THE JOURNAL OF RAPTOR RESEARCH A QUARTERLY PUBLICATION OF THE RAPTOR RESEARCH FOUNDATION, INC. VOL. 38 SEPTEMBER 2004 NO. 3 J. RaptorRes. 38(3):195-207 ¸ 2004 The Raptor ResearchFoundation, Inc. MODELING RAPTOR MIGRATION PATHW_AYS USING A FLUID-FLOW ANALOGY DAVID BRANDES 1 LafayetteCollege, Easton, PA 18042 U.S.A. DANIEL W. OMBALSKI U8 Filt• Inc., StateCollege, PA 16801 U.S.A. ABSTRACT.--Wedescribe an approach to mathematical modeling of raptor migration under conditions in which terrain updrafts are the primary sourceof lift. The model is basedon the analogyof laminar fluid flow to raptor migration, with the assumptionthat migration flux at a particular location is pro- portional to terrain conductivityand the local energy gradient driving migration. The terrain conduc- tivity parameter is taken to be the relative updraft strength, which is calculatedusing wind direction, terrain slope, and terrain aspectdata determined from a digital-elevationmodel of the area of interest. By imposing a directional energy gradient (a preferred axis of migration [PALM])across the resulting conductivityfield, flow (i.e., migration) is generated, and the predominant migration paths through the region are determined. We apply the model by simulating the spring migration of Golden Eagles (Aquila chrysaetos)through central Pennsylvaniaunder eight different wind scenarios.The locationsof the simulatedmigration tracksdepended on wind direction, PAM direction, and the spatialarrangement and orientation of terrain features. Migration tracks showed a marked tendency to converge toward a small number of prefbrred pathwaysas the migration proceeds.The overall pattern of simulated mi- gration wasconsistent with availablecount data. Model restiltsshowed that south/southeastand north/ northwestwinds provided the best conditions for rapid migration acrossthe region, as was suggested by field data. KEYWORDS: GoldenEagle; Aquila chrysaetos;migration; pathways; modeling;, updrafts; terrain. MODELACION DE LAS RUTAS DE MIGRACION MEDIANTE EL USO DE UN MODELO DE ANAL- OG• DE FLUIDODE FLUJO RESUMEN.--Describimosun enfoque para la modelaci6n matematica bajo condiciones en las cuales las corrientes termicas son la fuente primaria para elevarse.E1 modelo esta basadoen la analogia del fiqjo laminar aplicado a la migraci6n de rapacesasumiendo que el fiujo migratorio en una localidad particular es proporcional a la conductividaddel terreno y el gradiente de energia local que guia la migraci6n. E1 parametro de conductividad del terreno es asumido como la fuerza relativa de las corrientes ascendentesel cual es calculado usando la direcci6n del viento, la pendiente del terreno, determinados a partit de un modelo de elevaci6n digital del area de interis. Mediarite la imposici6n de un gradiente direccional de energia (un eje seleccionado de migraci6n) a trav6s del campo de conductividad resultante, el fiqjo (migraci6n) es generado, y las rutas de migraci6n son determinadas a travis de la regi6n. Aplicamos el modelo para simular la migraci6n de primavera del aguila dorada (Aquila chrysaetos)a travis del centro del Pennsylvaniabajo ocho escenariosde vientos diferentes. Las localidadesde simulaci6n de las rutas dependieron de la direcci6n del viento, del eje de migraci6n y del arreglo espacial y de la orientacion de las caracterlsticasdel terreno. Las rutas de migracion E-mail address:[email protected] 195 196 BRANDESAND OMBP•LSK• VOL. 38, NO. 3 mostraton una marcada tendencia hacia la cobertura de un pequefio numero de rutas preferidas tal como la migraci6n ocurre. E1 patrtn general de la migracitn simulada es consistentecon los datos de los conteosdisponibles. Los resultadosdel modelo demuestran que los vientos sur/surestey norte/ noresteproveen las mejores condicionespara una r•pida migracitn a travfisde la regi6n tal como lo sugieren los datos del campo. [Traduccitn de Cfisar M•rquez] It is widely held among raptor biologists and cussedby Haugh (1974). Updrafts are often the hawk watchersthat some mountain ridges concen- primary sourceof lift during early spring and late trate raptors in greater numbers than others (e.g., fall migration in temperate latitudes,and on over- Haugh 1974). Why this is true is an interesting castdays. We apply the model to simulatethe early question, especiallyin areas like the northern Ap- spring Golden Eagle (Aquila ch•Tysaetos) migration palachians,where there is a network of ridges and through central Pennsylvania and determine its many hawk watcheswhere counts have been con- primary migration pathwaysthrough the region. ducted (Zalles and Bildstein 2000). The factors The paper concludeswith a discussionof the util- that concentrate migrant raptors on one ridge, ity, limitations,and possibleadditional applications while leavinga nearbyridge of similarmorphology of the model. with few migrants, have only been discussedin a METHODS qualitativesense. Detailed quantitativemodeling of topographic and landscape structure effects on Model Equations. Fluid flows are modeled with the continuity equation (massconservation) and a momen- raptor-migrationpathways at the regional scalehas tum equation or equation of motion. Continuity alsoap- not been conducted. plies to migrating raptors, meaning that the same num- Here we describe a raptor migration model ber of raptors arrives at a location as leavesthat location based on a fluid-flow analogy and digital-elevation For laminar flow (e.g., groundwater flow), the equation data. The overall premise is that raptors migrating of motion is linear and is known as Darcy's law (Bear 1972): over the landscapeare analogousto fluid flowing Oh through a variablyconductive medium. Fluid flows q•: K-- (1) are driven by directionally-orientedenergy gradi- Os ents, much as raptors are driven by an innate urge where q• is the flux velocity (dimensionsof length [L]/ to migrate in a particular direction, termed the time IT]) in the s-direction(s is a spatialcoordinate x, y, preferred axisof migration (PAM; Kerlinger 1989). or z having dimension L), Kis the hydraulic conductiwty Fluid flows tend to channel along connectedpath- (dimensions of L/T), a material property which de- scribesthe easeof flow, and h is the fluid energyper unit waysof high conductivity(i.e., "the path of least weight (the symbolh is usedbecause energy is expressed resistance"). Similarly, raptors use pathwaysalong as an equivalent height (h) of water with dimension L) which migration can be obtained at the lowesten- Restated in words, the velocity of fluid in a particular ergy cost, by using thermals, ridge updrafts, and direction is directlyproportional to the local conductiwty (K) and the energy gradient in that direction. Analogous other sourcesof lift (Kerlinger 1989). Productive equations describea variety of physicaltransport phe- inland count sites like Hawk Mountain, PA (Broun nomena (e.g., heat, electricity). 1935) and the Goshute Mountains, NV (Hoffman The first assumptionin our model is that an equation 1985), are often on long or converginggeographic of the same form applies for raptor migration; that •s, features known as "leading lines" (Mueller and migration velocityat a particular location is directly pro- portional to "terrain conductivity"(defined below), and Berger 1967). In our analogy,these can be thought the magnitude of the local energy gradient driving rin- of as thin layersof sand (high conductivity)in an gration. Clearly, raptor migration is complex and qmte otherwise silty or clayeymedium (low conductivi- possiblynonlinear, but it is not unreasonableto assume ty). Becausethe equationsof groundwaterflow are this form as a first approximation. For example, migra- tion should be rapid where terrain conductivityis high well developed,the analogyis useful in creatinga in the desired direction (i.e., the PAM) and the energy quantitative model of raptor migration. gradient (i.e., urge to migrate) is strong.Conversely, m•- The model is applicableto conditionswhere up- gration should be slowwhere terrain conductivityis low drafts or deflection currents resulting from hori- and the energy gradient is weak. Other combinationsof conductivityand energy lead to intermediate migration zontal surfacewinds deflecting off sloping terrain rates. are the primary sourceof lift, rather than thermals In groundwater flows (and presumablyraptor migra- or other ephemeral atmosphericphenomena dis- tion acrossa diverselandscape), the parameter Kis high- SEPTEMBER2004 MODELING RAPTOR MIGRATION 197 ly variable in spaceand may range over severalorders of is relatively fiat. For a particular wind direction, we use magnitude. For simulating flow through such a domain, the product of two parametersto determine the relauve it is necessaryto divide the region of interest into a grid updraft strength (conductivity)at each location: (1) the of conductivityvalues, and approximate the derivative cosine of the angle between the terrain aspect and the term in equation (1) (Oh/Os)by diffk•renceat each point. wind direction (ranging from 0 fbr parallel winds to 1 For example, the velocityfrom point i to pointj waswrit- for perpendicularwinds), and (2) the terrain slope.Th•s ten as: algorithm determines the relative conductivity of dif•br- ent points of the landscape as a function of wind direc- - hl = (2) tion. In general, steeplysloping ridges that are oriented perpendicular to the wind direction will provide con- where/• • is a meanconductivity fbr flowbetween points nected areas of high conductivity. i and j (it is standardpractice in groundwatermodeling