Turbulent Channel Flows on a Rotating Earth - American ...

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JOURNAL OF PHYSICAL OCEANOGRAPHY

VOLUME 39

Turbulent Channel Flows on a Rotating Earth ROBERT A. HANDLER, RICHARD P. MIED, GLORIA J. LINDEMANN,

AND

THOMAS E. EVANS

Remote Sensing Division, Naval Research Laboratory, Washington, D.C. (Manuscript received 15 October 2007, in final form 2 October 2008) ABSTRACT This paper deals with flow in a rectilinear channel on a rotating earth. The flow is directed perpendicular to the background planetary vorticity; both an analytical theory and numerical simulations are employed. The analytical approach assumes the existence of an eddy viscosity and employs a perturbation expansion in powers of the reciprocal of the Rossby number (Ro). At lowest order, a cross-channel circulation arises because of the tilting of the planetary vorticity vector by the shear in the along-channel direction. This circulation causes a surface convergence, which achieves its maximum value at a channel aspect ratio (5 width/depth) of approximately 10. The location of the maximum surface convergence moves from near the center of the channel to a position very near the sidewalls as the aspect ratio increases from O(1) to O(100). To include the effects of turbulence, direct numerical pseudospectral simulations of the equations of motion are employed. While holding the friction Reynolds number fixed at 230.27, a series of simulations with increasing rotation (Ro 5 ‘, 10, 1.0, 0.1) are performed. The channelwide circulation cell observed in the analytical theory occurs for the finite Rossby number, but is displaced by lateral self-advection. In addition, turbulence-driven corner circulations appear, which make the along-channel maximum velocity appear at a subsurface location. The most interesting effect is the segregation of the turbulence to one side of the channel, while the turbulence is suppressed on the opposite side.

1. Introduction Coriolis acceleration plays an important role in geophysical flows because one effect of a rotating reference frame is to couple motions in perpendicular directions. This is particularly significant for channel flows, as was first discussed by Benton (1956). He showed that fluid flowing in a circular pipe, rotating with rapid constant angular velocity about an axis perpendicular to the flow direction, generates secondary circulations transverse to the flow along the channel axis. Later, Benton and Boyer (1966) demonstrated that these conclusions were valid for a much broader class of rotating flows. The physics responsible for the cross-channel circulations is straightforward: The vertical shear of the along-channel flow rotates the background vorticity of the rotating reference frame into the along-channel direction, and the resulting streamwise vorticity is manifested as a cross-channel circulation. Examining this

Corresponding author address: Dr. Robert A. Handler, Code 7230, Remote Sensing Division, Naval Research Laboratory, Washington, DC 20375. E-mail: [email protected] DOI: 10.1175/2008JPO3938.1 Ó 2009 American Meteorological Society

mechanism as the cause of along-channel surface convergence fronts in estuaries, Mied et al. (2000, 2002) showed that, for several different bottom bathymetry shapes, the expected cross-channel circulation accounts for regions of convergence and divergence on the surface of an estuarine flow with an imposed steady pressure gradient. Handler et al. (2001) showed that, even in the presence of an oscillatory axial pressure gradient introduced to mimic tidal forcing, this result was quite robust. In particular, the axial surface convergence zone present to one side of the channel flow was reestablished rapidly on the opposite side soon after the driving pressure gradient changed direction. This adaptation to the reversing pressure gradient is so rapid for realistic estuarine Rossby and Reynolds numbers that a crosschannel secondary circulation cell is present for ;80% of the tidal cycle. The works by Benton (1956), Benton and Boyer (1966), Mied et al. (2000, 2002), and Handler et al. (2001) address how the planetary vorticity tilting mechanism generates lateral circulations in flows without a free surface. When a free surface is present, Li and ValleLevinson (1999) show that the interaction of the tide in an estuary with cross-channel bathymetry generates

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regions of along-channel surface convergence. Winant (2007) has formulated an analytic model incorporating rotation, bottom bathymetry, and a free surface simultaneously and observes that the joint effects of planetary vorticity tilting and tide/bathymetry mechanisms are present. Specifically, the lateral circulation is established and reverses its direction of rotation as the tide changes from ebb to flood, as reported by Handler et al. (2001). Simultaneously, the free surface tide/ bathymetry interaction sets up a depth-dependent circulation, which appears to be the manifestation of the free surface convergence observed by Li and ValleLevinson (1999) in their depth-averaged model. Together, all of these works indicate that the tide/bathymetry and planetary vorticity tilting mechanisms can act separately or together to generate transverse circulations. In the present work, however, we focus solely upon the vortex tilting mechanism. Laboratory experiments on flow through rotating channels have also resulted in observations of these secondary cross-channel flows (Benton and Boyer 1966; Johnson and Ohlsen 1994). These two laboratory observations are governed by the tilting of the background planetary vorticity by the vertical shear discussed above but, because of the scales involved, the process is modified by the action of molecular viscosity. Similarly, the modeling studies (Mied et al. 2000, 2002; Handler et al. 2001; Winant 2007) parameterize the turbulence with a laminar-like eddy viscosity coefficient so that possible modifications to the flow by turbulent transport are absent. The influence of a fully developed turbulent flow on the generation of streamwise vorticity is thus not discernable from either the laboratory experiments or the models. Interestingly, some observational evidence exists for the generation of secondary (cross channel) flows in large, rapidly flowing natural channels. Johnson and Sanford (1992) present cogent evidence for the presence of a large vertical shear-driven cross-channel circulation cell on the order of 300 m in height contained in the deepest portion of the Faroe Bank Channel where it is ;20 km wide. Similarly, Signorini et al. (1997) display measurements in a wide, deep Arctic canyon with dimensions ;40 km 3 ;250 m. Although quite complicated, the data reveal a cross-channel flow with strength proportional to the magnitude of the vertical shear of the along-canyon flow. These experimental measurements suggest that the cross-channel circulations might be present in turbulent flows with very large Reynolds numbers, and it is informative to examine the transition from the nonrotating turbulent regime to one in which background planetary rotation plays a key role in the dynamics. Because of the

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present impossibility of performing direct numerical simulations on flows with Reynolds numbers on the order of millions, we attempt to make inroads into the problem by addressing turbulent flows having Reynolds numbers on the order of a few thousand. While not precisely representing naturally occurring flows, we believe that they give a hint of real-world behavior and serve as a useful starting point in understanding these very large-scale turbulent flows. We begin first by developing a linear analytical model in which the fluid is assumed to have a uniform viscosity and a large Rossby number.

2. Linear theory Most estuaries have shoals bordering a deeper central portion, which may have complicated bathymetry. In previous works (Mied et al. 2000; Handler et al. 2001), we have attempted to model this depth configuration by using a shallow region of uniform depth with a central Gaussian thalweg. In another study, Mied et al. (2002) obtained an exact solution for an estuary with an elliptical bottom profile. In all three cases, the steady alongchannel flow exhibits a single energetic core and a single steady cross-channel circulation cell. All three of these papers deal with analytical and/or numerical simulations, which parameterize the turbulence with an eddy viscosity. In the present work, we seek a geometry allowing an approximate analytical solution, while also allowing us to perform direct numerical turbulence simulations in order that a comparison can be made. We consider a channel of rectangular cross section referenced to the Cartesian (x, y, z) coordinate system (see Fig. 1) with corresponding unit vectors (i, j, k) in which the horizontal coordinates are x and y, and the vertical coordinate z is positive in a direction opposite that of gravity. A driving pressure gradient acts in the positive y direction and an anticlockwise frame rotation is taken along the z coordinate. The coordinate origin is defined at the shear-free boundary1 such that the three solid boundaries of the channel, for which no-slip boundary conditions on the velocity components are appropriate, are located on the planes x 5 6a / 2 and z 5 2b, where a and b are the width and height of the channel, respectively, and the velocity components are defined as (u, y, w) in this system. An identical physical situation and coordinate system is used for the direct numerical simulations described in section 3. The along-channel momentum equation for v and an equation for the streamfunction c(x, z), where

1 The terms shear free and free surface are used interchangeably, even though our free surface is not actually deformable.

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e ›c , 2 ›z viscous effects by Re21=2c, and the driving pressure gradient in the along-channel direction (y) by 1 j. The width and depth of the channel are used to define the aspect ratio of the channel (a [ a/b), and the factor of 2 is introduced to nondimensionalize the vertical coordinate to facilitate the solution of the biharmonic equation resulting from (1b). Thus, the nondimensional coordinate ranges for x and z will always be [21/2, 1/2] and [21/2, 0], respectively. In addition, =2 [

FIG. 1. The coordinate system for the direct numerical simulations and the linear theory. For the direct simulations, the coordinates x, y, and z are made nondimensional with the channel height, and the origin of the system is indicated by the point O. The driving pressure gradient acts in the positive y direction; the shearfree surface is located at z 5 1, and the no-slip boundaries are located on planes x 5 0, x 5 p, and z 5 0. The system is rotating with angular velocity Ve as shown. The domain size is given by the length scales Lx/b5 p, Lz/b 5 1, and Ly/b 5 8p/3 as shown. Virtual walls are used in the planes x 5 0 and x 5 p, whereas the no-slip condition is imposed exactly at the bottom wall given by the plane z 5 0. For the linear theory, the coordinate origin is located at (x, z) 5 (p/2, 1) in this coordinate system.

1 1 u 5 cz , w 5 cx 2 a are given by (Mied et al. 2000): 0 5 J (c , y)

e ›c 1 2 1 = y 1 1 j, 2 ›z Re

(1a)

e ›y 1 4 1 = c, 2 ›z Re

(1b)

0 5 J (c , =2 c) 1

where the x, z, y, and c have been nondimensionalized pffiffiffiffiffiffi by a, 2b, u* 5 Fb and u*b, where F is spatially and temporally invariant body force per unit mass, which drives the flow in the along-channel (y) direction. In these expressions, we represent nonlinear advection by the terms J(c,y) and J(c, =2c), Coriolis effects by

1 ›2 1 ›2 1 , 4 › z2 a2 › x2

(2a)

1 bf 5 . Ro u*

(2b)

e5

Here Ro is the Rossby number, f is the Coriolis parameter, and the Reynolds number is Re 5

u* b 1 5 , n d

(2c)

where v is the kinematic viscosity. We seek solutions for large Rossby number (Ro), that is, e 1. Oceanographers are accustomed to Rossby numbers, which are typically ; O(1021). In estuaries however, this parameter is not small. In Table 1, for example, we estimate values of Ro from several estuarine studies. We compute the Rossby number by citing values using 1) the channel width a and maximum velocity y MAX and 2) the friction velocity u* and b. The first is the standard open ocean definition, while the second is used in this work for estuaries influenced by friction. The Coriolis parameter is just f 5 2Ve sinf, where f is the latitude angle and Ve is the earth’s angular rotation rate. However, estimating the physically relevant values of b is not straightforward in the case of complicated bathymetric profiles. For rivers with a single deep channel [e.g., the Conway River, see Nunes and Simpson (1985)], the depth estimate is obvious. For cases with two or more deep channels near the river mouth (e.g., the York River), we base our estimate on the approximate width and depth of the deepest, most dominant trough. Finally,

TABLE 1. Rossby numbers, estimated two different ways for several rivers. Estimates for the width a, depth b, maximum channel velocity ymax, and friction velocity u* are made as described in the text. Estuary (reference)

yMAX (m s21)

u* (m s21)

a (km)

b (m)

yMAX fa

u* fb

York River (Huzzey and Brubaker 1988) James River (Valle-Levinson et al. 2000) Piscataqua River (Swift et al. 1996) Conway River (Nunes and Simpson 1985)

0.4 0.2 0.8 1.0

0.04 0.02 0.08 0.10

3 2 0.2 0.14

12 13 15 5.5

1.5 1.1 40.2 61.3

37.8 17.6 53.7 155.4

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we approximate the friction velocity by u* ’ 1021 yMAX, although this is only approximate and depends on channel geometry (Trowbridge et al. 1999). Although it is clear that there is significant variability among these values for different Ro, most of them are sufficiently large that an expansion in powers of e 5 Ro21 5 bf/u* can be formally justified, because Ro . 10 in all cases. It is also interesting to note that even with the open ocean definition of Ro, Ro . 1 in all cases. Accordingly, we expand the along-channel velocity and streamfunction in power series in e, y (x , z) ; y 0 (x , z) 1 e y 1 (x , z) 1 e2 y 2 (x , z) 1 (3a) c (x , z) ; e c1 (x , z) 1 e2 c2 (x , z) 1 .

(3b)

Substitution of (3a) and (3b) into (1a) and (1b) yields two equations for the leading order terms in (3a) and (3b): d =2 y 0 5 1, d =4 c1 5

1 › y0 . 2 ›z

(4a)

y0

1 › y 0 (x , 0) 5 0. 6 , z 5 y 0 (x , 1/2) 5 2 ›z

y0 5 yH 1 yP .

(7a)

Because of the constant on the rhs of Eq. (4a), we have some flexibility in the selection of the form of the particular solution. The choice

yP 5

4 d

1 z2 8 2

(7b)

(5)

The conditions that the side, bottom, and free surface be a streamline are written as 1 1 5 0, c1 6 , z 5 c1 (x , 0) 5 c1 x , 2 2

tion, this allows ›u/›x to be nonzero at the surface. So, while our rigid-lid condition does not permit surface displacements, it has the shear-free property of a free surface. We also remark that the rigid-lid condition differs markedly from the no-slip condition employed at the sidewalls and bottom because the no-slip designation implies no motion of any kind (u 5 y 5 w 5 0). One normally refers to surface regions for which ›u/›x . 0 as divergent and those for which ›u/›x . 0 as convergent. In this work, we will mostly be concerned with regions of convergence, and for convenience we refer often to the magnitude of the convergence |›u/›x|. Below, we derive a solution with the physically appealing property of comprising a broad slablike viscous flow over a wide channel, but explicitly including viscous boundary layers at the sidewalls. In appendix A, we derive the solution from first principles without this physical insight; the results are identical. The solution of (4a) is the sum of homogeneous and particular parts, and is written as

(4b)

The viscous boundary conditions at the channel sidewalls and bottom, along with a stress-free condition at the free surface for the along-channel flow, are written as

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(6a)

while › c1 (61/2 , z) › c1 (x , 1/2) ›2 c1 (x , 0) 5 5 50 ›x ›z › z2 (6b) ensures that the viscous boundary condition at the sidewalls and bottom, and the stress-free condition at the free surface, are met. We note that the boundary conditions presented above are commonly referred to as ‘‘rigid lid’’ conditions in the sense that w 5 0 at the surface (z 5 0). However, ›u/›z 5 0 is also enforced there as well. Thus, although no vertical motion is allowed at the surface, the x component of velocity (u) is allowed to assume any value consistent with the governing equations. In addi-

has the useful property of representing a one-dimensional viscous flow over a flat bottom, and is the solution of (4a) for an infinitely wide channel (a 5 ‘). For the homogeneous solution of (4a), a separable solution of the type yH } cosh (m a x) cos (2 m z) satisfies the boundary conditions (6a) along the free surface at (z 5 0) and the bottom (z 5 21/2) if we specify m5

(2 i 1 1) p , 2

i50, 1, 2, ...

The solution (7a) may thus be written 4 1 z2 y0 5 d 8 2 ‘

1

(2 i 1 1) a p x cos (2 i 1 1) p z å Ai cosh 2 i50

. (8)

The Ai are determined by satisfying the sidewall boundary conditions y 0(61/2, z) 5 0 in (5), using the orthogonality properties of the cosine over the interval [21/2, 1/2]. Then (8) becomes

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tesimally narrow channels (a 1) allow the side boundary layers to merge and viscosity to dominate. In this limit, we expect the velocity to vanish, as is seen in Fig. 2. Therefore, as a is increased, the distance between the sidewall boundary layers increases, the viscous influence decreases, and the surface centerline velocity increases, as shown in the graph. The solution of the biharmonic Eq. (4b) for the crosschannel secondary circulation (c1) with the boundary conditions, (6a) and (6b), is significantly more involved. We represent the solution in the x, z plane by ‘

FIG. 2. Dependence of the maximum along-channel velocity y max on aspect ratio a for d 5 1/Re 5 1.0, obtained from the firstorder analytical solution (9).

" ‘ 4 1 z2 y0 5 d 8 2 i50

4 (1)i (2 i 1 1) p a 3 3 (2 i 1 1) p cosh 4 (2 i 1 1) a p x cos (2 i 1 1) p z . 3 cosh 2

å

(9) For large aspect ratio channels (a 1) this solution for y 0 has the Poiseuillian behavior, y0 5

4 1 z2 , d 8 2

over a large flat expanse (the central portion of the channel) with the addition of an infinite sum of boundary layer constituents of the form 2 3 (2 i 1 1) a p x 6cosh 7 2 7 lim 6 4 a!‘ (2 i 1 1) p a 5 cosh 4 (2 i 1 1) a p 1 5 exp 7x 2 2

(10)

at the left and right channel sides (x 5 71/2), respectively. We remark that our solution (9), which was guided by physical insight, is identical to (A10), which was obtained by straightforward formal means. Isotachs of this along-channel flow are U-shaped contours, with a maximum value of y MAX (see Fig. 2) at the free surface in the channel center (0, 0). Figure 2 shows the behavior of this maximum centerline surface velocity for unit Re (1/d 5 1) as a function of a. For infinitely broad channels, the centerline surface velocity is 1/2, consistent with (7b). On the other hand, infini-

c1 (x , z) 5

‘

å å Bm n Cm (x) Sn(z),

(11)

m51 n51

where Cm and Sm are the Harris–Reid orthonormal functions (Harris and Reid 1958; Reid and Harris 1958). Since this assumed solution satisfies each of the four boundary conditions, (6a) and (6b), we need only determine the coefficients Bm n. Without actually determining the coefficients, we can discern some of the properties of the solution. Equation (4b) indicates that shape of the solution is determined only by the vertical distribution of the shear. Moreover, the shapes of the contours are independent of the direction of flow. That is, contours have the same shape irrespective of whether the flow is directed out of the page or into it; only the sign of c1 is changed. As noted previously, we nondimensionalize x by a and z by 2b. However, we must take care in exhibiting the relevant quantities since one of the main goals of the work is to determine the dependence of the surface convergence magnitude on the channel width (a) for fixed channel height (b). For the purpose of this discussion, we designate xp and up as the dimensional x coordinate and dimensional x component of the velocity, respectively. Nondimensional quantities are defined as x 5 xP /a, x 5 xP /b, and u 5 uP/u*, where (as defined above) u* depends only upon b (with F fixed) pffiffiffiffiffiffiffi through u* 5 F b. Consistent with our goal, the quantity › u/› x 5 (b/u*) › uP /› xP depends only upon a, as long as the other quantities (e.g., b, F, v, and f ) remain fixed. This quantity is exhibited in Fig. 4a. In all other figures associated with the linear theory (Figs. 3 and 4b), the length scales used are x 5 xP /a and z 5 zp /2b, where zP is the dimensional vertical coordinate. From this point on, unless there is a reason to do so for clarity, we will neither use coordinates with overbars (e.g., x) nor will we distinguish between dimensional and nondimensional quantities. The details of this calculation are given in appendix B, and a salient result is a contour plot of the streamfunction (Fig. 3) for a 5 10 and d 5 1. The function is symmetric

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FIG. 3. The streamfunction (streamlines) c(x,z) ; ec1 5 Ro21c1 calculated from (11) for a channel of aspect ratio a 5 10.0 and d 5 1/Re 5 1.0. Here we take e 5 1 for simplicity. The figure is oriented so that the driving pressure gradient, and hence the along-channel velocity, is positive into the page (away from the viewer); arrows indicate the sense of rotation of the flow. The contour interval is 22 3 1024.

about the channel centerline (x 5 0) because only those Harris–Reid functions satisfying the boundary conditions, (6a) and (6b), at x 5 60.5 are included. These retained functions are even over the interval 20.5 # x # 0.5, and therefore yield a solution symmetric about x 5 0. In the middle region of the channel at the surface (z 5 0), we expect the velocity to be larger, because there is no horizontal wall to retard the flow by viscous stresses. This effect can be seen in Fig. 3 also, as the near-surface streamlines are more closely spaced than anywhere else in the central water column away from the sidewalls, which denotes an increased velocity there. In Mied et al. (2000), the importance of surface convergence and divergence regions (regions of nonzero ›u/›x) generated by the Coriolis-induced circulation exhibited in Fig. 3 was discussed in some detail. We mentioned in that work that surface contaminants could potentially accumulate at convergence regions and thereby produce a variety of phenomena, detectable with radar and other remote sensing technologies. It is therefore of some interest to examine the properties of these regions in the context of the linear theory. In Fig. 4a, we exhibit the maximum absolute value of the surface convergence, j› u/› xjMAX as a function of the aspect ratio. Naturally, since the flow is perfectly symmetric with respect to the midpoint of the channel (x 5 0) this also represents the maximum absolute value of either the surface convergence or the divergence. We see that a plot of j› u/› xjMAX as a function of a exhibits a local maximum near a ’ 10. This occurs because of two competing effects: 1) As the aspect ratio increases with fixed channel depth, the lateral length scale increases and, hence, the variation of the cross-stream velocity

FIG. 4. (a) Dependence of the maximum absolute value of the convergence at the free surface j› u/› xjMAX as a function of aspect ratio a. Note that here the x coordinate is made nondimensional by using the channel height b and d 5 e 5 1. (b) The location of the maximum strain rate at the free surface (shown in Fig. 3a) xmax, where xmax is made nondimensional using the channel width a. This location can never exceed 0.5.

components decrease in x and 2) with increasing aspect ratio, the along-channel flow, which drives the circulation and associated surface convergence strength, increase as shown in Fig. 2. Channels having small a are narrow and tall, the lateral boundary layers (10) overlap, and the flow is dominated by the viscous flow in the sidewall boundary layers. As we show above (see Fig. 2), this small a case has a low axial velocity y. Since c is linearly proportional to y at this order, we expect u and its x derivative to be small; indeed, we observe this in Fig. 4a. On the other hand, as a ! ‘, (9) and (10) indicate that the flow over most of the channel reduces to the simple slab flow (7b). There, this one-dimensional flow and its associated cross-channel c depend only on z so that the convergence vanishes identically. In fact, Fig. 2 reveals that the maximum along-channel transport asymptotes to 0.5 in this limit. With a constant surface velocity and an increasing length scale, the cross-channel convergence decreases, which is seen in Fig. 4a at large a. These heuristic arguments seem to plausibly explain the

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existence of the maximum surface strain exhibited in Fig. 4a. Interestingly, the maximum absolute value of the surface convergence magnitude can never occur at the sidewall and, thus, must occur somewhere else on the free surface. This is true since at the boundaries (x 5 61/2), we have from continuity ›u/›x 1 (a/2) ›w/›z 5 0. Because w is independent of z, there as a consequence of the no-slip condition: ›u/›x 5 0 at the wall. Since symmetry requires ›u/›x to be zero at x 5 0 at the free surface, the assertion above is evidently correct. With this in mind, we plot the location of the maximum surface convergence as a function of the aspect ratio in Fig. 4b. It is evident that for small aspect ratio channels, the position of the maximum convergence occurs more toward the center of the channel. However, at a 5 100 the maximum occurs very near the sidewall, though careful examination of the behavior of the convergence result clearly shows that it is in fact zero at the no-slip sidewall boundary in every case.

3. Direct numerical simulations of turbulence in a rotating channel with a shear-free surface and three no-slip sidewalls In the first part of this paper, we describe an analytic approach to determine the effects of rotation on openchannel flows with closed lateral boundaries. The viscosity in the analytic model is assumed to be a constant and may be taken as a turbulent eddy viscosity. Flows in real channels are fully turbulent, and in these cases the eddy viscosity may be spatially inhomogeneous and anisotropic (Tennekes and Lumley 1972; Bernard and Wallace 2002). Consequently, we have performed a series of direct numerical simulations (DNS) in a rotating open channel geometry. In such simulations, no turbulence models are used. Instead, the molecular viscosity alone represents viscous effects, and the flow evolves naturally to a fully developed turbulent state. Another limitation of the analytic model is that it is valid only for large Rossby numbers, while the DNS approach removes any such restrictions on rotation rate. With these simulations, we hope to determine, in a general qualitative sense, the nature of these flows with realistic turbulent mixing. We solve the Navier–Stokes equations in a coordinate system (see Fig. 1) identical to that used for the linear theory. In these simulations, all length scales are made nondimensional by the channel height b, the velocity pffiffiffiffiffiffi components by u* 5 Fb, and time by b/u*, where, as in section 2, F is the spatially and temporally invariant driving body force. The corresponding nondimensional domain lengths are given by Lx/b 5 p, Ly/b 5 (8/3)p,

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and Lz/b 5 1 in the x, y, and z directions, respectively. These scales lead to the nondimensionalized Navier– Stokes and continuity equations du 1 2 1 5 $p 1 = u k 3 u11 j dt Re Ro

(12)

and $ u 5 0,

(13)

where d/dt is the derivative following the motion, p is a nondimensional pressure, and j and k are unit vectors in the y and z directions, respectively. The Reynolds number Re 5 u*b/n, and the Rossby number Ro 5 u*/fb are defined in section 2 with the exception that the kinematic viscosity n may be interpreted here as the molecular viscosity, as opposed to an eddy viscosity. To characterize the flow meaningfully, we must choose an appropriate Reynolds number. When the flow has reached a statistically steady state, the driving pressure gradient (force per unit mass) must be balanced by the sum of the forces on all three no-slip channel boundaries. A simple force balance then yields r F b 5 t b,

(14)

where t is the average tangential stress on the three noslip sidewalls, and b 5 (1 1 2b/Lx ).

(15)

It follows that pthe ffiffiffiffiffiffiffi friction Reynolds number Rt 5 utb/n, where ut 5 t /r, is pffiffiffi Rt 5 (1/ b)Re.

(16)

It is well known that a fully developed, self-sustaining turbulence can be attained in nonrotating wall-bounded flows only for friction Reynolds numbers greater than ; 100 (Orszag and Kells 1980; Orszag and Patera 1981). In the current study, therefore, we set our target Rt 5 180, which implies that the value of the Reynolds number used in our computations must be Re 5 230.27 from (16) since b 5 1 1 2/p in our case. This Reynolds number is held constant in all simulations and ensures the development of a vigorous turbulent flow in the case of no frame rotation. We have chosen four Rossby number values, Ro 5 ‘, 10, 1, and 0.1, which represent a variation from essentially no rotation to high rotation rates. The system given by (12) and (13) is solved using a pseudospectral method (Handler et al. 1999) in which Fourier expansions are used in the horizontal (x 2 y) plane, while Chebyshev expansions are used in the vertical (z) coordinate.

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The no-slip boundary conditions on the sidewalls (u 5 y 5 w 5 0 at x 5 0 and x 5 p) are enforced using a virtual surface approach2 (Goldstein et al. 1993, 1995), while the no-slip condition is enforced exactly on the plane z 5 0. Although this method is documented in these references, we include a brief description here for completeness. In the virtual surface approach, a body force f vs 5 g (x, t) d (x xs ) is added to the rhs of (12), where the position vector xs locates the point at which the force acts, and g (x, t) is a feedback forcing. The object of the approach is to determine g (x, t) such that the fluid motion is brought to rest (for a no-slip boundary) at xs . To accomplish this, we require the force to adapt to the local flow field and define it as ðt g (x, t) 5 a ~

~ u(x, t), u (x, t9) dt9 1 b

0

~ are chosen to be negative where the quantities a ~ and b constants. This force field represents an explicit feedback of the velocity field information into the equations of motion to bring the velocity to rest at the required location. The boundary conditions at the free surface (z 5 1) are shear free (›u/›z 5 ›y/›z 5 w 5 0). The computational resolution is 128 3 128 3 65 in the x,y, and z directions, respectively. The numerical methods will not be discussed further, except to state that our codes have been used successfully for many previous applications described in the works cited above. Examples of the results from the simulations appear in Figs. 5–7. In Fig. 5, the mean value of the alongchannel (y) component of velocity y is shown for each Rossby number. Here the mean, denoted by the overbar, signifies integration over the y coordinate and an average over all available realizations of the flow. In these calculations, averages were taken over 10 flow realizations spaced 2(b/u*) apart. It is evident that in the case of infinite Rossby number (Fig. 5a) that the velocity profile is nearly symmetric with respect to the cross-stream (x) coordinate. Any lack of symmetry is certainly due to the limited integration time used. It is also noteworthy that the mean velocity isotachs exhibit a subsurface maximum that lies near, and somewhat above, the domain center (z 5 1/2). This is due to the existence of two weak counterrotating vortices (not shown) near the free surface at x 5 0 and p. Their ro2

Approximately 3% of the domain volume is taken up by the virtual wall. If this is taken into account, a more accurate estimate for the shear-based Reynolds number given by (16) is Rt 5 178.92.

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tation pumps fluid along the shear free surface toward the center, forming a surface convergence region at x 5 p/2. In the vicinity of this convergence, fluid with high alongchannel momentum ( y momentum) is pumped downward into the channel interior, thus giving rise to the subsurface maximum in y . The existence of such corner eddies is well known (Gessner and Jones 1965; Gessner 1973; Madabhushi and Vanka 1991; Huser and Biringen 1993), and is entirely due to the existence of anisotropies in the Reynolds stress tensor. Since our interest here is on the effects of rotation on the turbulence and not on the much weaker secondary flows induced solely by turbulence, we will not discuss this topic further, except to note that this flow (Fig. 5a) is a fully developed turbulent one, in which the spatial scales of the turbulence are being computed directly without the use of any turbulence model. An important effect of rotation in this geometry is to establish a mean circulation in the x 2 z plane (Benton 1956; Benton and Boyer 1966; Mied et al. 2000, 2002; Handler et al. 2001). In these works, it was shown that the vorticity in the along-channel (y-direction) is driven by the term (1/Ro) ›y/›z, which represents the tilting of the planetary vorticity by the mean shear. It is clear from the sign of the shear in our case that a clockwise circulation in the x 2 z plane should be produced. We emphasize, however, that this clockwise secondary flow, displayed in Fig. 7a can exist in the absence of turbulence, unlike the turbulence-induced corner eddies mentioned above. The region of the domain over which this effect operates is given approximately by the Ekman layer thickness dE 5 (2 Ro/Re)1/2, where here dE is the layer thickness made nondimensional by b. Thus, we would expect that as the frame rotation rate increases, the rotation-induced secondary flow would be increasingly confined to thinner and thinner layers. The effect of the clockwise circulation cell described above can be seen in Fig. 5b for (Ro 5 10). Here, it is apparent that the mean circulation, which acts to advect the mean flow near the shear-free boundary toward the x 5 p plane, has caused the maximum value of y to move from the center, where it is located in the case of infinite Rossby number, toward the right. At higher rotation rates (Ro 5 1) the effect of this rotationinduced secondary circulation is even more evident (see Fig. 5c). Here it is observed that the maximum in y has shifted even farther to the right; in addition, the isotachs at the bottom of the channel exhibit a retroflection, indicating the development of a visible Ekman layer. At even higher rates of rotation (Ro 5 0.1, see Fig. 5d) the isotachs have become nearly vertical and the Ekman layer has thinned to perhaps no more than about onetenth the channel height.

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FIG. 5. The mean along-channel velocity y for Re 5 230.27. The Rossby numbers are (a) Ro 5 ‘, (b) Ro 5 10, (c) Ro 5 1, and (d) Ro 5 0.1. The contour interval 5 0.5 in all cases. The thin gray regions in this and subsequent figures near boundaries x 5 0 and x 5 p indicate the presence of a virtual surface.

This is readily understood from the behavior of the governing equations in the small Ro (quasigeostrophic) limit. Robinson (1978) gives an elegant and intuitively appealing development of the Ro 1 limit, which we will not reproduce. It is sufficient to note, however, that in the interior of the fluid (away from the viscous bottom layer), an expansion of the pressure and velocity

components in powers of the Rossby number reduces Eqs. (12) and (13) at O(Ro0) to (0 )

0 5 $p(0) k 3 uH 1 1j, 05

› p(0) , ›z

(17a) (17b)

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FIG. 6. Instantaneous profiles of the along-channel velocity y in the plane of the free surface for Re 5 230.27. Rossby numbers are (a) Ro 5 ‘, (b) Ro 5 10, (c) Ro 5 1, and (d) Ro 5 0.1. The driving pressure gradient acts in the positive y direction (from left to right).

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FIG. 7. (a) Flow vectors ( u, y) for the mean flow in the x2z plane for Re 5 230.27 and Ro 5 1. (b) For the same Reynolds number and Rossby number as in (a), the instantaneous along-channel velocity y for an arbitrary x2z slice. The contour interval 5 0.5. (0 )

$ u H 5 0.

(18)

Here, the Ekman number Ek [ Ro/Re, the quantities ( )(0) denote the leading terms in the small Ro expansion, and the subscript H denotes the horizontal components. Equations (17) and (18) are the same as those of Proudman (1916). They state that in the limit of vanishingly small Ro, the inviscid interior motion is independent of the vertical coordinate z, and corresponds to the well-known Taylor–Proudman columns (Taylor 1917, 1921). Inclusion of the viscous terms in (17a) superimposes on the motion a bottom Ekman layer with thickness of order Ek1/2 (see, e.g., Pedlosky 1979). This situation corresponds to the one portrayed in Fig. 5d. There, we see that the isotachs are indeed vertical lines, with a thin bottom Ekman layer. In addition, it is important to note that these same salient features of a nearly depth-independent motion within the body of the channel and a thin Ekman layer at the channel bottom are also evident in the Ro 5 1 case (Fig. 5c), even though the Rossby number is not small. The second effect of rotation in these flows is to stabilize the underlying turbulence. This stabilizing effect is closely related to the so-called inertial stability criterion. Charney (1973) gives an elegant treatment of the interplay between rotation and stratification in the general stability problem. For the unstratified case, only

the inertial instability mechanism can operate. In such cases, it is customary to define the unstable side of the flow as the pressure side, and the stable side as the suction side (Lezius and Johnston 1976). In our case, since the dominant component of Coriolis force (1/Ro y) points in the positive x direction, the stable and unstable sides are defined by the planes x 5 0 and x 5 p, respectively. A standard interpretation of this instability is that a fluid particle slightly displaced from its equilibrium position near the unstable side will be subject to a pressure gradient, which will drive it away from its original position. Thus, the instability is fundamentally a competition between pressure and Coriolis forces. Linear stability theory (see Lezius and Johnston 1976) establishes a boundary between stable and unstable regions as the position in the flow (in our case in the x direction) at which the planetary vorticity exactly cancels the along-channel component of the flow vorticity. It should be emphasized, however, that this inertial instability criterion might only be roughly valid for fully turbulent flows, which are obviously far from the laminar state for which this criterion was initially established. However, recent direct numerical simulations by Kristoffersen and Andersson (1993) have clearly demonstrated the stabilizing effects of rotation for a fully turbulent two-walled channel flow. We therefore expect that for our three-walled channel, as the Rossby number

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decreases, the flow in the vicinity of pressure side at x 5 p should remain fully turbulent as the suction-side flow approaches a more laminar state. The stabilizing effects of frame rotation and other aspects of the turbulence structure can be seen more clearly by examining the instantaneous isotachs of y at the free surface in Fig. 6. For the case of no rotation (Fig. 6a) the isotachs form what appear to be inclined shear layers on the no-slip walls given by the planes x 5 0 and p. These shear layers are qualitatively identical to those observed on the no-slip boundary given by the plane z 5 0. Similar structures are also observed at Ro 5 10 (Fig. 6b). These inclined shear layers are well known (Robinson 1991), and have been linked to processes associated with turbulence self-maintenance in wall-bounded turbulent flows (Bernard et al. 1993). At Ro 5 10, there appears to be no visible evidence of rotation-induced stabilization, but at Ro 5 1, Fig. 6c shows a dramatic example of the stabilization of the flow on the suction side (x 5 0) and the destabilization on the pressure side (x 5 p). Indeed, on the stable side there is virtually no evidence of turbulence, whereas on the unstable side the turbulence appears vigorous. We also note the remarkably distinct boundary between the turbulent and nonturbulent regions. These results are in reasonable agreement with the DNS results of Kristoffersen and Andersson (1993) who observed a nearly relaminarized flow on the suction side at the highest rotation rate that they explored. Finally, at the highest rates of rotation in these numerical experiments (Ro 5 0.1) the turbulence appears to have been largely suppressed at the free surface (Fig. 6d). Figure 7 contains additional details of the flow for the case Ro 5 1 in which the turbulence appears very vigorous on only the pressure side. Figure 7a shows the mean flow vector field in the x 2 z plane [i.e., the vectors given by ( u,w)]. It is evident that the mean flow is composed primarily of a clockwise circulation, which fills the channel and is caused by the tilting of the planetary vorticity as described above. However, an anticlockwise vortex also appears in the upper right corner of the flow. Near the free surface, the interaction of this vortex with the larger-scale clockwise flow forms an obvious convergence region at the free surface somewhat to the right of the center. The location of this convergence also seems to correspond to the boundary between the turbulent and nonturbulent flow exhibited clearly in Fig. 6c. In Fig. 7b it is seen that the contours of instantaneous isotachs of y, while having features similar to those shown in Fig. 5c, also reveal the instantaneous structure of the turbulence. On the right boundary in particular, a ‘‘mushroom shaped’’ structure is observed, similar to those revealed by the flow visuali-

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zations of Kristoffersen and Andersson (1993). In performing three-dimensional visualizations of this case (not shown here) we have seen that these structures are associated with counterrotating vortices, whose remarkably long coherence lengths can be nearly equal to the domain length in the along-channel direction. These are likely related to the roll cells predicted by the stability theory of Lezius and Johnston (1976) and visualized in the experiments of Johnston et al. (1972).

4. Summary and conclusions The principal results of this work are 1) development of the first analytic solution, as far as we are aware, for viscous rotating flow with a shear-free upper boundary in a rectangular basin; 2) analytic results, which reveal the surprising fact that there is a maximum in the surface convergence occurring at a channel aspect ratio of about 10; and 3) direct numerical simulations (DNS) of the same flow, which can also be used as a check on our own analytic solutions for flows that parameterize the turbulence with a constant eddy viscosity. The analytical solution relies upon the fact that the Poiseuille flow in a nonrotating channel may be taken as the leading term in a perturbation expansion in powers of the reciprocal of the Rossby number. The analytic solution results in a flow with closed streamlines, representing a circulation caused the vertical shear in the Poiseuille flow tilting the background Coriolis vector into the direction of the along-channel flow. These solutions for the cross-channel circulation have a surface convergence that assumes a maximum for a channel aspect ratio of around 10. This occurs because surface convergence increases in strength with increasing channel width for small aspect ratio, but decreases when it is large. Additionally, the position of the maximum surface convergence is close to channel center for small aspect ratio but moves toward the sidewall for large aspect ratios. While the analytical theory replicates the crosschannel circulation owing to planetary vorticity tilting, it is incapable of capturing the effects generated by the presence of turbulence. For this, we employ a series of direct numerical simulations of rotating channel flow bounded by three no-slip sidewalls and a shear-free top boundary. These simulations were performed at a fixed Reynolds number and fixed aspect ratio a [ a/b 5 p. Reference to Figs. 5a–d indicates that the influence of viscous effects is confined to a sidewall region, ; O(0.02p), or a relatively small fraction of the channel width p. This suggests that flows for this value of a are not greatly influenced by the proximity of the lateral boundaries and may further imply that the phenomena

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displayed in Figs. 5–7 are at least qualitatively similar to those present in channels with the very large a typical of geophysically relevant estuarine channels. Indeed, some encouraging points of agreement may be found between in situ observations and our numerical and analytical results. The surface convergence in estuaries (Nunes and Simpson 1985; Huzzey and Brubaker 1988; Swift et al. 1996; Valle-Levinson et al. 2000), where the aspect ratios are estimated to be a ’ 25.5, 250, 13.3, and 154 respectively, has been found to occur somewhere to the right of the channel center (when looking down channel). Our analytical results are, in fact, in good agreement with these observations. It is interesting to note that the effects reported here are also present if the free surface is displaced by gravity waves. Winant (2007) has developed a theory of the three-dimensional co-oscillating tidal motions in a long, rotating estuary. He finds that, when the current moves into the estuary, a cross-channel circulation similar to that in Figs. 3 and 7 is generated. However, the Winant (2007) results show that the transverse circulation vortex is approximately centrally located in the channel and that the isotachs of the along-channel velocity appear to be symmetric with respect to the channel centerline. The contrast with the results of our DNS calculations is most readily seen in the along-channel isotachs shown in Figs. 5 a–d, which exhibit a noticeable displacement to one side of the channel. This lateral bias in their location may be due in part to our flatbottomed bathymetry but is more likely attributed to the lateral advective effects in our simulations. Handler et al. (2001) observe this same displacement phenomenon in channels with a bathymetric trough, which is qualitatively similar to the bathymetry employed by Winant. We conclude that the reason for the difference in the along-channel velocity maximum and transverse circulation locations in Winant’s linear solution and ours is principally due to the presence of lateral advection in our simulations. Although our Reynolds number and aspect ratio are fixed, the Rossby number is varied from infinity (no rotation) to small values representing high rotation rates. At infinite Rossby number, the flow is fully turbulent, and the mean velocity exhibits a subsurface streamwise velocity maximum, apparently caused by weak corner eddies generated by Reynolds stress anisotropies. As the Rossby number is decreased, the subsurface maximum is seen to move toward the right due to advection by a clockwise circulation generated by the tilting of the planetary vorticity. Unlike the weak secondary corner eddies, which depend on Reynolds stress anisotropies for their maintenance, the origin of this channelwide circulation is in no way related to the

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existence of turbulence, and its presence in the theoretical model corroborates this. This is a unique aspect of our simulations, and it appears also to be an important mechanism in redistributing the mean streamwise momentum. In addition, this flow is also subject to the stabilizing effects of rotation. At Rossby numbers of order one, the simulations clearly show a strong segregation of the flow into turbulent and nonturbulent regions, in agreement with previous simulations and experiments. Thus, three-walled rotating open channel flow is apparently an interesting case in which three previously known physical processes appear to act simultaneously: 1) turbulence-driven corner circulation, 2) planetary vorticity reorientation, and 3) rotationinduced turbulence suppression. Our goal in this work has been to understand the physical origins of several different aspects of channel flow in problems of geophysical interest. The direct numerical simulations of the turbulence are an encouraging step, but the Reynolds numbers have been on the order of only several hundred. To reach geophysically relevant Reynolds numbers on the order of millions, recourse to large-eddy solution techniques may be mandatory. Nevertheless, the emergence of the observed phenomena at Re ; O(200) constitutes an encouraging first step. Acknowledgments. This work was supported by ONR 6.1 research projects, ‘‘Coastal ocean dynamics from geostationary remote sensing data’’ (Work Unit 728743) and ‘‘Non-equilibrium processes at the air–sea interface’’ (Work Unit 72-8759).

APPENDIX A Solution of the Poisson Equation for Along-Channel Flow The along-channel flow y 0 satisfies a Poisson Eq. (4a), 1 ›2 1 ›2 d = y0 5 d 1 y 0 5 1, 4 › z2 a2 › x2 2

(A1)

and must also satisfy boundary conditions (5), ensuring that the flow comes to rest at the sides and bottom of the channel while being stress free at the surface. Thus, we write 1 › y 0 (x, 0) 5 0. (A2) y 0 6 , z 5 y0 (x, 1/2) 5 2 ›z The solution y 0 is the sum of homogeneous (y H) and particular (y P) parts,

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y0 5 yH 1 yP .

(A3)

The homogeneous constituent must have the property that it yields zero when differentiated twice. This requirement can clearly be satisfied by the sum of any linear function of x and y, as well as a general separable solution satisfying the boundary conditions (A2) (Hildebrand 1962). Thus, we write yH 5 a 1 b x 1 c y 1 d x z ‘

1

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HANDLER ET AL.

(2 i 1 1) a p x cos (2 i 1 1) p z. å Ai cosh 2 i50 (A4)

The particular part can be satisfied by a polynomial that, when differentiated twice, will yield the rhs of (A1). The general form is thus yP 5 e x2 1 f z2 ,

Ai 5

a2 2

1 f . 1 d 2

(A9)

This is the along-channel velocity, which satisfies (4a) in section 2 of this paper, as well as obeying the viscous boundary conditions at the walls and bottom and the stress-free condition at the free surface.

APPENDIX B Solution of the Biharmonic Equation for the Cross-Channel Flow The biharmonic Eq. (4b)

(A5b) d =4 c1 5

(A6b)

1 › y0 2 ›z

(B1)

is formally solved by representing the solution (11) as ‘

c1 (x , z) 5

(A6a)

and c 1 d x 5 0,

.

å

(A5a)

Application of (A2) at the sidewalls (x 5 61/2) and free surface (z 5 0) yields equations constraining a,b,c,d,e,f, and Ai, which can be manipulated to show b1dz5 0

(2 i 1 1) p3 cosh [(2 i 1 1)p a/4]

Again, applying (A2) at z 5 21/2 and also finding that f 5 22/d, the solution may thus be written as ( ‘ 4 1 z2 4 (1)i y (x, z) 5 d 8 2 i50 (2 i 1 1)3 p3 cosh [(2 i 1 1)a p x/ 2] 3 cos [(2 i 1 1) p z] . cosh [(2 i 1 1) a p/ 4] (A10)

and substitution of (A5a) into (A1) constrains e and f so that e5

8 (1)i f 3

‘

å å Bm n Cm (x) Sn(z). m51 n51

(B2)

The Cm (x) and Sn (z) are the Harris–Reid orthonormal functions (Harris and Reid 1958; Reid and Harris 1958) defined by

respectively. Since (A6a) is to be valid for all z, we take d 5 0. Then (A6a) and (A6b) require b 5 c 50 as well. Application of (A2) at z 5 21/2 yields

Cm (x) 5

cosh lm x cos lm x , cosh lm /2 cos lm /2

(B3)

a2 1 f f . a5 1 2 4 8 d

Sm (z) 5

sinh mm z sin mm z , sinh mm /2 sin mm /2

(B4)

(A7) and obey the boundary conditions

The solution (A3) with (A4), (A5a), (A7), and b 5c 5 d 5 0 can now be written as a2 1 f 1 1 2 2 1 x 1f 1z y5 2 4 4 2 d

å

d Cm (61/2) 5 0 dx

(B5)

Sm (61/2) 5

d Sm (61/2) 5 0, dz

(B6)

and

‘

(2i 1 1) a p x cos (2i 1 1) p z. Ai cosh 1 2 i50

Cm (61/2) 5

(A8)

By again applying (A2) to (A8) at (x 5 61/2, multiplying (A8) by cos(2j 1 1)p z, and integrating over [21/2, 1 1/2], we calculate

provided the lm and mn satisfy tanh lm /2 tan lm /2 5 coth mn /2 cot mn /2 5 0. (B7)

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Solutions of the transcendental relation (B7) yield a countable infinity of lm and mn for m, n 5 1,2,3, . . . Direct computation reveals that the functions (B3) and (B4) possess the orthonormal property ð 11/2 ð 11/2 Cm (x) Cr (x) dx 5 Sm (x) Sr (x) dx 5 dm r . 1/2

1/2

(B8) The boundary conditions (6a) and (6b) on c are 1 1 5 0, c1 6 , z 5 c1 (x , 0) 5 c1 x , 2 2

(B9)

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(B5). In addition, (B6) guarantees that Sn(z) will satisfy the boundary conditions at z 5 21/2). However, because Sn(z) are odd functions in z, they satisfy the homogeneous requirements in (B9) and (B10) at z 5 0 as well. Thus, the solution (B2) satisfies all eight boundary conditions in Eqs. (B9) and (B10) so that we need determine only the value of the coefficients Bm n. Substituting (B2) and (9) into (B1), multiplying both sides of the resulting equation by Cp(x)Sq(z), integrating over the intervals 21/2 # (x, z) # 11/2, and using the orthonormal relations (B8) yields the system of algebraic equations ‘

‘

å å Bm n Rm n p q 5 d22 I m51 n51

and › c1 (61/2 , z) › c1 (x , 1/2) ›2 c1 (x , 0) 5 5 5 0. ›x ›z › z2 (B10)

8

‘

å d2 i50

(1)i II III (2 i 1 1)2 p2 cosh [(2 i 1 1) p a/4]

[ T p q, (B11)

Clearly, the four boundary conditions, (B9) and (B10), on the vertical end walls (x 5 61/2) are satisfied by

Rm n p q 5

where

Cm p Sn q lm4 mn4 d , 1 d 1 m p n q 16 a4 2 a2

8 > 2 l tanh lp /2 lp2 tanh2 lp /2 (m 5 p) > < p Cm p 5 8 lm2 lp2 (lm tanh lm /2 lp tanh lp /2) > > : l 4 l 4 m

p

8 2 2 > > > : m 4 m 4 q

11 ð /2

I5

11 ð /2

Cp (x) dx 1/2

cosh

(n 6¼ q)

,

lp mq2

,

(2 i 1 1) a p x Cp (x) dx, 2

1/2

5

(n 5 q)

n

8(tanh lp /2) (mq coth mq /2 2)

1/2

11 ð /2

II 5

z Sq (z) dz 5

(m 6¼ p),

32 lp2 [16 lp4 (2 i 1 1)4 a4 p4 ]

2 lp tanh (lp/2) cosh

(2 i 1 1) a p (2 i 1 1) a p (2 i 1 1) (a p) sinh 4 4

and III 5

Gathering all of the above terms, we write

4 mq3 (1)i coth mq /2 mq4 (2 i 1 1)4 p4

.

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Tp q 5

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HANDLER ET AL.

16 tanh (lp /2) (mq coth mq /2 2) d2 lp mq 2

8 9 ‘ (2 i 1 1)a p = 322 lp2 mq3 coth mq /2 < 2 lp tanh lp /2 (2 i 1 1) a p tanh 4 . : i50 (2 i 1 1)2 [16 l 4 (2 i 1 1)4 a4 p4 ][m 4 (2 i 1 1)4 p4 ]; d2 p2

å

p

The above integrals can either be calculated directly or evaluated using the tables of integrals in Harris and Reid (1958) and Reid and Harris (1958). Equations (B11) and (B12) are approximated by truncating the series as

(B12)

q

M

N

å å Bm n Rm n p q 5 T p q .

m51 n51

By substituting values of p and q over the intervals 1 # p # M and 1 # q # N and limiting the i in Tp,q to 0 # i # 20, we obtain the linear system of equations

B1 1 R1 1 1 1 1 B1 2 R1 2 1 1 1 B1 3 R1 3 1 1 1 1 B2 1 R2 1 1 1 1 B2 2 R2 2 1 1 1 5 T 1 1 B11 R1 1 1 2 1 B1 2 R1 2 1 2 1 B1 3 R1 3 1 2 1 1 B2 1 R2 1 1 2 1 B2 2 R2 2 1 2 1 5 T 1 2 .. .. . . B1 1 R1 1 M N 1

The values of Bm n from this calculation are used in (B2) to specify the cross-channel flow c. REFERENCES Benton, G. S., 1956: The effect of the Earth’s rotation on laminar flow in pipes. J. Appl. Math., 23, 123–127. ——, and D. Boyer, 1966: Flow through a rapidly rotating conduit of arbitrary cross-section. J. Fluid Mech., 26, 69–79. Bernard, P. S., and J. M. Wallace, 2002: Turbulent Flow: Analysis, Measurement, and Prediction. Wiley, 497 pp. ——, J. M. Thomas, and R. A. Handler, 1993: Vortex dynamics and the production of Reynolds stress. J. Fluid Mech., 253, 385–419. Charney, J. G., 1973: Planetary fluid dynamics. Dynamic Meteorology, P. Morel, Ed., Reidel, 97–351. Gessner, F. B., 1973: Origin of secondary flow in turbulent flow along a corner. J. Fluid Mech., 58, 1–25. ——, and J. B. Jones, 1965: On some aspects of fully developed turbulent flow in rectangular channels. J. Fluid Mech., 23, 689–713. Goldstein, D., R. Handler, and L. Sirovich, 1993: Modeling a noslip flow boundary with an external force field. J. Comput. Phys., 105, 354–366. ——, ——, and ——, 1995: Direct numerical simulation of a turbulent flow over a modeled riblet covered surface. J. Fluid Mech., 302, 333–376. Handler, R. A., J. R. Saylor, R. I. Leighton, and A. L. Rovelstad, 1999: Transport of a passive scalar at a shear-free boundary in fully developed open channel flow. Phys. Fluids, 9, 2607–2625.

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——, R. P. Mied, and T. E. Evans, 2001: Convergence fronts in tidally forced rotating estuaries. J. Geophys. Res., 106, 27 145– 27 162. Harris, D. L., and W. H. Reid, 1958: On orthogonal functions which satisfy four boundary conditions. I. Tables for use in Fourier-type expansions. Astrophys. J., Suppl. Ser., 3, 429– 447. Hildebrand, F. B., 1962: Advanced Calculus for Applications. Prentice-Hall, 646 pp. Huser, A., and S. Biringen, 1993: Direct numerical simulation of turbulent flow in a square duct. J. Fluid Mech., 257, 65–95. Huzzey, L. M., and J. M. Brubaker, 1988: The formation of longitudinal fronts in a coastal plain estuary. J. Geophys. Res., 93, 1329–1334. Johnson, G. C., and T. B. Sanford, 1992: Secondary circulation in the Faroe Bank Channel outflow. J. Phys. Oceanogr., 22, 927– 933. ——, and D. R. Ohlsen, 1994: Frictionally modified rotating hydraulic channel exchange and ocean outflows. J. Phys. Oceanogr., 24, 66–78. Johnston, J. P., R. M. Halleent, and D. K. Lezias, 1972: Effects of spanwise rotation on the structure of two-dimensional fully developed turbulent channel flow. J. Fluid Mech., 56, 533–557. Kristoffersen, R., and H. I. Andersson, 1993: Direct numerical simulation of rotating flow in a rotating channel. J. Fluid Mech., 256, 163–197. Lezius, D. K., and J. P. Johnston, 1976: Roll cell instabilities in rotating laminar and turbulent channel flows. J. Fluid Mech., 77, 153–175. Li, C., and A. Valle-Levinson, 1999: A two-dimensional analytic tidal model for a narrow estuary of arbitrary lateral depth

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variation: The intratidal motion. J. Geophys. Res., 104, 23 525–23 543. Madabhushi, R. K., and S. P. Vanka, 1991: Large eddy simulation of turbulence-driven secondary flow in a square duct. Phys. Fluids, 3, 2734–2745. Mied, R. P., R. A. Handler, and T. E. Evans, 2000: Longitudinal convergence fronts in homogeneous rotating channels. J. Geophys. Res., 105, 8647–8658. ——, ——, and T. F. Donato, 2002: Regions of estuarine convergence at high Rossby number: A solution in estuaries with elliptical cross sections. J. Geophys. Res., 107, 3206, doi:10.1029/2001JC001050. Nunes, R. A., and J. H. Simpson, 1985: Axial convergence in a well-mixed estuary. Estuarine Coastal Shelf Sci., 20, 637–649. Orszag, S. A., and L. C. Kells, 1980: Transition to turbulence in plane Poiseuille and plane Couette flow. J. Fluid Mech., 96, 159–205. ——, and A. T. Patera, 1981: Calculation of von Karman’s constant for turbulent channel flow. Phys. Rev. Lett., 12, 832–835. Pedlosky, J., 1979: Geophysical Fluid Dynamics. Springer-Verlag, 624 pp. Proudman, J., 1916: On the motion of solids in a liquid possessing vorticity. Proc. Roy. Soc. London, 92A, 408–424. Reid, W. H., and D. L. Harris, 1958: On orthogonal functions which satisfy four boundary conditions. II. Integrals for use with Fourier-type expansions. Astrophys. J., Suppl. Ser., 3, 448–452.

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Robinson, A. R., 1978: Boundary layers in homogeneous-rotating fluids. Boundary Layers in Homogeneous and StratifiedRotating Fluids, J. S. Fein, Ed., University Press of Florida, 1–68. Robinson, S. K., 1991: Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech., 23, 601–639. Signorini, S. R., A. Mu¨nchow, and D. Haidvogel, 1997: Flow dynamics of a wide Arctic canyon. J. Geophys. Res., 102, 18 661– 18 680. Swift, M. R., D. W. Fredriksson, and B. Celikkol, 1996: Structure of an axial convergence zone from acoustic Doppler current profiler measurements. Estuarine Coastal Shelf Sci., 43, 109–122. Taylor, G. I., 1917: Motion of solids in fluids when the flow is not irrotational. Proc. Roy. Soc. London, 93A, 99–113. ——, 1921: Experiments with rotating fluids. Proc. Roy. Soc. London, 100A, 114–121. Tennekes, H., and J. L. Lumley, 1972: A First Course in Turbulence. The MIT Press, 300 pp. Trowbridge, J. H., W. R. Geyer, M. M. Bowen, and A. J. Williams III, 1999: Near-bottom turbulence measurements in a partially mixed estuary: Turbulent energy balance, velocity structure, and along-channel momentum balance. J. Phys. Oceanogr., 29, 3056–3072. Valle-Levinson, A., C. Li, K.-C. Wong, and K. M. M. Lwiza, 2000: Convergence of a lateral flow along a coastal plain estuary. J. Geophys. Res., 105, 17 045–17 061. Winant, C. D., 2007: Three-dimensional tidal flow in an elongated, rotating basin. J. Phys. Oceanogr., 37, 2345–2362.