shape of the current profile was altered by the

interaction of kelp mimics with waves. The bottomboundary layer was thinner and the maximum currentoccurred lower in the water column, suggesting thatinteraction between model kelp and waves affected thevertical distribution of drag and/or the vertical transport ofmomentum. Increased mean shear and turbulence in thebottom boundary layer due to interaction of waves with kelpresulted in larger positive Reynolds stresses for experimentswith waves that for those without (Fig. 3d). In the upperwater column, Reynolds stresses were negative over a widerrange of z for experiments with waves due to the increasedvertical extent of shear due to the surface canopy (Fig. 3d).

The momentum budget (Eq. 4) was dominated by thealong-channel pressure gradient, which drives the current,and kelp drag, which resists the current (Fig. 4). Whenwaves were present, the time-averaged drag was abouttwice as large as when there were no waves, due toincreased relative motion between kelp and water (Eq.7).The stress gradient term that represents the verticalmixing of momentum was an order of magnitude smallerthan the other terms, indicating that it has little influenceon the shape of current profiles.

The drag parameter (cda)eff, a measure of time-averageddrag on currents, was larger for experiments with wavesthan for experiments with a current alone throughout mostof the water column (Fig. 5). Because drag is a quadraticfunction of the relative velocity between kelp and water,any relative motion associated with waves increases (cda)eff

(Eq. 7). The (cda)eff values for experiments with andwithout waves meet in the upper water column, presumably

Fig. 3. Profiles of time-averaged current (a,c) and Reynoldsstress (b,d) for experiments without and with model kelp.Measurements with and without waves are shown on the samepanel. Symbols are spatial averages of measured quantities, solidlines are cubic spline fits, and dotted lines are spatial means plusand minus one spatial standard deviation. Measurements havebeen normalized by the depth-averaged velocity U.

Waves and currents in a kelp forest 795

Page 7: Interaction of waves and currents with kelp forests ......Interaction of waves and currents with kelp forests (Macrocystis pyrifera): Insights from a dynamically scaled laboratory

because the surface canopy is able to move with waveorbital motion to a large extent. Values of (cda)eff predictedusing the no-wave cda profile and a rigid model for kelpmotion are larger than observed (cda)eff in the upper watercolumn because the kelp is able to move with the flow andthe rigid model overestimates drag. (cda)eff predicted fromthe rigid model are smaller than observed (cda)eff in thelower water column because wave motion increases thedrag force on model kelp but its buoyancy is unchanged;therefore, waves cause the model kelp to sit lower in thewater column and the cda distribution is shifted downward.The depth-integrated drag on currents was a factor of 2larger for experiments with waves; the bulk drag coeffi-cients required to parameterize depth-integrated drag (CD,Eq. 6) were 0.2 and 0.4 for experiments without and withwaves, respectively.

Effects of model kelp on wave orbital motion andStokes drift—Wave orbital velocities were also altered bymodel kelp (Fig. 6). The amplitudes of wave orbitalvelocities for experiments without kelp closely matchedpredictions from linear wave theory. With model kelp,amplitudes of horizontal wave orbital velocity componentswere smaller than linear wave theory predictions and moreuniform throughout the water depth. The difference wasgreatest (, 8%) in the upper water column where solidobstacles were most closely spaced. The vertical componentof wave orbital velocities was unchanged in the lower watercolumn, but reduced by , 5% in the upper water columnwhen kelp was present. For experiments without kelp, thecorrelation between horizontal and vertical wave orbitalvelocities S~uu~wwT was consistent with a partial standing wavedue to a slight wave reflection from the downstream end ofthe flume. We were unable to determine from themeasurements whether the model kelp caused a significantchange in the wave stress gradient that could affect thecurrent, like that described in Luhar et al. (2010).

The reduction in wave orbital velocities in the upperwater column could be due either to forces model kelp exerton moving water or vertical mixing of the momentumassociated with wave orbital velocities. Because spatially

averaged Reynolds stresses were not phase-dependent andReynolds stress gradients were small compared with otherterms of the momentum budget, it is unlikely that verticalturbulent mixing could have a significant effect on waveorbital motion. The possible effects of drag and inertialforces on wave properties and wave orbital motion areassessed in the Discussion.

Depth-integrated Stokes drift (Eq. 10) was reduced by20% by model kelp, due both to the reduction in waveorbital velocity amplitude and the smaller vertical shear(Fig. 7). Additionally, because Uw was more uniform overthe water depth, Stokes drift was more uniform over thewater depth when model kelp was present. The results inFig. 7 were for the single wave amplitude and frequencyused in the laboratory experiments (fv2/g 5 0.04, f/H 50.05), and trends may be different for other waveconditions.

Fig. 4. Terms of the time-averaged momentum budgetcalculated from measurements (a) without waves, and (b) withwaves. All momentum budget terms have been normalized byU2H21 where U is depth-averaged current and H is water depth.

Fig. 5. Effect of kelp and waves on quadratic drag parametercda that parameterizes drag on current. Shown are (cda)eff

calculated from the momentum budget without and with waves,along with predicted effective (cda)eff for a rigid array with thesame frontal area distribution as model kelp. In each case, (cda)eff

is non-dimensionalized by multiplying by the stipe bundlediameter (d).

Fig. 6. Effect of kelp on wave orbital velocities. Plotted arethe amplitudes of stream-wise and vertical components of waveorbital velocities vs. height above bottom. Symbols are spatialmeans, solid lines are spline fits, and dotted lines are spatial meansplus and minus one spatial standard deviation. Blue dashed linesare wave orbital velocity amplitudes predicted by linear wavetheory. Wave orbital velocities have been non-dimensionalized bydividing by the wave amplitude (f) and wave frequency (v).

796 Rosman et al.

Page 8: Interaction of waves and currents with kelp forests ......Interaction of waves and currents with kelp forests (Macrocystis pyrifera): Insights from a dynamically scaled laboratory

Effects of model kelp and waves on turbulence andmixing—The flow was more turbulent for experimentswith waves (Fig. 8e), suggesting that relative motionbetween kelp and water during the passage of waves leadsto greater wake production of turbulence, particularly inthe surface canopy where the biomass is greatest. Thisincreased turbulent energy, combined with larger shear inthe bottom boundary layer (Fig. 3c), resulted in largerpositive Reynolds stresses in the lower water column(Fig. 3d). The increased vertical extent of shear due tothe surface canopy in the upper part of the water column

caused larger negative Reynolds stresses over a wider rangeof z (Fig. 3d).

Reynolds stresses and turbulent kinetic energy werephase-dependent at individual measurement locations,suggesting that wakes were shed off model kelp atparticular wave phases. However, the way in which thesequantities varied with wave phase depended on across-flume measurement position, probably due to differentdownstream distances from obstacles generating wakes.When turbulent correlations were spatially averaged, therewas no observable phase-dependence over a wave cycle.

Turbulence integral length scales were almost an order ofmagnitude smaller for experiments with model kelp thanfor those without and were more uniform over the waterdepth (Fig. 8d). Integral length scales were similar to stipebundle diameters, suggesting that turbulence was generatedpredominantly in kelp wakes rather than by mean shear inboundary layers. For experiments without kelp, maximumturbulent length scales occurred in the middle of the watercolumn and had a parabolic shape, similar to expectationsfor open channel flow. In the lower water column,turbulent integral length scales were similar for experimentswith and without waves. In the upper water column, lengthscale estimates were , 30% smaller when waves werepresent, although this may be an artifact of assuming thatturbulence was advected past the sensor in only the stream-wise direction (Eq. 11).

The effects of wake-generated turbulence on verticalmixing can be assessed by considering the eddy viscosity, nt,which relates turbulent fluxes to mean gradients (Eq. 13).There was some discrepancy between eddy viscosity values

Fig. 7. Stokes drift us calculated from cubic-splined waveorbital velocity measurements with and without kelp, and theprofile predicted by linear wave theory. Because the linear wavetheory prediction is proportional to the wave amplitude (f)squared, wave frequency (v), and wavenumber (k), uS has beennormalized by dividing by f2vk.

Fig. 8. Profiles of turbulence properties for experiments with and without waves, with and without model kelp. Panels are verticalprofiles of spatially averaged (a,d) turbulent integral length scales, (b,e) turbulent kinetic energy, and (c,f) eddy viscosities. Eddyviscosities were calculated directly from cubic-splined Reynolds stress and velocity profiles (solid lines, Eq. 13) and deduced from scalingarguments (symbols, Eq. 14). Scaling estimates were multiplied by a factor of 0.25 in (c) and (f).

Waves and currents in a kelp forest 797

Page 9: Interaction of waves and currents with kelp forests ......Interaction of waves and currents with kelp forests (Macrocystis pyrifera): Insights from a dynamically scaled laboratory

calculated from the definition and those calculated fromturbulence length scales and kinetic energy (Fig. 8f). Thisdiscrepancy is probably a result of fairly large uncertaintiesin both methods used to calculate eddy viscosities: valuescalculated from the definition (Eq. 13) are very sensitiveto the spline fits, while values calculated from Eq. 14 arescaling estimates and the constant of proportionality isunknown. It is probably reasonable to compare eddyviscosity values calculated using the same method. Scalingestimates suggest that peak values of eddy viscosity in themiddle of the water column were 25–50% smaller whenkelp was present (Fig. 8c, f). Although turbulent energywas increased by interaction of flow with model kelp, theintegral length scales were smaller, and thus turbulence wasless effective at mixing. Eddy viscosities were , 20–50%higher for experiments with waves due to larger turbulentkinetic energies arising from increased relative motionbetween kelp and water.

Discussion

We now assess the limitations of the laboratory findingsand present analytical models that can be used tounderstand laboratory observations and predict patternsthat might occur in real kelp forests.

Effect of kelp forests and waves on currents—Inlaboratory experiments, the bulk drag coefficients (CD)for experiments without and with waves were estimated tobe 0.2 and 0.4, respectively. The larger value for theexperiment with waves was due to greater relative motionbetween water and model kelp, combined with a quadraticdependence on relative velocity in the drag law. Rosmanet al. (2010) estimated CD , 0.08 from current measure-ments within a kelp forest in Santa Barbara, California(Gaylord et al. 2007), which was significantly smaller thanvalues calculated from the laboratory measurements.Although the frond density in the Santa Barbara kelpforest (2.7–5 m22) was larger than that simulated with thelaboratory model (1 m22), fronds were counted at 1 mabove bottom and frond lengths and blade distributionswere not reported. It is possible that the kelp surfacecanopy was less dense in the Santa Barbara kelp forest thanwas represented by the laboratory model even though thefrond density was greater. Another factor that couldcontribute to the observed differences in CD is thedifference in wave propagation directions relative to thecurrent. In real kelp forests, waves propagate at a range ofdifferent angles to currents; therefore, their effect on time-averaged drag on currents may be smaller than predicted inour laboratory experiments in which waves propagatedparallel to the current.

To assess the effect of angle between waves and currenton kelp forest drag estimates, we consider the general caseof waves propagating at an angle to the current. If wavespropagate at angle h to the current, the effective cda thatparameterizes the drag on the current at a given heightabove bottom is

cdað Þeff~2fD

�uu2~cda 1z

~bb{ _BB

�uu

!cos h

" #

1z2~bb{ _BB

�uu

!cos hz

~bb{ _BB

�uu

!2

z~ww{ _ZZ

�uu

� �224

35

1=2ð15Þ

Here, b and B.

are the wave orbital velocity and kelpvelocity, respectively, in the direction of wave propagation.Beyond several 10s of meters into a kelp forest, the velocityprofile shape is fully developed and the effective bulk dragcoefficient that describes the drag on the depth-averagedcurrent U (Rosman et al. 2010) is

CD~

1

H

�ff D dz

U2~

H3

2

ðH

cdað Þeff{1=2

dz

24

35

{2

ð16Þ

CD was computed from Eq. 15 and Eq. 16 using cda fromlaboratory experiments without waves, and b from linearwave theory with kH 5 0.68. For kelp motion, we considertwo scenarios, as described above, a rigid canopy thatrepresents an upper limit on the effect of waves on drag,and a more realistic flexible model in which kelp moveswith wave orbitals at surface and kelp velocity decayslinearly to zero at the bottom where it is fixed to thesubstrate. These CD were compared with the bulk dragcoefficient with no waves.

This analysis suggests that if wave orbital velocities aresmaller than or equal to the current, CD is not increasedsignificantly by surface waves (Fig. 9). However, if waveorbital velocities are much larger than currents, as is often thecase, drag on the current is increased. This effect is largest forwaves travelling parallel to the current, but time-averageddrag can be significantly increased even by waves propagat-ing perpendicular to the current. The drag coefficientspredicted for waves travelling perpendicular to the currentwere about a factor of two smaller than those predicted forwaves travelling parallel to the current. Also, the increase inCD due to waves was a factor of two smaller when the morerealistic flexible model was used for kelp motion rather thanthe rigid model. While the results in Fig. 9 are specific to thecda profile and kH value in the laboratory experiments, thegeneral pattern of increasing bulk drag coefficient withincreasing ratio of wave velocity to current are expected tohold across a range of wave and current conditions.

The effect of wave-enhanced drag on currents within andaround kelp forests is not clear from this analysis becausebottom friction is also increased by waves (Grant andMadsen 1979). Because the bottom is fixed while kelp isflexible, it is possible that the increase in bottom frictiondue to longer period waves is actually greater than theincrease in water-column drag due to kelp. The effect oncurrents within and around kelp forests would likelydepend on mechanisms driving the flow, local bathymetry,and spatial patterns in bottom roughness. Field studies ofspatial patterns in currents in and around kelp forestsacross a range of wave conditions would be valuable;

798 Rosman et al.

Page 10: Interaction of waves and currents with kelp forests ......Interaction of waves and currents with kelp forests (Macrocystis pyrifera): Insights from a dynamically scaled laboratory

however, it may be difficult to isolate the effects of wave–current interaction on drag because of the range ofdifferent forcing mechanisms that drive currents in thecoastal ocean.

Effects of kelp forests on wave orbital motion—In thelaboratory experiments, wave orbital velocities were , 8%

smaller than linear wave theory predictions in the model kelpsurface canopy, where solid obstacles were most denselyspaced. To investigate the reasons for this change in waveproperties caused by model kelp, we consider equations thatdescribe phase-dependent water motion under waves. Thederivation and solution of these equations is laid out in theWeb Appendix, Solution of the phase-averaged conservationof mass and momentum equations, (www.aslo.org/lo/toc/vol_58/issue_3/0790a.pdf). The effect of the kelp is throughdrag (fD) and inertial force (fI) terms that appear in thephase-averaged momentum equation:

LS~uuTLt

acceleration

~ {1

r

LS~ppTLx

pressure gradient term

{ ~ff Dxdrag term

{ ~ff Ixinertial force term

ð17Þ

If there are no obstacles in the flow, the drag and inertialterms are zero and Eq. 17 reduces to the relationship fromlinear wave theory (Kundu 1990).

The drag term fD 5 fD 2 f D is the part of the total drag thatis phase-dependent and fluctuates around a mean of zero:

~ff Dx~1

2cda �uuz~uu{ _XX

� ��uuz~uu{ _XX� �2

z ~ww{ _ZZ� �2

h i1=2

{ �uuz~uu{ _XX� �

�uuz~uu{ _XX� �2

z ~ww{ _ZZ� �2

h i1=2

) ð18Þ

The inertial force term is

~ffIx~ckm

d~uu

dt{ €XX

� �ð19Þ

where c is the solid volume divided by the fluid volume atheight z.

The terms in Eq. 17 were calculated from measured waveorbital velocities. For fDx and fIx, we used the two simplemodels for kelp motion described previously: the rigidmodel to give upper limits for these forces; and the morerealistic flexible model in which the kelp was fixed at thebase, moved exactly with water at the surface, and variedlinearly between the two. Profiles of cda from the

Fig. 9. Effect of waves on bulk drag coefficient CD. Contours are the ratio of CD in the presence of waves to CD without waves,plotted vs. angle between waves and current (h), and wave orbital velocity amplitude divided by current speed (fv/U). Panels show resultsfor (a) rigid, and (b) flexible kelp. Both panels use cda profiles from laboratory experiments and wave orbital velocities from linear wavetheory. Values shown are for kH 5 0.68, corresponding to the laboratory model and equivalent full-scale kelp forest.

Fig. 10. Relative sizes and phases of terms in the phase-dependent momentum budget (Eq. 17) at 30 cm above bottom (z/H 5 0.75). Drag and inertial force terms were calculated using (a)rigid, and (b,c) flexible models for kelp motion. Panels (a) and (b)show the lower bounds on fIx corresponding to kmc 5 cdad, and(c) shows the upper bound kmc 5 10cdad. All momentum budgetterms have been normalized by fv2. Drag and inertial force termsare plotted as forces acting on the fluid (2fDx, 2fIx) and have beenmultiplied by 5 for visualization purposes.

Waves and currents in a kelp forest 799

Page 11: Interaction of waves and currents with kelp forests ......Interaction of waves and currents with kelp forests (Macrocystis pyrifera): Insights from a dynamically scaled laboratory

unidirectional momentum budget were used for theanalysis. To bound the range of ckm, we consider twocases: a lower bound, ckm 5 cdad, that is valid for cylinderarrays, and an upper bound, ckm 5 10cdad, moreappropriate for streamlined obstacles such as flat platesoriented parallel to the flow (Gaylord et al. 1994). Theacceleration was calculated from measured wave orbitalvelocities by finite differencing. The pressure-gradient termwas calculated from the sum of the other terms in Eq. 17.

The inertial force term is in phase with, but opposite indirection to, the pressure gradient and acceleration terms(Fig. 10); hence, its main effect is to decrease the amplitudeof wave orbital velocities. The drag force is 90u out of phasewith the pressure gradient and acceleration terms; there-fore, the drag force has only a small effect on themagnitude of velocity fluctuations but alters their phase.The drag and inertial force terms predicted using the rigidcanopy model with kmc 5 cdad were an order of magnitudesmaller than the acceleration term (Fig. 10a). When theflexible model was used, these terms were about an order ofmagnitude smaller still (Fig. 10b), suggesting that forcesexerted by kelp on water had little potential to affect waveorbital motion. When an inertial force coefficient moretypical of streamlined objects was used (kmc 5 10cdad), theinertial force term was again about an order of magnitudesmaller than the acceleration term. The inertial force maytherefore have contributed to the , 10% reduction in waveorbital velocities observed in the surface canopy inlaboratory experiments. The phase shift due to the dragforce predicted from this analysis was too small for us toresolve in the measurements.

To predict changes to wave properties that may occuracross the wider range of conditions that occur in real kelpforests, we considered a simplified scenario in which thephase-dependent conservation of mass and momentum

equations (A1–3) could be solved analytically. To solve theequations, we used a similar approach to Mendez et al.(1999); the details are included in the Web Appendix. Alinear approximation was used for the drag term:

~ffDx~KD ~uu{ _XX� �

ð20Þ

It was assumed that kelp motion was proportional to watermotion, and the two were related by X

.5 (1 2 a)u, where a

5 1 corresponds to rigid or fixed kelp and a 5 0corresponds to kelp moving exactly with wave orbitalmotion. To solve the equations analytically, it was alsonecessary to assume that aKD and akmc were constant overthe water depth.

The solution for the wave orbital velocities is

~uu~{Gfvcosh Gkz

sinh GkHei(kx{vt)~~uulwtGei(k{k�)x ð21Þ

~ww~ifvsinh Gkz

sinh GkHei(kx{vt)~~wwlwte

i(k{k�)x ð22Þ

where ulwt and wlwt are the linear wave solutions withoutkelp (Kundu 1990), and

G~

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(1zackm){ aKD=vð Þ i

(1zackm)2z aKD=vð Þ2

sð23Þ

G is a factor that modifies wave orbital velocities from thelinear wave theory solution by altering both their amplitudeand phase. Because |G| , 1, the horizontal wave orbital velocityamplitude is reduced by the kelp. k* is the wavenumber for awave with the same frequency without kelp, and is related to kby k* 5 Gk. The new dispersion relation is

v2~gGk tanh GkH~gk� tanh k�H ð24Þ

Because |G| , 1, the wavenumber k is larger than it would be inthe absence of kelp and the wave speed c 5 v/k is smaller; thatis, the waves are shorter and travel slower.

The dimensionless quantities ackm and aKD/v representthe ratios of the inertial force and drag terms, respectively,to the fluid acceleration. As these ratios approach zero, Gapproaches unity and Eqs. 21–24 approach the relation-ships from linear wave theory. As the inertial forceparameter ackm increases, the magnitude of G decreases;hence, Uw decreases (Fig. 11). As the drag parameter aKD/v increases, the magnitude of G decreases only slightly butthe phase of G is shifted, resulting in phase shifts in waveorbital velocities. If phase shifts in vertical and horizontalorbital velocity components are different, kelp drag couldcause an effective wave stress, S~uu~wwT, that affects currents(Luhar et al. 2010).

For the laboratory experiments, using parameterscharacteristic of the lower water column (z/H 5 0.2, a ,0.8, ckm , cdad), we estimate aKD/v , 0.08 and ackm ,0.008. A , 0.5% reduction in horizontal wave orbitalvelocity amplitude and a phase shift of , 2.5u (t/T , 0.007)are predicted by the model (Fig. 11). If parameters morecharacteristic of the surface canopy are used (z/H 5 0.8,a , 0.2, ckm , 10cdad), then aKD/v , 0.05 and ackm , 0.2.

Fig. 11. Predicted effects of drag and inertial forces on waveorbital motion. Vertical black contours are change in magnitudeand horizontal grey contours are change in phase of horizontalwave orbital velocity relative to linear wave theory. Verticalcontours can also be interpreted as the ratio of wavelength in thekelp forest to that with no kelp. The x- and y-axes aredimensionless parameters that represent the sizes of the inertial(ackm) and drag (aKD/v) force terms relative to the fluidacceleration.

800 Rosman et al.

Page 12: Interaction of waves and currents with kelp forests ......Interaction of waves and currents with kelp forests (Macrocystis pyrifera): Insights from a dynamically scaled laboratory

In this case, a , 7% reduction in horizontal wave orbitalvelocity amplitude and a phase shift of 1u (t/T , 0.003) arepredicted. Although the values of aKD/v and ackm areknown only approximately and many assumptions havebeen made, the model reproduces the general trendsobserved in laboratory experiments. It is expected that inreal kelp forests, aKD/v , 0.1 and ackm , 0.2; therefore,wave orbital velocity phase shifts of , 3u and reductions inwave orbital velocity amplitudes of , 10% are predicted bythe analytical model. Field measurements of wave orbitalvelocity profiles over the complete water depth are neededto confirm whether the predicted changes in waveproperties occur within real kelp forests.

Effect of kelp–wave interaction on turbulence—Differenc-es between turbulence characteristics observed in labora-tory experiments with and without model kelp are alsoexpected to occur in real kelp forests. As water movesrelative to kelp fronds, eddies are generated in kelp wakes.Large turbulent eddies are broken up into smaller, wake-scale eddies that are only capable of mixing over smallerspatial scales. Turbulent kinetic energies within kelp forestsare expected to be larger than under similar current andwave conditions without kelp, and larger during wavyperiods than calm periods. Eddy viscosities are expected tobe smaller within kelp forests than would be predicted forsimilar flow conditions if kelp were absent, due to thecombination of smaller turbulent length scales and largerturbulent kinetic energy.

Currents are typically reduced by a factor of 1.5–10within kelp forests (Jackson 1998; Gaylord et al. 2007;Rosman et al. 2007). Because turbulent kinetic energyscales with U2, turbulence generation from the mean flowcould be quite small within kelp forests (Rosman et al.2010). However, waves propagate through kelp forestswith minimal damping (Elwany et al. 1995), and turbu-lence is generated by relative motion between kelp andwater due to waves even if currents are small. Laboratoryresults suggest that vertical mixing due to wake turbulenceis largest within and immediately below the surfacecanopy. Wake turbulence generated by interaction be-tween kelp and waves could be the main source of verticalmixing within kelp forests, particularly in dense kelpstands where currents are small. Although field verifica-tion of these eddy viscosity models would be valuable,measurement of turbulence within real kelp forests ischallenging due to biases from instrument noise andsurface waves, patchiness of kelp individuals and turbu-lence, and interruption of acoustic beams by kelp fronds.Measurements do not currently exist to compare withpredictions from the laboratory model.

Summary of main findings and their implications—Hy-drodynamics within real kelp forests are complex, a widerange of flow conditions occur, and kelp forest structure isvariable. Laboratory experiments have enabled a moredetailed mechanistic study of interactions between acurrent, waves, and a kelp forest than would be possiblein the field. Because laboratory results were highly idealized

and Reynolds numbers were smaller than field values, caremust be taken when interpreting laboratory results. Herewe summarize the main findings of this study as they relateto hydrodynamics within real kelp forests.

Relative motion between water and kelp associated withocean swell is expected to increase the drag on currentswithin kelp forests. Bulk drag coefficients for real kelpforests will depend on the relative sizes of wave orbitalvelocities and current, the angle between waves andcurrent, the wave frequency, and the physical structure ofthe kelp forest. For typical conditions, increases of 50–300% in CD due to waves are likely. Because bottomfriction is also increased by waves and spatial flow patternsadjust to the distributions of drag and bottom friction, afocused field study would be needed to assess theimplications of this change in drag for damping of currentswithin kelp forests.

Laboratory measurements and analyses suggest that aswaves propagate through kelp forests, drag and inertialforces reduce the amplitudes of wave orbital velocities andslightly shift their phase. Our analyses also suggest thatwave propagation is slowed and wavelengths are shortenedwithin kelp forests. Although these changes are predicted tobe small (, 10% reduction in wave orbital velocityamplitudes and phase speeds, , 3 degree phase shift), thischange in wave properties could generate wave stresses thataffect vertical transport of momentum and scalars. Fieldmeasurements of wave orbital velocities within real kelpforests are needed to confirm these results.

Although previous field studies have shown that currentsare reduced within kelp forests (Jackson 1998; Gaylordet al. 2007), ocean swell propagates through kelp forests withlittle damping (Elwany et al. 1995). Wake production ofturbulence due to relative motion between kelp and waterunder waves may therefore be the main source of turbulentkinetic energy and associated mixing within dense kelpstands. Even with this additional mechanism for turbulencegeneration, our laboratory results suggest that eddy viscos-ities within kelp forests are unlikely to exceed those outsidekelp forests because large turbulent eddies are broken up intowake-scale eddies that are less effective at mixing.

Our laboratory results indicate that interactions betweenkelp and waves must be taken into account when developingmodels for drag and mixing within kelp forests. Fieldmeasurements of wave orbital motion and turbulence inthese systems would be valuable to confirm laboratoryfindings and validate the simple analytical models developedhere. Additionally, measurements of spatial patterns incurrents in and around kelp forests across a range of differentwave and current conditions would enable us to betterevaluate the performance of the simple model for wave–current interaction in a kelp forest developed in this study.

AcknowledgmentsWe thank the students and post docs from the Environmental

Fluid Mechanics Laboratory at Stanford for their help construct-ing the artificial kelp. We also gratefully acknowledge Bob Brownand Bill Sabala for technical support. We thank two anonymousreviewers and the associate editor, Jim Leichter, for theirsuggestions to improve the manuscript.

Waves and currents in a kelp forest 801

Page 13: Interaction of waves and currents with kelp forests ......Interaction of waves and currents with kelp forests (Macrocystis pyrifera): Insights from a dynamically scaled laboratory

Support for this work came from the National Science Founda-tion; grant 0335346 from the Fluid Dynamics program funded thelaboratory experiments and grant 1061108 from the PhysicalOceanography program funded the analysis and publication.

References

DAYTON, P. K., M. J. TEGNER, P. E. PARNELL, AND P. B. EDWARDS.1992. Temporal and spatial patterns of disturbance andrecovery in a kelp forest community. Ecol. Monogr. 62:421–445, doi:10.2307/2937118

DEAN, R. G., AND R. A. DALRYMPLE. 1991. Water wave mechanicsfor engineers and scientists. Prentice Hall.

DENNY, M. W., B. P. GAYLORD, AND E. A. COWEN. 1997. Flow andflexibility II. The roles of size and shape in determining waveforces on the bull kelp Nereocystis luetkeana. J. Exp. Mar.Biol. Ecol. 200: 3165–3183.

ELWANY, M. H. S., W. C. O’REILLY, R. T. GUZA, AND R. E. FLICK.1995. Effects of Southern California kelp beds on waves.J. Waterway Port Coastal Ocean Eng. 121: 143–150,doi:10.1061/(ASCE)0733-950X(1995)121:2(143)

FRAM, J. P., H. L. STEWART, M. A. BRZEZINKSKI, B. GAYLORD,D. C. REED, S. L. WILLIAMS, AND S. MACINTYRE. 2008.Physical pathways and utilization of nitrate supply to thegiant kelp Macrocystis pyrifera. Limnol. Oceanogr. 53:1589–1603, doi:10.4319/lo.2008.53.4.1589

GAYLORD, B., C. A. BLANCHETT, AND M. W. DENNY. 1994.Mechanical consequences of size in wave-swept algae. Ecol.Monogr. 64: 287–313, doi:10.2307/2937164

———, M. W. DENNY, AND M. A. R. KOEHL. 2003. Modulation ofwave forces on kelp canopies by alongshore currents. Limnol.Oceanogr. 48: 860–871, doi:10.4319/lo.2003.48.2.0860

———, D. C. REED, P. T. RAIMONDI, L. WASHBURN, AND S.MCLEAN. 2002. A physically based model of macroalgal sporedispersal in the wave and current-dominated nearshore.Ecology 83: 1329–1352, doi:10.1890/0012-9658(2002)083[1239:APBMOM]2.0.CO;2

———, AND OTHERS. 2007. Spatial patterns of flow and theirmodification in and around a giant kelp forest. Limnol.Oceanogr. 52: 1838–1852, doi:10.4319/lo.2007.52.5.1838

GRAHAM, M. H. 1997. Factors determining the upper limit ofgiant kelp Macrocystis pyrifera Aghard along the Montereypeninsula, central California, USA. J. Exp. Mar. Biol. Ecol.218: 127–149, doi:10.1016/S0022-0981(97)00072-5

———. 2003. Coupling propagule output to supply at the edgeand interior of a giant kelp forest. Ecology 84: 1250–1264,doi:10.1890/0012-9658(2003)084[1250:CPOTSA]2.0.CO;2

GRANT, W. D., AND O. S. MADSEN. 1979. Combined wave andcurrent interaction with a rough bottom. J. Geophys. Res. 84:1797–1808, doi:10.1029/JC084iC04p01797

HEPBURN, C. D., J. D. HOLBOROW, S. R. WING, R. D. FREW, AND

C. L. HURD. 2007. Exposure to waves enhances the growthrates and nitrogen status of the giant kelp Macrocystispyrifera. Mar. Ecol. Prog. Ser. 399: 99–108, doi:10.3354/meps339099

JACKSON, G. A. 1984. Internal wave attenuation by coastal kelpstands. J. Phys. Oceanogr. 14: 1300–1306, doi:10.1175/1520-0485(1984)014,1300:IWABCK.2.0.CO;2

———. 1998. Currents in the high drag environment of a coastalkelp stand off California. Cont. Shelf Res. 15: 1913–1928.

———, AND C. D. WINANT. 1983. Effects of a kelp forest oncoastal currents. Cont. Shelf Res. 2: 75–80, doi:10.1016/0278-4343(83)90023-7

KUNDU, P. K. 1990. Fluid mechanics. Academic Press.LUHAR, M., S. COUTU, E. INFANTES, S. FOX, AND H. NEPF. 2010.

Wave-induced velocities inside a model seagrass bed. J.Geophys. Res. 115: C12005, doi:10.1029/2010JC006345

MENDEZ, F. J., I. J. LOSADA, AND M. A. LOSADA. 1999.Hydrodynamics induced by wind waves in a vegetation field.J. Geophys. Res. 104: 18383–18396, doi:10.1029/1999JC900119

NEPF, H. M. 1999. Drag, turbulence and diffusion in flow throughemergent vegetation. Water Resour. Res. 35: 479–489,doi:10.1029/1998WR900069

POPE, S. B. 2000. Turbulent flows. Cambridge Univ. Press.RAUPACH, M. R., AND R. H. SHAW. 1982. Averaging procedures

for flow within vegetation canopies. Boundary Layer Me-teorol. 22: 79–90, doi:10.1007/BF00128057

ROSMAN, J. H., J. R. KOSEFF, S. G. MONISMITH, AND J. GROVER.2007. A field investigation into the effects of a kelp forest(Macrocystis pyrifera) on coastal hydrodynamics and trans-port. J. Geophys. Res. 112: C02016, doi:10.1029/2005JC003430

———, S. G. MONISMITH, M. W. DENNY, AND J. R. KOSEFF. 2010.Currents and turbulence within a kelp forest (Macrocystispyrifera): Insights from a dynamically scaled laboratorymodel. Limnol. Oceanogr. 55: 1145–1158, doi:10.4319/lo.2010.55.3.1145

SEYMOUR, R. J., M. J. TEGNER, P. K. DAYTON, AND P. E. PARNELL.1989. Storm wave induced mortality of giant kelp, Macro-cystis pyrifera, in Southern California. Estuar. Coast. ShelfSci. 28: 277–292, doi:10.1016/0272-7714(89)90018-8

STEVENS, C. L., AND C. L. HURD. 1997. Boundary layers aroundbladed aquatic macrophytes. Hydrobiologia 346: 119–128,doi:10.1023/A:1002914015683

UTTER, B. U., AND M. W. DENNY. 1996. Wave-induced forces onthe giant kelp Macrocystis pyrifera (Agardh): Field test of acomputational model. J. Exp. Mar. Biol. Ecol. 199: 2645–2654.

YAO, Y. 1992. Theoretical and experimental studies of wavemak-ing by a large oscillating body in long tanks, including non-linear phenomena near resonance. Ph.D. thesis. Univ. ofCalifornia, Santa Barbara.

Associate editor: James J. Leichter

Received: 30 June 2012Accepted: 20 January 2013Amended: 15 January 2013

802 Rosman et al.