Two-Layer Analysis of Steady Circulation in Stratified Fjords

495

TWO-LAYER ANALYSIS OF STEADY CIRCULATION IN STRATIFIED FJORDS C.E. PEARSON and D.F. WINTER Department of Aeronautics and Astronautics and Department of Oceanography, University of Washington, Seattle, WA 98195

ABSTRACT In deep, narrow, fjord-type inlets with copious runoff near the head, freshwater inflow produces a surface slope and a pressure gradient which drives a brackish, near-surface layer seaward; at greater depths a denser, saline layer derived from oceanic water moves landward. We describe in this paper a self-consistent twolayer representation of this mode of inlet flow, generally referred to as "estuarine circulation." Our approach is different from other layered analyses in that the present model includes important effects of variations in mass density, channel width and depth, and a l s o allows for turbulent and advective exchange between the deep and near-surface layers. The starting point of the analysis is a set of equations expressing incompressibility and conservation of mass and horizontal momentum in each zone. Transfer of mass and momentum across the interface between the layers is parameterized by two interzonal exchange flux rates, FU and Fd, representing the upward and downward rate of fluid flow per square meter of interfacial area. When the time-averaged mass density variations can be estimated from field data, then the flux rates, FU and Fa, can be inferred entirely from known or measurable quantities. Two integrals of the motion are immediately available, and the mathematical problem is reduced to solving a pair of nonlinear equations for the layer cross-sectional areas. By way of illustration, the procedure is applied to Knight Inlet, a deep, stratified fjord on the Pacific Northwest coastline. TWO-LAYER ANALYSIS Inlets with appreciable fresh water runoff frequently exhibit a two-layer circulation pattern in which a brackish near-surface layer moves seaward while a deeper saline layer moves landward. The near-surface layer salinity generally increases in the seaward

496

-- - LAYER 2

-- - -

LAYER

I

. . . : ..., ......:

Figure 1.

'; :

.

' '

Sketch of an inlet cross-section (a) and longitudinal section (b), depicting the two major flow layers and illustrating geometrical quantities. The horizontal scale is compressed.

't

I I

I

+.-aX+l

I

I

I

I

...... ...

Figure 2.

Sketch of fluid sections used to derive equations of conservation of volume, mass, and horizontal momentum.

497

direction because of entrainment of water from the lower layer. The water in the lower layer is derived largely from sea water external to the inlet, although it may be freshened somewhat during its progress by brackish water from above due to turbulent mixing between the layers.

We consider here a simplified model of the

estuarine circulation mode, constructed from an hydraulic analysis similar to that developed by Stoker (1957) for river flow and later adapted to flow in stratified estuaries by Vreugdenhil (1970), Long ( 1 9 7 5 ) , and others. The present analysis extends previous layer models by including bathymetric effects and by allowing for interlayer exchange; the discussion is restricted to time-averaged conditions (over a tidal cycle, say). It is assumed that the inlet is sufficiently narrow that a one-dimensional treatment is appropriate, in the sense that all quantities in each layer depend only on the distance x fro= the seaward end (Fig. 1). The z-axis is directed vertically upward from a horizontal datum level. Subscripts 1 and 2 are used to denote variables in the lower and upper layers, respectively. It is assumed that the fresh water in the inlet is introduced exclusively into the upper layer with zero horizontal velocity (except at x =L) at a cumulative volumetric rate given by R(x); (thus, R(x) denotes the total influx between x and L). Because of the typically small depth of the near-surface layer, its breadth b(x) may be taken as constant over its depth. Let d 2 be an appropriate fixed reference depth, such as the layer thickness at the inlet mouth, and denote by hl, h2 the displacements of the lower and upper surfaces, respectively (Fig.1). The cross-sectional area of the near-surface layer, denoted by A2(x), can be written as

Next, suppose that the deep layer has cross-sectional area A(x) relative to the datum level: the breadth of its top surface will be b(x). The actual cross-section of the deep layer can then be represented as

(It will be convenient subsequently to refer to a maximum depth dl (x) of the deep layer when the fluid is at rest.)

Finally, the

498

horizontal velocities of the deep and near-surface layers will be denoted by u,(x) and u2(x), respectively; each is positive when directed landward. Equations expressing incompressibility (or conservation of volume) and conservation of mass and horizontal momentum in each layer can now be written down. In so doing, it is necessary to allow for (upward) convecti-Je motion and for turbulent mixing between the layers. In models where flow parameters are continuous functions of depth, vertical transport is described by vertical flow velocities and turbulent fluxes, the latter involving eddy coefficients of viscosity and diffusion. Since turbulent exchange mechanisms in estuarine flow are not at all well understood, the eddy coefficients are usually assigned values or functional forms which are adjusted until calculated flow and hydrographic patterns are similar to those observed in practice. In layered models of coastal plain and salt wedge estuaries, the approach has not been much different; the traditional procedure is to assign adjustable values to friction coefficients at a layer interface until some correspondence is achieved between computation and observation. In the present work, convective and turbulent transfer between the layers is represented by two interzonal exchange flux rates denoted by FU and Fd. The symbol FU represents the upward volume rate of flow of fluid from the deep layer to the near-surface layer per square meter of interfacial area. Likewise, Fd denotes the downward volumetric flux rate from the upper to the lower layer. Equations expressing incompressibility or volume conservation for each layer are derived from considerations of volume flow rates in and out of sectional "slices" of thickness Ax, illustrated in Fig. 2.

For the deep and near-surface layers respectively, one

obtains (UIA1)'

=

b(Fd

-

Fu)

(3)

and

where a prime denotes differentiation with respect to x. Similarly, the equations expressing conservation of mass in the lower and upper layers are (PIUIA1)' =

b(P2Fd

- PIFU)

(5)

499

and

where p o denotes the mass density of fresh water.

In writing down

Eqs.

(5) and ( 6 ) , it has been assumed that horizontal diffusion is negligible compared with advective transport. This is a common assumption in deep inlet studies and is based upon the findings of certain oceanographic field studies of fjord waters (e.g., McAlister et al., 1959; Dyer, 1973). On the other hand, it may not be justified in the vicinity of a long shallow sill where tidally induced longitudinal dispersion of mass can become competitive with advection. To derive the momentum equations, we use the fact that the net force on each slice of fluid, in Fig. 2 , must equal the net rate of efflux of momentum from that slice. Consider first the lower slice. The net force on it is made up of a pressure imbalance across its two faces, the pressure force on the sides due to changes in breadth, the horizontal component of the pressure force acting on the interface, and the frictional stress T ~ ( x )acting on the wetted perimeter Cw:

where p is the pressure, g the acceleration of gravity, and B(x,z) the breadth of the lower layer at vertical position z . The usual convention of shallow water theory is adopted to the effect that p results from hydrostatic forces only, so that

In carrying out the differentiation indicated for the first term of the momentum conservation equation above, there is no contribution from the x-dependence of the lower limit, since B(x,-[d2+dl]) = 0. Furthermore, there is no contribution from this term if channel has

500

a flat bottom since, in that case, the momentum equation contains an additional term of equal magnitude and opposite sign. The final result can be written as

r2

where y is defined by the relation

y(-d2

-

1 -2d 1) A

=

zB(x,z) dz

-d2 - d1

Clearly, in the case of a rectangular channel, y =l. A similar line of argument leads to the equation of horizontal momentum applicable to the near-surface layer:

+ - TWb

2A2

+

POU2

R'

'

where ~ ~ ( is x )the stress exerted on the upper surface by wind. Equations (3) - ( 6 ) , ( 9 ) , and (10) can be regarded as a system of equations for the six unknown dependent variables p l , p 2 , A1, Two integrals can be obtained immediately by A 2 , ul, and u 2 . adding Eqs. (3) and ( 4 ) ,

and

and Eqs. (5) and ( 6 ) , respectively:

501

These two equations will be used subsequently to eliminate a pair of dependent variables from the system. We suppose now that p1 and p 2 (and therefore also p l ' and p 2 ' ) are known from field measurements as functions of x; R(x) is also

supposed known. The quantities (u A ) and (u2A2) can then be deter1 1 mined by solving the linear equation system (12) and (13): u,A,

=

u A

=

I

I

2 2

R(P2 - Po) p1

-

02

5), we obtain expressions for the

Using Eq. (14) in Eqs. ( 3 ) and exchange flux rates FU and Fd:

I

L

From Eqs. (14) and ( 1 5 ) , u1 and u2 can be expressed in terms of A1 and A2. Since Eqs. (16) and (17) determine Fd and F U , it now follows that Eqs. (9) and (11) constitute a coupled pair of differential equations for A1 and A

2

of the form

allAll

+

a12A2'

=

bl

a21A1'

+ a22A2'

=

b2

where the a. and b . coefficients are certain functions of A 1' A2' +j and axial distance x. (The coefficients are readily derived from information provided above, but inasmuch as they are quite lengthy, the expressions are not presented here.) The system (18) and (19) can be readily integrated by, say, a conventional fourth order Runge-Kutta method to obtain Al and A2 as functions of x. Of course, initial conditions at x = O must be specified; because the choice of the datum level is arbitrary, we can set h2 = O at x = O , and the value of hl at x = O can be obtained from the measured

502

salinity (or, preferably,. horizontal current) profiles. In turn, these two values determine A1 and A2 at x = O . Once A1(x) and A2(x) have been calculated, ul(x) and u2(x) may be obtained Trom Eqs. (14) and (15), and the solution is complete.

An example is provided in

the last section. STABILITY The numerical values of the coefficients in Eqs. (18) and (19) are frequently such that the two equations are very similar to one another, and consequently it is advantageous to solve the modified system allAl' + a12A2'

=

bl

(20)

where the second of these equations is obtained by analytically subtracting Eq. (19) from Eq. (18). Even so, the solution of the modified system is fairly sensitive to the quantity ( p l - p 2 ) , as might be expected from its appearance in denominators in Eqs. (14) through (17), and from the fact that pl- p 2 is generally small compared with either p1 or p 2 . It turns out that for sufficiently small values of ( p 1 - p 2 ) the two equations (20) and (21) can cease to have a solution. This can occur in the neighborhood of a sill, for example, where intense turbulent mixing may take place by means of physical processes whose effects are not adequately represented in the present model. In order to gain further insight into such a situation, we confine our attention to the simple case in which R , B, and A are constant. Then Eqs. (20) and (21) become

+ - P2bFd PIAl

(u2

-

ul)

-

bul

- (Fd A1

- Fu)

503

U

2

A1,

+

I

- u2) +

(ul

[s +1 P2A2

-

[q+ bul 1 1

(Fu - Fa) bu2

PIAl

Typical numerical values are such that, to a first approximation, Eq. (22) states that A 1 + A 2 is approximately constant and equal to (say) K. Then in Eq. ( 2 3 ) , A1' can be replaced by - A 2 ' . Next we write p1 = p 2 + A p and assume that Ap and p 2 ' are approximately constant over some range of x-values of interest. In most examples we have considered, A1 has been much greater than A 2 , and the dominant terms on the right-hand side of Eq. (23) have been the first three.

It then turns out that Eq. (23) can be approximated by

If p l - p o is treated as approximately constant, integration of Eq. (24) yields

where

- E X , where Write p1 - p10 approximate a by

-E

is the (constant) value of p l l

and

504

where we have inserted representative values for ( p l - po) and for (in metric units). Then Eq. (25) may be written as PO

where A

is close to the value of A2 at x = O . The left-hand side 20 of Eq. (28) has a minimum value when A2 = and, consequently, from Eq. (28) the maximum value of x must be given by

3%

If the average depth of the channel is D , then from Eq. (27) it follows that the maximum length over which the quasi-laminar flow model can be valid is given by

where h is the thickness of the upper layer at x = O . This equa0 tion yields results compatible with numerical solutions of Eqs. (20) and (21) in appropriate cases, in the sense that the thickness of the upper layer, as computed from Eqs. (20) and (21) vanishes near this value of x. APPLICATION TO KNIGHT INLET, BRITISH COLUMBIA The method is illustrated by application to Knight Inlet in the southern portion of the mainland British Columbia coastline. Knight Inlet is a deep fjord located about 320 km northwest of Vancouver, British Columbia. It is approximately 110 km long and has an average width of about 3 km (see map in Fig. 3). Knight Inlet is a positive, high-runoff fjord estuary which is in communication with the Pacific Ocean by way of Johnstone Strait and Queen Charlotte Strait. The inlet is divided into two basins by a threshold of approximately 65 m depth, located at 75 km from the head. The outer basin, which is the shallower of the two, has a second 65 m deep threshold at the confluence of the inlet with marine waters in Johnstone Strait. The outer basin is characterized by depths in the range 150 to 200 m, and has an irregular

505

KNIGHT INLET BRITISH COLUMBIA

STATION

I

2

3

3 ! ’ 2 4 5

6

7

8

9

1011

IMEASURED)

E 200 c I

a

600 MAXIMUM CHANNEL DEPTH (MODEL SIMULATION)

F i g u r e 3.

Map of K n i g h t I n l e t , B r i t i s h Columbia, t o g e t h e r w i t h o b s e r v e d l o n g i t u d i n a l s e c t i o n ( a f t e r P i c k a r d and R o d g e r s , 1959), and maximum d e p t h v a r i a t i o n u s e d i n model c a l c u l a t i o n s .

506

shoreline indented by passes leading to adjacent bodies of water. The inner basin is deep over most of its extent, with the maximum depth in excess of 550 m. A longitudinal section of Knight Inlet is shown in Fig. 3; in the same figure is shown an idealized representation of the inner basin and inner sill depth profile, based on elementary functions (trigonometric and quadratic forms), which was used in the model calculations below. Most of the fresh water in Knight Inlet is derived from the Klinaklini and Franklin Rivers which discharge into the head. Although runoff is significant throughout most of the year, its intensity exhibits a seasonal variation characterized by a JanuaryMarch minimum and a May-July maximum, the latter being due to summertime melting of snow which fell on the adjacent mountains during the winter. A secondary runoff maximum may occur during the rainy season in October-November. The Klinaklini and Franklin Rivers are not routinely gauged, but estimates of fresh water discharge for Knight Inlet are available (Trites, 1955; Pickard, 1961), which indicate an annual mean of 410 m3 sec-l, and high runoff rates in the neighborhood of 600 and 750 m3 sec-l during the months of June and July, respectively. Inlets with the yearly discharge pattern of Knight Inlet have been termed Type A.l (stored runoff) fjords by Pickard (1961). Typically, the surface salinity is well below 2 o/oo near the head when runoff is intense and it remains at low values over substantial portions of the inlet. Figures 4a and 4b show continuous profiles of salinity obtained during 3-4 June 1951 and 15 July 1953, as presented by Trites (1955). Pronounced stratification over the major portion of the inlet is evident in these traces; the data from Stations 3-5 near the sill can only be regarded as qualitative since it is now known that aliassing is a serious problem throughout the region which includes these stations, (D.M. Farmer and J.D. Smith, personal communication, 1977). Model calculations were performed for comparison with the Trites' data from June and July, and the computed results for the salinity distributions are exhibited in Figs. 4c and 4d. The depth profile in Fig. 3 was used, together with an assumed constant main channel width of 2 km. The mean monthly cumulative fresh water discharges R for June and July were taken to be independent of x and equal to 600 and 750 m3 sec-1 , respectively. Average upper and lower layer salinities were estimated from the data. The thickness of the near surface layer near the head was assigned a value of I m

507

SALINITY

(o/oo)

L

I

20

I

I

KNIGHT INLET, JUNE 3-4,1951

Figure 4.

I

KNIGHT INLET, JULY 15, 1953

Depth profiles of salinity measured on (a) 3-4 June 1951 (1955); and (b) 15 July 1953, as presented by Trites model salinity profiles for conditions in (c) June 1951 and ( d ) July 1953.

508

for June and 6 m for July, as suggested by the measurements, and the integration of Eqs. (20) and (21) was carried out in the direction of decreasing x. It can be seen from Fig. 4 that the general features of the salinity and zonal thickness data are adequately represented over much of the inlet length, except near the sill where aliassing is known to occur. Figure 5 shows the calculated axial variation of upper layer thickness, zonal velocities, and the interzonal exchange rate FU for the month of June (Fd was equal to zero in this calculation since p1 was taken to be independent of x). Unfortunately, we lack appropriate field data for making meaningful comparisons between time- and depth-averaged horizontal current profiles and calculated values of u in the upper and lower zones. Field data described by Pickard and Rodgers (1959) suggests that, in the absence of wind stress, near the Knight Inlet sill the average horizontal current speed may be as high as a few decimeters per second at the surface, but it decreases rapidly in magnitude toward zero below the halocline. A consideration by Winter (1973) of distance and velocity scales appropriate to near-surface circulation in stratified fjords suggests that the characteristic transport speed uo in the upper zone is u0

=

Ro/bo

z0 IS 0

(31)

'

where Ro is a representative cumulative runoff value, b 0 is an effective channel width, zo is a representative thickness of the upper zone, and u0 is the fractional salinity excursion from great depths to the surface in the inlet segment under consideration. we assign representative mid-inlet values of R = 6 0 0 m3 sec-1,

If

0

bo = Z O O 0 m, z o = 8 m, and o0=2/3, then the characteristic transport speed for Knight Inlet during the month of June turns out to be u = 0.056 m sec-l, which is in agreement with the calculated value 0

of u (x) in the vicinity of Stations 5 and 7, but lower than might 2 be inferred from Pickard and Rodgers' measurements. If the characteristic horizontal scale xo is determined by a balance between mixing and advective salt transport, as assumed by Winter (1973), then

509

-

-E

K N I G H T INLET (JUNE) 10

0

01

I

30

20

I

1

1

1

1

50

40 1

I

1

1

60 1

I

8 0 (km)

70 I

I

1

'

"

I

a , In

E

1 6 ~ -

F i g u r e 5.

C a l c u l a t e d a x i a l v a r i a t i o n s of u p p e r l a y e r t h i c k n e s s , a v e r a g e u p p e r and l o w e r l a y e r s p e e d s , and upward f l u x

rate F

U

i n Knight I n l e t , B . C . ,

f o r June c o n d i t i o n s .

C o o r d i n a t e x i s measured p o s i t i v e landward from S t a t i o n 3 , seaward of t h e s i l l .

510

where K is a representative value of the vertical eddy diffusivity. 0 From the equation of continuity it follows that the characteristic vertical velocity w 0 is of the order of

w0 =

uo zo/xo

.

(33)

Trites (1955) has estimated that KO in Knight Inlet for early summer conditions is of the order of 2 x 1 0-4 m2 sec-’; it follows from -1 the equations above that x0 = 18000 m and wo = 2 . 5 x m sec . The latter quantity is of the same order of magnitude as the computed values of the upward interzonal flux rate FU. In this same connection, it is also interesting to note that from physical model experiments, Keulegan (1949) established an empirical relationship between the entrainment velocity and the longitudinal velocity of the upper layer: F~

=

3.5 x 1 0 - ,~ ~ ~

(34)

in the present notation. The value 0 . 0 5 6 m sec-’ for u 0 leads to an estimate of 2 x lob5 m sec-I for FU, which is of the same order as calculated mid-inlet values (Fig. 5). It would appear from the figure that in Knight Inlet FU attains its maximum value at x = 0 , i.e., at the seaward end of the main inlet segment.

However, this

feature is an artifact of the functional representation of the upper layer mass density variation p 2 (x). Field data indicate that in reaches of the inlet seaward of x = O , p 2 ’ eventually decreases faster than ( p - P , ) ~ , and it follows from Eq. (17) that FU ultimately declines to smaller values. Figure 6 presents longitudinal variations of surface salinity and salinity at depth as functions of x, where x is measured positive landward from Station 3 , located a few kilometers seaward of the inner sill.

The points in the figure represent the average of

field measurements acquired during the month of June in the years 1951, 1973, and 1974. It would appear from the figure (and from other field data) that mixing processes taking place near the inner threshold cause the salinity of the surface layer to increase substantially from about 30 km upstream from the sill. Figure 7 shows averages of salinity measurements at selected depths at fixed stations in the inlet during the months of March and June in the three years 1951, 1973, 1974.

The runoff during

511

I

DEEP ZONE

3

Figure 6.

4

5

6

7

9 STATION NUMBER

Longitudinal variations of surface salinity and salinity at depth for March and June conditions in Knight Inlet. The data points are averages of field measurements acquired in the years 1951, 1973, and 1974.

March is estimated to be of the order of 250 m3 sec-l.

It may be

noted that the scatter in the measurements in the pycnocline leads to an apparent distortion of the variation of salinity with depth, as exemplified by the traces in Fig. 4 . Nevertheless, calculations based on the averaged measurements seem to give reasonable representations of the upper layer thickness variations for both runoff regimes in Knight Inlet. It can be seen from Fig. 5 that, during high runoff, the upper layer thickness increases monotonically in the seaward direction from 6 or 7 m near the head to about 20 meters on the seaward side of the sill. Throughout the greater part of the inlet, however, the calculated interface depth is between 6 and 10 m, in accord with observation. On the basis of a somewhat different analysis (lower layer at rest, critical flow at the mouth, specified entrainment rate), Long (1975) has calculated a halocline depth of 21 m for Knight Inlet, but this appears to be somewhat of an overestimate. Only on the seaward side of the sill does the time-averaged upper layer thickness attain a value of the order of 20 meters. It is an interesting fact that when integrations of Eqs. (20) and (21) are performed with the same longitudinal density profiles p l and p 2 ,

512

KNIGHT INLET (MARCH 1 ST4.3 ( O h m i

ST4.5iZOkmi

STA.7 (40krr.l

ST4.9!60kml

ST4.11(75km1

KNIGHT INLET (JUNE) STA 3 ( 0 km)

STA 5 ( 2 0 k m )

STA 7 (40 km)

STA 9 ( 6 0 k m )

I

401

60

Figure 7.

n

I

j

I

I I

I I

I

!

!

Measured and calculated salinity profiles at f i v e stations along Knight Inlet for average conditions during March and June.

513

but with somewhat different starting values of upper layer thickness, the calculated upper layer thicknesses from mid-inlet toward the head are still of the order of a decameter or less. We conclude from the results for Knight Inlet, and from calculations performed for several other fjords along the British Columbia coastline, that the present model can provide reasonable, self-consistent representations of the main features of time-averaged estuarine circulation in stratified inlets. ACKNOWLEDGMENTS We are grateful to Dr. David M. Farmer for several useful discussions of the physical oceanography of British Columbia inlets. The research described in this paper was partially supported by the National Coastal Pollution Research Program of the Environmental Protection Agency under Grant No. R-801320, and by the National Science Foundation, Oceanography Section, under Grant No. OCE7680720. A travel grant from the National Science Foundation made it possible to present this work at the Ninth International Lisge Colloquium on Ocean Hydrodynamics. Salinity data for Knight Inlet were obtained from Data Reports prepared by the Institute of Oceanography, University of British Columbia, Vancouver, B.C. REFERENCES 1.

2. 3.

4.

5. 6. 7. 8.

Dyer, K. R., 1973. Estuaries: A Physical Introduction. John Wiley, London, 140 pp. Keulegan, G. H., 1949. Interfacial instability and mixing in stratified flows. Journal of Research, National Bureau of Standards, 43: 487-500. Long, R. R., 1975. Circulations and density distributions in a deep, strongly stratified, two-layer estuary. Journal of Fluid Mechanics, 71: 529-540. McAlister, W. B., Rattray, M., Jr., and Barnes, C. A., 1959. The dynamics of a fjord estuary: Silver Bay, Alaska. Department of Oceanography Technical Report No. 62, University of Washington, Seattle, Washington, 70 pp. Pickard, G. L., 1961. Oceanographic features of inlets in the British Columbia mainland coast. Journal of the Fisheries Research Board of Canada, 18: 907-999. Pickard, G. L. and Rodgers, K., 1959. Current measurements in Knight Inlet, British Columbia. Journal of the Fisheries Research Board of Canada, 16: 635-678. Stokes, J. J., 1957. FJater Waves. Interscience, New York, 567 pp. Trites, R. W., 1955. A study of the oceanographic structure in British Columbia inlets and some of the determining factors. Ph.D. Thesis, Institute of Oceanography, University of British Columbia, Vancouver, B.C., 125 pp.

514 9. 10.

V r e u g d e n h i l , C . B . , 1 9 7 0 . C o m p u t a t i o n of g r a v i t y c u r r e n t s i n estuaries. D e l f t Hydraulics Laboratory P u b l i c a t i o n N o . 86, 1 0 8 pp. Winter, D. F . , 1973. A s i m i l a r i t y s o l u t i o n f o r s t e a d y s t a t e g r a v i t a t i o n a l c i r c u l a t i o n i n f j o r d s . E s t u a r i e s and C o a s t a l M a r i n e S c i e n c e , 1: 387-400.

C o n t r i b u t i o n N o . 1 0 0 1 , D e p a r t m e n t of O c e a n o g r a p h y , U n i v e r s i t y of W a s h i n g t o n , S e a t t l e , WA.