Beta-plane turbulence above monoscale topography
UUnder consideration for publication in the Journal of Fluid Mechanics Beta-plane turbulence above monoscaletopography
Navid C. Constantinou † , and William R. Young Scripps Institution of Oceanography, University of California San Diego, La Jolla,CA 92093-0213, USA(Received 16 December 2016; revised 18 April 2017; accepted xx)
Using a one-layer quasi-geostrophic model, we study the effect of random monoscaletopography on forced beta-plane turbulence. The forcing is a uniform steady wind stressthat produces both a uniform large-scale zonal flow U ( t ) and smaller-scale macroturbulencecharacterized by both standing and transient eddies. The large-scale flow U is retarded bya combination of Ekman drag and the domain-averaged topographic form stress producedby the eddies. The topographic form stress typically balances most of the applied windstress, while the Ekman drag provides all of the energy dissipation required to balancethe wind work. A collection of statistically equilibrated numerical solutions delineates themain flow regimes and the dependence of the time-average of U on parameters such asthe planetary vorticity gradient β and the statistical properties of the topography. Weobtain asymptotic scaling laws for the strength of the large-scale flow U in the limitingcases of weak and strong forcing.If β is significantly smaller than the topographic PV gradient then the flow consists ofstagnant pools attached to pockets of closed geostrophic contours. The stagnant deadzones are bordered by jets and the flow through the domain is concentrated into a narrowchannel of open geostrophic contours. In most of the domain the flow is weak and thusthe large-scale flow U is an unoccupied mean.If β is comparable to, or larger than, the topographic PV gradient then all geostrophiccontours are open and the flow is uniformly distributed throughout the domain. In thisopen-contour case there is an “eddy saturation” regime in which U is insensitive to largechanges in the wind stress. We show that eddy saturation requires strong transient eddiesthat act effectively as PV diffusion. This PV diffusion does not alter the kinetic energyof the standing eddies, but it does increase the topographic form stress by enhancingthe correlation between topographic slope and the standing-eddy pressure field. Usingbounds based on the energy and enstrophy power integrals we show that as the strengthof the wind stress increases the flow transitions from a regime in which most of the formstress balances the wind stress to a regime in which the form stress is very small andlarge transport ensues. Key words:
Geostrophic turbulence, Quasi-geostrophic flows, Topographic effects
1. Introduction
Winds force the oceans by applying a stress at the sea surface. A question of interestis where and how this vertical flux of horizontal momentum into the ocean is balanced.Consider, for example, a steady zonal wind blowing over the sea surface and exerting a † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . a o - ph ] A p r Constantinou and Young
Table 1: Various idealized topographies previously used in the literature.
Charney & DeVore (1979) cos ( m π x ) sin ( n π y )Charney et al. (1981) h ( x ) sin ( π y )Hart (1979) cos (2 π x ) (plus some remarks on h ( y ) cos (2 π x ))Davey (1980) triangular ridge: h ( x ) sin ( π y )Pedlosky (1981) cos ( m π x ) sin ( n π y )K¨all´en (1982) P ( r ) cos(3 φ ) (on the sphere)Rambaldi & Flierl (1983) sin (2 π x )Rambaldi & Mo (1984) sin ( π y ) sin (4 π x )Yoden (1985) cos ( m π x ) sin ( n π y )Legras & Ghil (1985) P ( r ) cos(2 φ ) (on the sphere)Tung & Rosenthal (1985) cos ( m π x ) sin ( n π y )Uchimoto & Kubokawa (2005) sin (2 π x ) sin ( π x ) force on the ocean. In a statistically steady state we can identify all possible mechanismsfor balancing this surface force by first vertically integrating over the depth of the ocean,and then horizontally integrating over a region in which the wind stress is approximatelyuniform. Following the strategy of Bretherton & Karweit (1975), we have in mind amid-ocean region which is much smaller than ocean basins, but much larger than thelength scale of ocean macroturbulence. The zonal wind stress on this volume can bebalanced by several processes which we classify as either local or non-local. The mostobvious local process is Ekman drag in turbulent bottom boundary layers. But in the deepocean Ekman drag is negligible (Munk & Palm´en 1951); instead the most important localprocess is topographic form stress (the correlation of pressure and topographic slope).Topographic form stress is an inviscid mechanism for coupling the ocean to the solidEarth. Non-local processes include the advection of zonal momentum out of the domainand, most importantly, the possibility that a large-scale pressure gradient is supported bypiling water up against either distant continental boundaries or ridge systems.In this paper we concentrate on the local processes that balance wind stress andresult in homogeneous ocean macroturbulence. Thus we investigate the simplest model oftopographic form stress. This is a single-layer quasi-geostrophic (QG) model, forced by asteady zonal wind stress in a doubly periodic domain (Hart 1979; Davey 1980; Holloway1987; Carnevale & Frederiksen 1987). A distinctive feature of the model is a uniformlarge-scale zonal flow U ( t ) that is accelerated by the applied uniform surface wind stress τ while resisted by both Ekman bottom drag µU and domain-averaged topographic formstress: U t = F − µU − (cid:104) ψη x (cid:105) . (1.1)In (1.1), F = τ / ( ρ H ) where ρ is the reference density of the fluid and H is themean depth. The eddy streamfunction ψ ( x, y, t ) in (1.1) evolves according to the quasi-geostrophic potential vorticity (QGPV) equation (2.3), η is the topographic PV and (cid:104) ψη x (cid:105) is the domain-averaged topographic form stress. (All quantities are fully defined insection 2.)This model may be pertinent to the Southern Ocean. There, the absence of continentalboundaries along a range of latitudes implies that a large-scale pressure gradient cannotbe invoked in balancing the zonal wind stress. However, we emphasize that the model eta-plane turbulence above monoscale topography U on wind stress forcing F . Theparameters β ∗ and F ∗ are defined in section 3.2. The box encloses the two points discussesin sections 3.4 and 3.5.in (1.1) and (2.3) may also be relevant in a small region of the ocean away from anycontinental boundaries, where we expect a statistically homogeneous eddy field. Althoughthe model has been derived previously by several authors, it has never been investigatedin detail except under severe low-order spectral truncation, and only for the simplestmodel topographies summarized in table 1. Here, we delineate the various flow regimes ofgeostrophic turbulence above a homogeneous, isotropic and monoscale topography e.g.,the topography shown in figure 2.Similar models were developed in meteorology in order to understand stationary wavesand blocking patterns. Charney & DeVore (1979) introduced a reduced model of theinteraction of zonal flow and topography and demonstrated the possibility of multipleequilibrium states, one of which corresponds to a topographically blocked flow. Charney& DeVore (1979) paved the way for the studies summarized in table 1, which are directedat understanding the existence of multiple stable solutions to systems such as (1.1).This meteorological literature is mainly concerned with planetary-scale topography e.g.,note the use of low-order spherical harmonics and small wavenumbers in table 1. Here,reflecting our interest in oceanographic issues, we consider smaller scale topography suchas features with 10 to 100 km scale i.e., topography with roughly the same scale as oceanmacroturbulence. Despite this difference, we also find a regime with multiple stable statesand hysteresis (section 4).Figure 1 summarizes our main result by showing how the time-mean large-scale flow ¯ U varies with increasing wind stress forcing F . The two solution suites shown in figure 1represent two end-points corresponding to either closed geostrophic contours (small valueof β ∗ , which is the ratio of the planetary PV gradient to the r.m.s. topographic PVgradient) or open geostrophic contours (large β ∗ ). In both cases there are two flow regimesin which the flow is steady without transient eddies: the “lower branch” and the “upperbranch” (indicated in figure 1). The mean flow ¯ U varies linearly with F on both the lowerand the upper branch. On the upper branch, form stress (cid:104) ψη x (cid:105) is negligible and U ≈ F/µ .On the lower branch the forcing F is weak and the dynamics is linear. Furthermore, forboth small and large β ∗ the transition regime between the upper and lower branches isterminated by a “ drag crisis ” at which the form stress abruptly vanishes and the system Constantinou and Young jumps discontinuously to the upper branch. The lower and upper branches, and the dragcrisis, are largely anticipated by results from low-order truncated models.A main novelty here, associated with geostrophic turbulence, is the phenomenology of thetransition regime: the lower branch flow becomes unstable at a critical value of F . Furtherincrease of F above the critical value results in transient eddies and active geostrophicturbulence. The turbulent transition regime is qualitatively different for the two valuesof β ∗ in figure 1. For open geostrophic contours (large β ∗ ) the flow is homogeneouslydistributed over the domain and ¯ U is almost constant as the forcing F increases. For closedgeostrophic contours (small β ∗ ) the flow is spatially inhomogeneous and is channeledinto a narrow boundary layers separating almost stagnant “dead zones”; in this case ¯ U continues to vary roughly linearly with F . The representative transition-regime solutionsindicated in figure 1 are discussed further in sections 3.4 and 3.5.The insensitivity of time-mean large-scale flow ¯ U to the strength of the wind stress F for the large- β case in figure 1 is reminiscent of the “ eddy saturation ” phenomenonidentified in eddy-resolving models of the Southern Ocean (Hallberg & Gnanadesikan2001; Tansley & Marshall 2001; Hallberg & Gnanadesikan 2006; Hogg et al. et al. et al. et al. et al. et al. et al. U only doubles as F ∗ varies from 0 .
2. Formulation
We consider barotropic flow in a beta-plane fluid layer with depth H − h ( x, y ), where h ( x, y ) /H is order Rossby number. The fluid velocity consists of a large-scale zonal flow, U ( t ), along the x -axis plus smaller scale eddies with velocity ( u, v ); thus the total flow is U def = ( U ( t ) + u ( x, y, t ) , v ( x, y, t ) ) . (2.1)The eddying component of the flow is derived from an eddy streamfunction ψ ( x, y, t ) via( u, v ) = ( − ψ y , ψ x ); the total streamfunction is − U ( t ) y + ψ ( x, y, t ) with the large-scaleflow U ( t ) evolving as in (1.1). The relative vorticity is ζ = ψ xx + ψ yy , and the QGPV is f + βy + ζ + η (cid:124) (cid:123)(cid:122) (cid:125) def = q . (2.2)In (2.2), f is the Coriolis parameter in the center of the domain, β is the meridionalplanetary vorticity gradient and η ( x, y ) = f h ( x, y ) /H is the topographic contribution to eta-plane turbulence above monoscale topography topographic PV . The QGPV equation is: q t + J( ψ − U y, q + βy ) + D ζ = 0 , (2.3)where J is the Jacobian, J( a, b ) def = a x b y − a y b x . With Navier–Stokes viscosity ν and linearEkman drag µ the “dissipation operator” D in (2.3) isD def = µ − ν ∇ . (2.4)The domain is periodic in both the zonal and meridional direction, with size 2 π L × π L .In numerical solutions, instead of Navier–Stokes viscosity ν ∇ in (2.4), we use eitherhyperviscosity ν ∇ , or a high-wavenumber filter. Thus we achieve a regime in which therole of lateral dissipation is limited to removal small-scale vorticity: the lateral dissipationhas a very small effect on larger scales and energy dissipation is mainly due to Ekmandrag µ . Therefore we generally neglect ν except when discussing the enstrophy balance,in which ν is an important sink.The energy and enstrophy of the fluid are defined through: E def = U (cid:124)(cid:123)(cid:122)(cid:125) def = E U + (cid:104)| ∇ ψ | (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) def = E ψ and Q def = βU (cid:124)(cid:123)(cid:122)(cid:125) def = Q U + (cid:104) q (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) def = Q ψ . (2.5)Appendix A summarizes the energy and enstrophy balances among the various flowcomponents.The model formulated in (1.1) and (2.3) is the simplest process model which can usedto investigate homogeneous beta-plane turbulence driven by a large-scale wind stressapplied at the surface of the fluid.Although the forcing F in (1.1) is steady, the solution often is not: with strong forcing theflow spontaneously develops time-varying eddies. In those cases it is useful to decomposethe eddy streamfunction ψ into time-mean “standing eddies”, with streamfunction ¯ ψ , andresidual “transient eddies” ψ (cid:48) : ψ ( x, y, t ) = ¯ ψ ( x, y ) + ψ (cid:48) ( x, y, t ) . (2.6)All fields can then be decomposed into time-mean and transient components e.g., U ( t ) =¯ U + U (cid:48) ( t ). A main question is how ¯ U depends on F , µ , β , as well as the statistical andgeometrical properties of the topographic PV η .The form stress (cid:104) ψη x (cid:105) in (1.1) necessarily acts as increased frictional drag on thelarge-scale mean flow U . This becomes apparent from the energy balance of the eddyfield, which is obtained through (cid:104)− ψ × (2.3) (cid:105) : U (cid:104) ψη x (cid:105) = (cid:104) µ | ∇ ψ | + νζ (cid:105) . (2.7)The right hand side of (2.7) is positive definite and thus U ( t ) is positively correlated withthe form stress (cid:104) ψη x (cid:105) i.e., on average the topographic form stress acts as an increaseddrag on the large-scale flow U .
3. Topography, parameter values and illustrative solutions
Although the barotropic quasi-geostrophic model summarized in section 2 is idealized,it is instructive to estimate U using numbers loosely inspired by the dynamics of theSouthern Ocean: see table 2. Without form stress, the equilibrated large-scale velocityobtained from the large-scale momentum equation (1.1) using F from table 2 is F (cid:14) µ = 0 .
77 m s − . (3.1) Constantinou and Young
Table 2: Numerical values characteristic of the Southern Ocean; f = − . × − s − and β = 1 . × − m − s − . The drag coefficient µ is taken from Arbic & Flierl (2004). domain size, 2 π L × π L L
800 kmmean depth H ρ − r.m.s. topographic height h rms
200 mr.m.s. topographic PV η rms = f h rms /H . × − s − Ekman drag coefficient µ . × − s − wind stress τ .
20 N m − forcing on the right of (1.1) F = τ / ( ρ H ) 4 . × − m s − topographic length scale (cid:96) η = 0 . L .
20 kmr.m.s. topographic slope h rms /(cid:96) η . × − a velocity scale β(cid:96) η . × − m s − non-dimensional β β ∗ = β(cid:96) η /η rms . × − non-dimensional drag µ ∗ = µ/η rms . × − non-dimensional forcing F ∗ = F/ ( µη rms (cid:96) η ) 2 . Figure 2: The structure and spectrum of the topography used in this study. Panel (a)shows the structure of the topography for a quarter of the full domain. Solid curves arepositive contours, dashed curves negative contours and the thick curves marks the zerocontour. Panel (b) shows the 1D power spectrum. The topography has power only withinthe annulus 12 (cid:54) | k | L (cid:54) × m s − ; this is larger by a factor of about twenty than the observedtransport of the Antarctic Circumpolar Current (Koenig et al. et al. The topography
If the topographic height has a root mean square value of order 200 m, typical ofabyssal hills (Goff 2010), then η − is less than 2 days. Thus even rather small topographicfeatures produce a topographic PV with a time scale that is much less than that of thetypical drag coefficient in table 2. This order-of-magnitude estimate indicates that theform stress is likely to be large. To say more about form stress we must introduce themodel topography with more detail.The topography is synthesized as η ( x, y ) = (cid:80) k e i k · x η k , with random phases for η k . eta-plane turbulence above monoscale topography η k to a relatively narrow annuluswith 12 (cid:54) | k | L (cid:54)
18. The spectral cut-off is tapered smoothly to zero at the edge ofthe annulus. In addition to being homogeneous and isotropic, the topographic model infigure 2 is approximately monoscale i.e., the topography is characterized by a single lengthscale determined, for instance, by the central wavenumber | k | ≈ /L in figure 2(b). Toassess the validity of the monoscale approximation we characterize the topography usingthe length scales (cid:96) η def = (cid:113) (cid:104) η (cid:105) (cid:14) (cid:104)| ∇ η | (cid:105) and L η def = (cid:114)(cid:68)(cid:12)(cid:12) ∇ ∇ − η (cid:12)(cid:12) (cid:69) (cid:14) (cid:104) η (cid:105) . (3.2)For the model in figure 2: (cid:96) η = 0 . L and L η = 0 . L . (3.3)(Recall the domain is 2 π L × π L .) Because (cid:96) η ≈ L η we conclude that the topography infigure 2 is monoscale to a good approximation and we use the slope-based length (cid:96) η asthe typical length scale of the topography.The isotropic homogeneous monoscale model adopted here has no claims to realism.However, the monoscale assumption greatly simplifies many aspects of the problembecause all relevant second-order statistical characteristics of the model topography canbe expressed in terms of the two dimensional quantities η rms and (cid:96) η e.g., (cid:104) ( ∇ − η x ) (cid:105) = (cid:96) η η . The main advantage of monoscale topography is that despite the simplicity ofits spectral characterization it exhibits the crucial distinction between open and closed geostrophic contours : see figure 3.3.2. Non-dimensionalization
There are four time scales in the problem: the topographic PV η − , the dissipation µ − ,the period of topographically excited Rossby waves ( β(cid:96) η ) − , and the advective time-scaleassociated with the forcing (cid:96) η µ/F . From these four time scales we construct the threemain non-dimensional control parameters: µ ∗ def = µ (cid:14) η rms , β ∗ def = β(cid:96) η (cid:14) η rms and F ∗ def = F (cid:14) ( µη rms (cid:96) η ) . (3.4)The parameter β ∗ is the ratio of the planetary vorticity gradient over the r.m.s. topographicPV gradient. There is a fourth parameter L/(cid:96) η that measures the scale separation betweenthe domain and the topography. We assume that as L/(cid:96) η → ∞ there is a regime ofstatistically homogeneous two-dimensional turbulence. In other words, as L/(cid:96) η → ∞ , theflow becomes asymptotically independent of L/(cid:96) η so that the large-scale flow ¯ U and otherstatistics, such as E ¯ ψ , are independent of the domain size L .Besides the control parameters above, additional parameters are required to characterizethe topography. For example, in the case of a multi-scale topography the ratio L η /(cid:96) η characterizes the spectral width of the power-law range. A main simplification of themonoscale case used throughout this paper is that we do not have to contend with theseadditional topographic parameters.3.3. Geostrophic contours
We refer to contours of constant βy + η as the geostrophic contours . Closed geostrophiccontours enclose isolated pools within the domain — see figure 3(a) — while open contoursthread through the domain in the zonal direction, connecting one side to the other — seefigure 3(c). The transition between the two limiting cases is controlled by β ∗ . Figure 3(b)shows an intermediate case with a mixture of closed and open geostrophic contours. Constantinou and Young
Figure 3: The structure of the geostrophic contours, βy + η , for the monoscale topographyof figure 2 and for various values of (a) β ∗ = 0 .
10, (b) β ∗ = 0 .
35 and (c) β ∗ = 1 .
38. Itis difficult to visually distinguish the geostrophic contours with β ∗ = 0 from those with β ∗ = 0 . β = 0. Then only the geostrophic contour η = 0 is open and all other geostrophic contours are closed. This intuitive conclusionrelies on a special property of the random topography in figure 2: the topography − η isstatistically equivalent + η . In other words, if η ( x, y ) is in the ensemble then so is − η ( x, y ).For further discussion of this conclusion see the discussion of continuum percolation byIsichenko (1992).If β is non-zero but small, in the sense that β ∗ (cid:28)
1, then most of the domain is withinclosed contours: see figure 3(a). The planetary PV gradient β is too small relative to ∇ η to destroy local pools of closed geostrophic contours. But β dominates the long-rangestructure of the geostrophic contours and opens up narrow channels of open geostrophiccontours.The other extreme is β ∗ (cid:29)
1; in this case, illustrated in figure 3(c), all geostrophiccontours are open. Because of its geometric simplicity the situation with β ∗ (cid:29) An example with mostly closed geostrophic contours: β ∗ = 0 . β ∗ = 0 . × U ( t ) and the form stress (cid:104) ψη x (cid:105) .After a spin-up of duration ∼ µ − the flow achieves a statistically steady state in which U ( t ) fluctuates around the time mean ¯ U . In figure 4(a) the form stress (cid:104) ¯ ψη x (cid:105) balancesalmost 98% of F , so that U ( t ) is very much smaller than F/µ in (3.1). The time-meanof the large-scale flow is ¯ U = 1 .
70 cm s − , which is 2.2% of the velocity F/µ in (3.1).Figure 4(b) shows the evolution of the energy: the eddy energy E ψ is about 50 timesgreater than the large-scale energy E U . With the decomposition of ψ in (2.6) the time-mean eddy energy E ψ is decomposed into E ¯ ψ + E ψ (cid:48) ; the dash-dot line in figure 4(b) isthe energy of the standing component E ¯ ψ : the transient eddies are less energetic than thestanding eddies. This is also evident by comparing the snapshot of the relative vorticity ζ eta-plane turbulence above monoscale topography β ∗ = 0 . F ∗ = 2 .
20 and µ ∗ = 10 − .Panel (a) shows the evolution of the large-scale zonal flow U ( t ) (dashed) and the formstress (cid:104) ψη x (cid:105) (solid). Panel (b) shows the evolution of E ψ (solid) and E U (dashed). Thedash-dot line in panel (b) is the energy level of the standing eddies, E ¯ ψ = (cid:104)| ∇ ¯ ψ | (cid:105) .Panel (c) shows a snapshot of the relative vorticity, ζ , (shaded) at µt = 10 in one-quarterof the domain overlying the topographic PV (solid contours are positive η and dashedcontours are negative). Panel (d) shows the time-mean ¯ ζ . A movie showing the evolutionof q = ζ + η and ψ − U y from rest is found in
Supplementary Materials .Figure 5: A solution with β ∗ = 0 . F ∗ = 2 .
20 and µ ∗ = 10 − . (a) The speed of thetime-mean flow, | ¯ U | is indicated by colors; the geostrophic contours βy + η are shown aswhite curves. (b) Surface plot of the total time-mean streamfunction, ¯ ψ − ¯ U y .0 Constantinou and Young in figure 5(c) with the time mean ¯ ζ in figure 5(d): many features in the snapshot are alsoseen in the time mean.Figures 4(c) and (d) show that there is anti-correlation between the time-mean relativevorticity and the topographic PV: corr(¯ ζ, η ) = − .
53, where the correlation between twofields a and b is corr( a, b ) def = (cid:104) ab (cid:105) (cid:46)(cid:112) (cid:104) a (cid:105)(cid:104) b (cid:105) . (3.5)Another statistical characterization of the solution is that corr( ¯ ψ, η x ) = 0 .
06, showingthat a weak correlation between the standing streamfunction ¯ ψ and the topographic PVgradient η x is sufficient to produce a form stress balancing about 98% the applied windstress.The most striking characterization of the time-mean flow is that it is very weak inmost of the domain: figure 5(a) shows that most of the flow through the domain ischanneled into a relatively narrow band centered very roughly on y/ (2 π L ) = 0 .
25: this“main channel” coincides with the extreme values of ζ and ¯ ζ evident in figures 4(c) and (d)(notice that figures 4(c) and (d) show only a quarter of the flow domain). Outside ofthe main channel the time-mean flow is weak. We emphasize that ¯ U = 1 .
70 cm s − is anunoccupied mean that is not representative of the larger velocities in the main channel:the channel velocities are 40 to 50 times larger than ¯ U .Figure 5(b) shows the streamfunction ¯ ψ ( x, y ) − ¯ U y as a surface above the ( x, y )-plane.The mean streamfunction surface appears as a terraced hillside: the mean slope of thehillside is − ¯ U and stagnant pools, with constant ¯ ψ − ¯ U y , are the flat terraces carved into thehillside. The existence of these stagnant dead zones is explained by the closed-streamlinetheorems of Batchelor (1956) and Ingersoll (1969). The dead zones are separated byboundary layers and the strongest of these boundary layers is the main channel whichappears as the large cliff located roughly at y/ (2 π L ) = 0 .
25 in figure 5(b). The mainchannel is determined by a narrow band of geostrophic contours that are opened by thesmall β -effect: this provides an open path for flow through the disordered topography.3.5. An example with open geostrophic contours: β ∗ = 1 . β ∗ = 1 .
38; this is the β ∗ = 1 .
38 “boxed” point indicated in figure 1. The geostrophiccontours are open throughout the domain. The most striking difference when compared tothe previous blocked case in section 3.4 is that there are no dead zones; the flow is moreevenly spread throughout the domain: compare figure 7 with figure 5. The time-meanstreamfunction in figure 7(b) is not “terraced”. Instead ¯ ψ − ¯ U y in figure 7(b) is bettercharacterized as a bumpy slope.The large-scale flow is ¯ U = 4 .
68 cm s − , which is again very much smaller than theflow that would exist in the absence of topography: ¯ U is only 6% of F/µ . The eddyenergy E ψ is roughly 15 times larger than the large-scale flow energy E U . Moreover, theenergy of the transient eddies, shown in figure 6(b), is in this case much larger thanthat of the standing eddies. This is also apparent by comparing the instantaneous andtime-mean relative vorticity fields in figures 6(c) and (d). In anticipation of the discussionin section 8 we remark that these strong transient eddies act as PV diffusion on thetime-mean QGPV (Rhines & Young 1982).In contrast to the example of section 3.4, the relative vorticity is positively correlatedwith the topographic PV: corr(¯ ζ, η ) = 0 .
23. Because of the strong transient eddies the signof corr(¯ ζ, η ) is not apparent by visual inspection of figures 6(c) and (d). The form-stresscorrelation is corr( ¯ ψ, η x ) = 0 .
15. Again, this weak correlation is sufficient to produce aform stress balancing 94% of the wind stress. eta-plane turbulence above monoscale topography β ∗ = 1 .
38 and open geostrophic contours. All other parametersas in figure 4. Panels also as in figure 4. Note that the color scale is different betweenpanels (c) and (d). A movie showing the evolution of q = ζ + η and ψ − U y from rest canbe found in
Supplementary Materials .Figure 7: A solution with β ∗ = 1 .
38. All other parameters are as in figure 5. (a) The speedof the time-mean flow, | ¯ U | (colors); the geostrophic contours βy + η are shown as whitecurves. (b) Surface plot of the total time-mean streamfunction, ¯ ψ − ¯ U y .
4. Flow regimes and a parameter survey
In this section we present a comprehensive suite of numerical simulations of (1.1)and (2.3) using the topography of figure 2(a). A complete survey of the parameter spaceis complicated by the existence of at least three control parameters (3.4). In the followingsurvey we use µ ∗ = 10 − , (4.1)and vary the strength of the non-dimensional large-scale wind forcing F ∗ and the non-dimensional planetary vorticity gradient β ∗ . Most the solutions presented use 512 grid2 Constantinou and Young
Figure 8: (a) The equilibrated large-scale mean flow ¯ U as a function of F ∗ for caseswith β ∗ = 0 and β ∗ = 1 .
38. Shown are results for three different monoscale topographyrealizations (each denoted with a different marker symbol: ∗ , (cid:52) , ◦ ) all with the spectrumin figure 2(c). Other parameters are in Table 2 e.g., µ ∗ = 10 − . Panel (b) shows adetailed view of the transition from the lower to the upper branch solution for the casewith β ∗ = 1 .
38 and panel (c) shows the hysteretic solutions for one of the topographyrealizations with β ∗ = 1 .
38. Dashed lines in panel (a) correspond to asymptotic expressionsderived in section 6 and dash-dotted lines in all panels mark the solution: U = F/µ .points; additionally, a few 1024 solutions were obtained to test sensitivity to resolution(we found none). Unless stated otherwise, numerical simulations are initiated from restand time-averaged quantities are calculated by averaging the fields over the interval10 (cid:54) µt (cid:54) Flow regimes: the lower branch, the upper branch, eddy saturation and the drag crisis
Keeping β ∗ fixed and increasing the wind forcing F ∗ from very small values we find thatthe statistically equilibrated solutions show either one of the two characteristic behaviorsdepicted in figure 8.For β = 0, or for values of β ∗ much less than one, we find that the equilibratedtime-mean large-scale flow ¯ U scales linearly with F ∗ when F ∗ is very small. On this lowerbranch the large-scale velocity is¯ U ≈ F (cid:14) µ eff , with µ eff (cid:29) µ . (4.2)In section 6 we provide an analytic expression for the effective drag µ eff in (4.2); thisanalytic expression is shown by the dashed lines in figure 8. As F ∗ increases, ¯ U transitionsto a different linear relation with ¯ U ≈ F (cid:14) µ . (4.3)On this upper branch the form stress is essentially zero and F is balanced by bare drag µ .For the β = 0 case shown in figure 8 the transition between the lower and upperbranch occurs in the range 0 . < F ∗ <
3; the equilibrated ¯ U increases by a factor ofmore than 200 within this interval. On the other hand, for β ∗ larger than 0.05, we finda quite different behavior, illustrated in figure 8 by the runs with β ∗ = 1 .
38. On thelower branch ¯ U grows linearly with F with a constant µ eff as in (4.2). But the linearincrease in ¯ U eventually ceases and instead ¯ U then grows at a much more slower rate as F increases. For the case β ∗ = 1 .
38 shown in figure 8, ¯ U only doubles as F is increased eta-plane turbulence above monoscale topography F ∗ = 0 . U is insensitiveto changes in F , with the “eddy saturation” regime of Straub (1993). As F increasesfurther the flow exits the eddy saturation regime via a “drag crisis” in which the formstress abruptly vanishes and ¯ U increases by a factor of over 200 as the solution jumpsto the upper branch (4.3). In figure 8 this drag crisis is a discontinuous transition fromthe eddy saturated regime to the upper branch. The drag crisis, which requires non-zero β , is qualitatively different from the continuous transition between the upper and lowerbranches which is characteristic of flows with small (or zero) β ∗ .Figure 8 shows results obtained with three different realizations of monoscale topographyviz., the topography illustrated in figure 2(a) and two other realizations with the monoscalespectrum of figure 2(b). The large-scale flow ¯ U is insensitive to these changes in topographicdetail; in this sense the large-scale flow is “self-averaging”. However, the location of thedrag crisis depends on differences between the three realizations: panel (b) of figure 8shows that the location of the jump from lower to upper branch is realization-dependent:the three realizations jump to the upper branch at different values of F ∗ .The case with β ∗ = 0 .
1, which corresponds a value close to realistic (cf. Table 2), doesshow a drag crisis, i.e., a discontinuous jump from the lower to the upper branch at F ∗ ≈ .
9; see figure 1. However, the eddy saturation regime, i.e., the regime in which ¯ U grows with wind stress forcing are at rate less than linear, is not nearly as pronounced asin the case with β ∗ = 1 .
38 shown in figure 8(a).4.2.
Hysteresis and multiple flow patterns
Starting with a severely truncated spectral model of the atmosphere introduced byCharney & DeVore (1979), there has been considerable interest in the possibility thattopographic form stress might result in multiple stable large-scale flow patterns whichmight explain blocked and unblocked states of atmospheric circulation. Focussing onatmospheric conditions, Tung & Rosenthal (1985) concluded that the results of low-ordertruncated models are not a reliable guide to the full nonlinear problem: although multiplestable states still exist in the full problem, these occur only in a restricted parameterrange that is not characteristic of Earth’s atmosphere.With this meteorological background in mind, it is interesting that in the oceanographicparameter regime emphasized here, we easily found multiple equilibrium solutions oneither side of the drag crisis. After increasing F beyond the crisis point, and jumpingto the upper branch, we performed additional numerical simulations by decreasing F and using initial conditions obtained from the upper-branch solutions at larger valuesof F . Thus we moved down the upper branch, past the crisis, and determined a rangeof wind stress forcing values with multiple flow patterns. Panel (c) of figure 8, with β ∗ = 1 .
38, shows that multiple states co-exist in the range 11 (cid:54) F ∗ (cid:54)
29. Note that forquasi-realistic case with β ∗ = 0 . . (cid:54) F ∗ (cid:54) . F is balanced by form stress: the example discussed in connection with figures 4 and 5is typical. On the other hand, the co-existing upper-branch solutions are steady (thatis ψ (cid:48) = U (cid:48) = 0) and nearly all of the wind stress is balanced by bottom drag so that µU/F ≈
1. 4.3.
A survey
In this section we present a suite of solutions, all with µ ∗ = 10 − . The main conclusionfrom these extensive calculations is that the behavior illustrated in figure 8 is representativeof a broad region of parameter space.4 Constantinou and Young
Figure 9: Panel (a) shows the ratio µ ¯ U /F as a function of the non-dimensional forcing, F ∗ , for seven values of β ∗ . The dashed lines are asymptotic results in (6.3). Panel (b) is adetailed view of the shaded lower part of panel (a), showing the eddy saturation regimeand the drag crisis. The dashed line is the asymptotic result in (6.5).Figure 10: (a) The index in (4.4) measures the strength of the transient eddies as afunction of the forcing F ∗ . Panel (b) is a detailed view of the shaded lower-left part ofpanel (a). The onset of transient eddies is signaled by the large jump in the fluctuationindex.Figure 11: (a) The equilibrated large-scale flow, ¯ U , scaled with β(cid:96) η as a function of thenon-dimensional forcing for various values of β ∗ . Dashed curves indicate upper branchanalytic result from section 7 (also scaled with β(cid:96) η ). (b) An expanded view of the shadedpart of panel (a) that shows the eddy saturation regime. eta-plane turbulence above monoscale topography ζ with the topographic PV η for β ∗ = 0, 0 .
10 and 1 .
38. For β ∗ = 0 the correlation corr(¯ ζ, η ) is always negative; for F ∗ = 10 − , corr(¯ ζ, η ) = − . × − . (b) Correlation of ¯ ψ with η x .Figure 9(a) shows the ratio µ ¯ U /F as a function of F ∗ for seven different values of β .The three series with β ∗ (cid:54) .
10 are “small- β ” cases in which closed geostrophic contoursfill most of the domain; the other four series, with β ∗ (cid:62) .
35, are “large- β ” cases in whichopen geostrophic contours fill most of the domain. For small values of F ∗ in figure 9(a) theflow is steady ( ψ (cid:48) = U (cid:48) = 0) and µ ¯ U /F does not change with F : this is the lower-branchrelation (4.2) in which ¯ U varies linearly with F with an effective drag coefficient µ eff . As F ∗ is increased, this steady flow becomes unstable and the strength of the transient eddyfield increases with F .Figure 9(b) shows a detailed view of the eddy saturation regime and the drag crisis.The dashed lines in the left of figures 9(a) and (b) show the analytic results derived insection 6. For the large- β cases the form stress makes a very large contribution to thelarge-scale momentum balance prior the drag crisis. We emphasize that although drag µ does not directly balance F in this regime, it does play a crucial role in producingnon-zero form stress (cid:104) ¯ ψη x (cid:105) . In all of the solutions summarized in figure 9, non-zero µ isrequired so that the flow is asymmetric upstream and downstream of topographic features;this asymmetry induces non-zero (cid:104) ¯ ψη x (cid:105) .In figure 10 we use (cid:112) U (cid:48) (cid:46) ¯ U (4.4)as an indication of the onset of the transient-eddy instability and as an index of thestrength of the transient eddies. Remarkably, the onset of the instability is roughly at F ∗ = 1 . × − for all values of β ∗ : the onset of transient eddies is the sudden increasein (4.4) by a factor of about 10 or 10 in figure 10(b). The transient eddies resultin reduction of µ ¯ U /F ; for the large- β runs, this is the eddy saturation regime. In thepresentation in figure 9(a) the eddy saturation regime is the decease in µ ¯ U /F that occursonce 0 . < F ∗ < . β ∗ ). The eddy saturation regime is terminatedby the drag-crisis jump to the upper branch where µ ¯ U /F ≈
1. This coincides withvanishing of the transients: on the upper branch the flow becomes is steady: ψ (cid:48) = U (cid:48) = 0:see figure 10(a).Figure 11 shows the eddy saturation regime that is characteristic of the three serieswith β ∗ (cid:62) .
35. Eddy saturation occurs for forcing in the range0 . (cid:47) F ∗ (cid:47)
30 ;6
Constantinou and Young in this regime the large-scale flow is limited to the relatively small range0 . β(cid:96) η (cid:47) ¯ U (cid:47) . β(cid:96) η . In anticipation of analytic results from the next section we note that in the relation above, β(cid:96) η is the speed of Rossby waves excited by this topography with typical length scale (cid:96) η .Figure 12 shows the correlations corr(¯ ζ, η ) and corr( ¯ ψ, η x ) as a function of the forcing F ∗ for three values of β ∗ . In most weakly forced cases ¯ ζ is positively correlated with η ; as the forcing F increases, ¯ ζ and η become anti-correlated. However, for β ∗ = 0 thecorrelation corr(¯ ζ, η ) is negative for all values of F : for the monoscale topography usedhere, the term (cid:104) η D¯ ζ (cid:105) , which is the only source of enstrophy in the time-average of (A 1 b )if β = 0, can be approximated as ( µ + ν/(cid:96) η ) (cid:104) ¯ ζη (cid:105) . Therefore in this case (cid:104) ¯ ζη (cid:105) must benegative (see the discussion in Appendix A).
5. A quasilinear (QL) theory
A prediction of the statistical steady state of (1.1) and (2.3) was first made by Davey(1980). In this section we present Davey’s quasilinear (QL) theory and in subsequentsections we explore its validity in various regimes documented in section 4. QL is anexploratory approximation obtained by retention of all the terms consistent with easyanalytic solution of the QGPV equation: see (5.1) below; terms hindering analytic solutionare discarded without a priori justification. We show in sections 6 and 7 that QL is ingood agreement with numerical solutions in some parameter ranges e.g., everywhere onthe upper branch and on the lower branch provided that β ∗ (cid:39)
1. With hindsight, andby comparison with the numerical solution, one can understand these QL successes aposteriori by showing that the terms discarded to reach (5.1) are, in fact, small relativeto at least some of the retained terms.Assume that the QGPV equation (2.3) has a steady solution and also neglect J( ψ, q ) =J( ψ, η ) + J( ψ, ζ ). These ad hoc approximations result in the QL equation
U ζ x + βψ x + µζ = − U η x , (5.1)in which U is determined by the steady mean flow equation F − µU − (cid:104) ψη x (cid:105) = 0 . (5.2)In (5.1) we have neglected lateral dissipation so that the dissipation is D = µ (seediscussion in section 2). Notice that the only nonlinear term in (5.1) is U ζ x . Regarding U as an unknown parameter, the solution of (5.1) is: ψ = U (cid:88) k i k x η k e i k · x µ | k | − i k x ( β − | k | U ) . (5.3)Thus the QL approximation to the form stress in (5.2) is (cid:104) ψη x (cid:105) = U (cid:88) k µk x | k | | η k | µ | k | + k x ( β − | k | U ) . (5.4)Inserting (5.4) into the large-scale momentum equation (5.2) one obtains an equation for U . This equation is a polynomial of order 2 N + 1, where N (cid:29) U . However, for the monoscale topography of figure 2, we usually findeither one real solution or three as F is varied: see figure 13(a). Only in a very limitedparameter region we find a multitude of additional real solutions: see figure 13(b). The eta-plane turbulence above monoscale topography µ ¯ U /F , as a function of forcing the F ∗ for the caseswith β ∗ = 0 .
10 and 1.38. The solid curves are the QL predictions using a single realizationto evaluate the sum in (5.4) and the dashed curves are the ensemble-average predictionsfrom (5.5); the markers indicate the numerical solution of the full nonlinear system (1.1)and (2.3). Panels (b) and (c) show a detailed view of the bottom right corner of panel(a); the resonances in the denominator of (5.4) come into play in this small region.fine-scale features evident in figure 13(b) vary greatly between different realizations ofthe topography and are irrelevant for the full nonlinear system.For the special case of isotropic monoscale topography we simplify (5.4) by convertingthe sum over k into an integral that can be evaluated analytically (see appendix B). Theresult is (cid:104) ψη x (cid:105) = µU (cid:96) η η µ (cid:96) η + (cid:0) β(cid:96) η − U (cid:1) + µ(cid:96) η (cid:113) µ (cid:96) η + (cid:0) β(cid:96) η − U (cid:1) . (5.5)Expression (5.5) is a good approximation to the sum (5.4) for the monoscale topography offigure 2(a) that has power over an annular region in wavenumber space with width ∆k L ≈
8. The dashed curves in figure 13 are obtained by solving the mean-flow equation (5.2)with form stress given by the analytic expression in (5.5); there is good agreement with thesum (5.4) except in the small regions shown in panels (b) and (c), where the resonancesof the denominator come into play. This comparison shows that the form stress producedin a single realization of random topography is self-averaging i.e., the ensemble averagein (5.5) is close to the result obtained by evaluating the sum in (5.4) using a singlerealization of the η k ’s.Figure 13 also compares the QL prediction in (5.4) and (5.5) to solutions of the fullsystem. Regarding weak forcing ( F ∗ (cid:28) µU/F for the case with β ∗ = 0 . ψ, η ) is discarded in (5.1). On the other hand, the QL approximation has somesuccess for the case with β ∗ = 1 .
38: proceeding in figure 13 from very small F ∗ , we findclose agreement till about F ∗ ≈ .
1. At that point the QL approximation departs fromthe full solution: the velocity U predicted by the QL approximation is greater than theactual velocity, meaning that the QL form stress (cid:104) ¯ ψη x (cid:105) is too small. This failure of theQL approximation is clearly associated with the linear instability of the steady solutionand the development of transient eddies: the nonlinear results for the β ∗ = 1 .
38 case infigure 13(a) first depart from the QL approximation when the index (4.4) signals the onset8
Constantinou and Young of unsteady flow. This failure of the QL theory due to transient eddies will be furtherdiscussed in section 8. For strong forcing ( F ∗ (cid:29) β ∗ (cid:39)
1, even provides a goodquantitative prediction of ¯ U .
6. The weakly forced regime, F ∗ (cid:28) In this section we consider the weakly forced case. In figures 8 and 9 this regime ischaracterized by the “effective drag” µ eff in (4.2). Our main goal here is to determine µ eff in the weakly forced regime.Reducing the strength of the forcing F ∗ to zero is equivalent to taking a limit in whichthe system is linear. This weakly forced flow is then steady, ψ (cid:48) = 0, and terms which arequadratic in the flow fields U and ψ , namely U ζ x and J ( ψ, ζ ), are negligible. Thus in thelimit F ∗ → ψ, η ) + βψ x + µζ = − U η x . (6.1)When compared to the QL approximation (5.1) we see that (6.1) contains the additionallinear term J ( ψ, η ) and does not contain the non-linear term U ζ x . We regard the righthand side of the linear equation (6.1) as forcing that generates the streamfunction ψ .6.1. The case with either µ ∗ (cid:29) or β ∗ (cid:29) (cid:96) η , the ratio of the terms on the left of (6.1) is: βψ x (cid:46) J ( ψ, η ) = O ( β ∗ ) and µζ (cid:46) J ( ψ, η ) = O ( µ ∗ ) . (6.2)If µ ∗ (cid:29) β ∗ (cid:29) ψ, η ) is negligible relative to one, or both, of the other twoterms on the left hand side of (6.1). In that case, one can neglect the Jacobian in (6.1) andadapt the QL expression (5.5) to determine the effective drag of monoscale topography as µ eff = µ + µη (cid:96) η µ (cid:96) η + β (cid:96) η + µ(cid:96) η (cid:113) µ (cid:96) η + β (cid:96) η . (6.3)In simplifying the QL expression (5.5) to the linear result (6.3) we have neglected U relative to either β(cid:96) η or µ(cid:96) η : this simplification is appropriate in the limit F ∗ →
0. Theexpression in (6.3) is accurate within the shaded region in figure 14. The dashed lines infigures 8 and 9(a) that correspond to the series with β ∗ (cid:62) .
35 indicate the approximation¯ U ≈ F/µ eff with µ eff in (6.3).6.2. The thermal analogy — the case with µ ∗ (cid:47) and β ∗ (cid:47) µ ∗ and β ∗ are order one or less the term J( ψ, η ) in (6.1) cannot be neglected.As a result of this Jacobian, the weakly forced regime cannot be recovered as a specialcase of the QL approximation. In this interesting case we rewrite (6.1) asJ ( η + βy, ψ − U y ) = µ ∇ ψ , (6.4)and rely on intuition based on the “thermal analogy”. To apply the analogy we regard η + βy as an effective steady streamfunction advecting a passive scalar ψ − U y . The eta-plane turbulence above monoscale topography β ∗ < µ ∗ < J ( ψ, η ) cannot be neglected. The expressionsshow the behavior of µ eff ∗ = µ eff /η rms in each parameter region.planetary vorticity gradient β is analogous to a large-scale zonal flow − β and the large-scale flow U is analogous to a large-scale tracer gradient; the drag µ is equivalent tothe diffusivity of the scalar. The form stress (cid:104) ψη x (cid:105) is analogous to the meridional fluxof tracer ψ by the meridional velocity η x . Usually in the passive-scalar problem thelarge-scale tracer gradient U is imposed and the main goal is to determine the flux (cid:104) ψη x (cid:105) (equivalently the Nusselt number). But here, U is unknown and must be determined bysatisfying the steady version of the large-scale momentum equation (5.2). Geostrophiccontours are equivalent to streamlines in the thermal analogy.With the thermal analogy, we can import results from the passive-scalar problem. Forexample, in the passive-scalar problem, at large P´eclet number, the scalar is uniformwithin closed streamlines (Batchelor 1956; Rhines & Young 1983). The analog of this“Prandtl–Batchelor theorem” is that in the limit µ ∗ → ψ − U y ,is constant within any closed geostrophic contour i.e., all parts of the domain containedwithin closed geostrophic contours are stagnant; see also the paper by Ingersoll (1969).This “Prandtl–Batchelor theorem” explains the result in figure 15, which shows a weaklyforced, small-drag solution with β ∗ = 0. The domain is packed with stagnant eddies(constant ψ − U y ) separated by thin boundary layers. The “terraced hillside” in figure 15(c)is even more striking than in figure 5: the solution in figure 5 has transient eddies resultinga blurring of the terraced structure. The weakly-forced solution in figure 15 is steadyand the thickness of the steps between the terraces is limited only by the small drag, µ ∗ = 5 × − .Isichenko et al. (1989) and Gruzinov et al. (1990) discuss the effective diffusivity of apassive scalar due to advection by a steady monoscale streamfunction. Using a scalingargument, Isichenko et al. (1989) show that in the high P´eclet number limit the effectivediffusivity of a steady monoscale flow is D eff = DP / , where D is the small moleculardiffusivity and P is the P´eclet number; the exponent 10 /
13 relies on critical exponentsdetermined by percolation theory. Applying Isichenko’s passive-scalar results to the β = 0form-stress problem we obtain the scaling µ eff = cµ / η / , and µ ¯ UF = 1 c (cid:18) µη rms (cid:19) / , (6.5)where c is a dimensionless constant. Numerical solutions of (1.1) and (2.3) summarized0 Constantinou and Young
Figure 15: Snapshots of flow fields for weakly forced simulations at F ∗ = 10 − withdissipation µ ∗ = 5 × − and β ∗ = 0. (a) The total velocity magnitude | U | . Theflow is restricted to a boundary layer around the dashed η = 0 contour. (b) Thetotal streamfunction ψ − U y for the solution in panel (a). (c) Surface plot of the totalstreamfunction, ψ − U y . The terraced hillside structure is apparent. (In panels (a) and (b)only one quarter of the domain is shown.)Figure 16: The large-scale flow for weakly forced solutions ( F ∗ = 10 − ) with β = 0 as afunction of µ ∗ . The dashed line shows the scaling law (6.5) with c = 1.in figure 16 confirm this remarkable “ten-thirteenths” scaling and show that the constant c in (6.5) is close to one. The dashed lines in figures 8 and 9(b), corresponding to thesolution suites with β ∗ (cid:54) .
1, show the scaling law (6.5) with c = 1.To summarize: the weakly forced regime is divided into the easy large- β case, in which µ eff in (6.3) applies, and the more difficult case with small or zero β . In the difficult case, eta-plane turbulence above monoscale topography β = 0 scaling lawin (6.5) is the main results in this case. The value of β ∗ separating these two regimesin the schematic of figure 14 is identified with the β below which (6.3) underestimates µU/F compared to (6.5). For the topography used in this work, and taking c = 1 in (6.5),this is β ∗ = 0 .
17. (If we choose c = 0 . β ∗ = 0 . β = 0 result in (6.5) works better than µ eff in (6.3) for β ∗ < .
35: see figure 9.
7. The strongly forced regime, F ∗ (cid:29) We turn now to the upper branch, i.e., to the flow beyond the drag crisis. In thisstrongly forced regime the flow is steady: ψ (cid:48) = U (cid:48) = 0 and the QL theory gives goodresults for all values of β ∗ .The solutions in figure 11(a) show that on the upper branch the large-scale flow ¯ U is much faster than the phase speed of Rossby waves excited by the topography, i.e.,¯ U (cid:29) β(cid:96) η . Therefore, we can simplify the QL approximation in (5.5) by neglecting termssmaller than β(cid:96) η /U . This gives: (cid:104) ψη x (cid:105) = µη (cid:96) η U + O ( β(cid:96) η /U ) . (7.1)This result is independent of β up to O ( β(cid:96) η /U ) . Using (7.1) in the large-scale zonalmomentum equation (5.2), while keeping in mind that 0 (cid:54) U (cid:54) F/µ , we solve a quadraticequation for U to obtain: µUF = 12 + (cid:115) − F ∗ . (7.2)The location of the drag crisis depends on β , and on details of the topography thatare beyond the reach of the QL approximation. But once the solution is on the upperbranch these complications are irrelevant e.g., (7.2) does not contain β . The dashed curvein figure 17 compares (7.2) to numerical solutions of the full system and shows closeagreement.We get further intuition about the structure of the upper-branch flow through the QLequation (5.1). For large U we have a two-term balance in (5.1) that gives ¯ ζ ≈ − η , sothat ¯ q is O ( (cid:96) η η − U − ). Figure 12(a) shows that on the upper branch the correlation of ¯ ζ with η is close to − ζ ≈ η .2 Constantinou and Young
8. Intermediate forcing: eddy saturation and the drag crisis
In sections 6 and 7 we discussed limiting cases with small and large forcing respectively.In both these limits the solution is steady i.e., there are no transient eddies. We now turnto the more complicated situation with forcing of intermediate strength. In this regimethe solution has transient eddies and numerical solution shows that these produce dragthat is additional to the QL prediction (see figure 13 and related discussion). The eddysaturation regime, in which U is insensitive to large changes in F ∗ (see figure 11), is alsocharacterized by forcing of intermediate strength: the solution described in section 3.5is an example. Thus a goal is to better understand the eddy saturation regime and itstermination by the drag crisis.8.1. Eddy saturation regime
As wind stress increases transient eddies emerge: in figure 10 this instability of thesteady solution occurs very roughly at F ∗ = 1 . × − for all values of β . The powerintegrals in appendix A show that the transient eddies gain kinetic energy from thestanding eddies ¯ ψ through the conversion term (cid:104) ¯ ψ ∇ · E (cid:105) , where E def = U (cid:48) q (cid:48) , (8.1)is the time-averaged eddy PV flux. The conclusions from appendix A are summarized infigure 21 by showing the energy and enstrophy transfers among the four flow components¯ U , U (cid:48) , ¯ ψ and ψ (cid:48) .Figure 18 compares the numerical solutions of (1.1) and (2.3) with the prediction ofthe QL approximation (asterisks ∗ versus the solid QL curve) for the case with β ∗ = 1 . (i) QL assumes steady flow and has no way of incorporating the effect of transient eddieson the time-mean flow and (ii)
QL neglects the term J( ¯ ψ, ¯ q ).We address these points by following Rhines & Young (1982) and approximating theeffect of the transient eddies as PV diffusion: ∇ · E ≈ − κ eff ∇ ¯ q . (8.2)In the discussion surrounding (A 6), we determine κ eff using the time-mean eddy energypower integral (A 5 b ). According to this diagnosis, the PV diffusivity is κ eff = µ (cid:104)| ∇ ψ (cid:48) | (cid:105) (cid:46) (cid:10) ¯ ζ ¯ q (cid:11) . (8.3)Figure 19(a) shows κ eff in (8.3) for the solution suite with β ∗ = 1 . κ eff , rather thandetermining it diagnostically from the energy power integral as in (8.3).With κ eff in hand, we can revisit the QL theory and ask for its prediction when theterm κ eff ∇ q is added on the right hand side of (5.1). This way we include the effect ofthe transients on the time-mean flow but do not include the effect of the term J( ¯ ψ, ¯ q ).The QL prediction is only slightly improved — see the dash-dotted curve in figure 18.To include also the effect of the term J( ¯ ψ, ¯ q ) we obtain solutions of (1.1) and (2.3) withadded PV diffusion in (2.3) with κ eff as in figure 19(a). We find that the strength of the eta-plane turbulence above monoscale topography β ∗ = 1 .
38 (shaded). Asterisks ∗ indicatenumerical solutions of (1.1) and (2.3). Circles ◦ show the numerical solutions of (1.1)and (2.3) with the added PV diffusion, κ eff ∇ q . The solid curve is the QL prediction (5.4)and the dashed-dot curve is the QL prediction with added PV diffusion.Figure 19: (a) The effective PV diffusivity, κ eff , diagnosed from (A 6 b ) for the seriesof solutions with β ∗ = 1 .
38. (b) The correlation corr( ¯ ψ, η x ) for this series of solutions(asterisks ∗ ) and for the solutions with parameterized transient eddies (circles ◦ ). (c) Sameas panel (b) but showing the strength of the transient eddies using the index (4.4). (c)Same as panel (b), but showing the energy of the standing eddies ¯ ψ . Also shown are thepower laws F ∗ and F ∗ as dashed black lines.transient eddies is dramatically reduced: see figure 19(c). Thus the approximation (8.2)with the PV diffusivity supplied by (8.3) is self-consistent in the sense that we do notboth resolve and parameterize transient eddies. Moreover, the large-scale flow ¯ U withparameterized transient eddies is in much closer agreement with ¯ U from the solutionswith transient eddies — see figure 18. This striking quantitative agreement as we vary F ∗ shows that at least in the case with β ∗ = 1 .
38 the transient eddies act as PV diffusion onthe time-mean flow.Thus we conclude, that in addition to β , the main physical mechanisms operating in theeddy saturation regime are PV diffusion via the transient eddies and the mean advectionof mean PV i.e., the term J( ¯ ψ, ¯ q ).4 Constantinou and Young
There are two remarkable aspects of this success. First, it is important to use κ eff ∇ (¯ ζ + η )in (8.2); if one uses only κ eff ∇ ¯ ζ then the agreement in figure 18 is degraded. Second,PV diffusion does not decrease the amplitude of the standing eddies: see figure 19(d).Furthermore, PV diffusion quantitatively captures corr( ¯ ψ, η x ): see figure 19(b).Unfortunately, the success of the PV diffusion parameterization does not extend tocases with closed geostrophic contours (small β ∗ ), such as β ∗ = 0 .
10. For small β ∗ the flowis strongly affected by the detailed structure of the topography. The solution described insection 3.4 shows that flow is channeled into a few streams and thus a parametrizationthat does account the actual structure of the topography is, probably, doomed to fail. Infact, for β ∗ = 0 . κ eff diagnosed according to (8.3) is negative because (cid:104) ¯ ζ ¯ q (cid:105) < β ∗ so that the geometry is dominated by open geostrophic contours.In the context of baroclinic models, eddy saturation is not captured by standardparameterizations of transient baroclinic eddies (Hallberg & Gnanadesikan 2001). Onlyvery recently have Mak et al. (2017) proposed a parameterization of baroclinic turbulencethat successfully produces baroclinic eddy saturation. Thus the success of (8.2) in thebarotropic context, even though it depends on diagnosis of κ eff via (8.3), is significant.8.2. Drag crisis
In this section we provide some further insight into the drag crisis. We argue that therequirement of enstrophy balance among the flow components leads to a transition fromthe lower to the upper branch as wind stress forcing increases. We make this argumentby constructing lower bounds on the large-scale flow ¯ U based on energy and enstrophypower integrals.We consider a “test streamfunction” that is efficient at producing form stress: ψ test = αη x , (8.4)with α a positive constant to be determined by satisfying either the energy or the enstrophypower integrals from appendix A. A maximum form stress corresponds to a minimumlarge-scale flow ¯ U min , which in turn can be determined by substituting (8.4) into thetime-mean large-scale flow equation (5.2): µ ¯ U min = F − α (cid:104) η x (cid:105) . (8.5)We can determine α so that the eddy energy power integral (A 5 b )+(A 5 a ) is satisfied:0 = ¯ U min α (cid:104) η x (cid:105) − µα (cid:104)| ∇ η x | (cid:105) − να (cid:104) ( ∇ η x ) (cid:105) . (8.6)The averages above are evaluated using properties of monoscale topography, e.g. (cid:104) ( ∇ η x ) (cid:105) = η (cid:96) − η . Solving (8.5) and (8.6) for α and ¯ U min we obtain a lower boundon the large-scale flow based on the energy constraint,¯ U (cid:62) ¯ U min E def = Fµ (cid:20) η µ + ν/(cid:96) η ) (cid:21) − . (8.7)Alternatively, one can determine α and ¯ U min by satisfying the eddy enstrophy powerintegral (A 8 a )+(A 8 b ). This leads to a second bound,¯ U (cid:62) ¯ U min Q def = Fµ (cid:34) − βη (cid:96) η µ + ν/(cid:96) η ) F (cid:35) . (8.8) eta-plane turbulence above monoscale topography ∗ ) and the QLprediction (5.4) (solid curve). The dashed line shows the lower bound (8.9).Thus U (cid:62) max (cid:0) ¯ U min E , ¯ U min Q (cid:1) . (8.9)The test function in (8.4) does not closely resemble the realized flow so the bound aboveis not tight. Nonetheless it does capture some qualitative properties of the turbulentsolutions.(Using a more elaborate test function with two parameters one can satisfy both theenergy and enstrophy power integrals simultaneously and obtain a single bound. However,the calculation is much longer and the result is not much better than the relativelysimple (8.9).)The lower bound (8.9) is shown in figure 20 for the case with β ∗ = 1 .
38 togetherwith the numerical solution of the full nonlinear equations (1.1) and (2.3) and the QLprediction. Although it cannot be clearly seen, the energy bound ¯ U min E does not allow¯ U to vanish completely, e.g. for the µ ∗ = 10 − we have that µ ¯ U min E /F = 2 × − . Onthe other hand, the dominance of the enstrophy bound ¯ U min Q at high forcing explains theoccurrence of the drag crisis: the enstrophy power integral requires that the large-scaleflow transitions from the lower to the upper branch as F ∗ is increased beyond a certainvalue.These bounds provide a qualitative explanation for the existence the drag crisis. Thecritical forcing predicted by the enstrophy bound dominance overestimates the actualvalue of the drag crisis. For example, for the case with β ∗ = 1 .
38 shown in figure 20,¯ U min Q becomes the lower bound at a value of F ∗ that is about 240 times larger than theactual drag crisis point. We have no reason to expect these bounds to be tight: theydo not depend on the actual structure of topography itself but only on gross statisticalproperties, e.g. η rms , (cid:96) η , L η . For example, a sinusoidal topography in the form of η = √ η rms cos ( x/(cid:96) η ) (8.10)has identical statistical properties as the random monoscale topography used in this paperand, therefore, imposes the same bounds. But with the topography in (8.10) there isa laminar solution with ψ y = 0 and, as a result, also J( ψ, q ) = 0. In this case the QLsolution (5.3) is an exact solution of the full nonlinear equations (1.1) and (2.3) and thebound (8.9) is tight.
9. Discussion and conclusion
The main new results in this work are illustrated by the two limiting cases described insections 3.4 and 3.5. The case in section 3.4, with β ∗ = 0 .
1, is realistic in that ballparkestimates indicate that topographic PV will overpower βy to produce closed geostrophic6 Constantinou and Young contours almost everywhere. The topographically blocked flow then consists of close-packed stagnant “dead zones” separated by narrow jets. Dead zones are particularlynotable in the steady solution shown in figure 15. But they are also clear in the unsteadysolution of figure 5. There is no eddy saturation in this topographically blocked regime: thelarge-scale flow ¯ U increases roughly linearly with F till the drag-crisis jump to the upperbranch. Because of the topographic partitioning into dead zones, the large-scale time-meanflow ¯ U is an unoccupied mean: in most of the domain there are weak recirculating eddies.The complementary limit, illustrated by the case in section 3.5 with β ∗ = 1 .
38, is whenall of the geostrophic contours are open. In this limit we find that: (i) the large-scaleflow ¯ U is insensitive to changes in F ; (ii) the kinetic energy of the transient eddiesincreases linearly with F : see figure 19(d). Both (i) and (ii) are defining symptoms ofthe eddy-saturation phenomenon documented in eddy resolving Southern-Ocean models.Constantinou (2017) identifies a third symptom common to barotropic and barocliniceddy saturation: increasing the Ekman drag coefficient µ increases the large-scale meanflow (Marshall et al. µ can berationalized by arguing that the main effect of increasing Ekman drag is to damp thetransient eddies responsible for producing κ eff in (8.3). Section 8 shows that decreasingthe strength of the transient eddies, and their associated effective PV diffusivity, alsodecreases the topographic form stress and thus increases the transport.The two limiting cases described above are not quirks of the monoscale topographyin figure 2: using a multiscale topography with a k − power spectral density, we findsimilar qualitative behaviors (not shown here), including eddy saturation in the limit ofsection 3.5. Moreover, the main controlling factor for eddy saturation in this barotropicmodel is whether the geostrophic contours are open or closed — the numerical value of β ∗ is important only in so far as β ∗ determines whether the geostrophic contours areopen or closed. For example, the “unidirectional” topography in (8.10) always has opengeostrophic contours; using this sinusoidal topography Constantinou (2017) shows thatthere is barotropic eddy saturation with β ∗ as low as 0 .
1. Thus it is the structure ofthe geostophic contours, rather than the numerical value of β ∗ , that is decisive as far asbarotropic eddy saturation is concerned.The explanation of baroclinic eddy saturation, starting with Straub (1993), is thatisopycnal slope has a hard upper limit set by the marginal condition for baroclinicinstability. As the strength of the wind is increased from small values, the isopycnal slopeinitially increases and so does the associated “thermal-wind transport”. (The thermal-windtransport is diagnosed from the density field by integrating the thermal-wind relationupwards from a level of no motion at the bottom.) However, once the isopycnal slopereaches the marginal condition for baroclinic instability, further increases in slope, and inthermal-wind transport, are no longer possible. At the margin of baroclinic instability, theunstable flow can easily make more eddies to counteract further wind-driven steepeningof isopycnal slope. This is the standard explanation of baroclinic eddy saturation in whichthe transport (approximated by the thermal-wind transport) is unchanging, while thestrength of the transient eddies increases linearly with wind stress.Direct comparison of the barotropic model with baroclinic Southern-Ocean models,and with the Southern Ocean itself, is difficult and probably not worthwhile exceptfor gross parameter estimation as in table 2. However several qualitative points shouldbe mentioned. Most strikingly, we find that eddy saturation occurs without baroclinicinstability and without thermal-wind transport. This finding challenges the standardexplanation of eddy saturation in terms of the marginal condition for baroclinic instability.Nonetheless, the onset of transient barotropic eddies, shown in figure 10(b), is also theonset of barotropic eddy saturation. This barotropic-topographic instability is the source eta-plane turbulence above monoscale topography κ eff in (8.3). Thus one can speculate that barotropiceddy saturation also involves a flow remaining close to a marginal stability condition.Substantiating this claim, and clarifying the connection between barotropic and barocliniceddy saturation, requires better characterization the barotropic-topographic instabilityand also of the effect of small-scale topography on baroclinic instability. The latter pointis fundamental: in a baroclinic flow topographically blocked geostrophic contours in thedeep layers co-exist with open contours in shallower layers. The marginal condition forbaroclinic instability in this circumstance is not well understood. The issue is furtherconfused because transient eddies generated by barotropic-topographic instability havethe same length scale as the topography, which can be close to the deformation lengthscale of baroclinic eddies.Further evidence for the importance of barotropic processes in establishing eddysaturation is provided by several Southern-Ocean type models. Abernathey & Cessi (2014)showed that an isolated ridge results in localized baroclinic instability over the ridge anda downstream barotropic standing wave. Relative to the flat-bottom case, transient eddiesare weak in most of the domain and the thermocline is shallow with small slopes; seeThompson & Naveira Garabato (2014) for further discussion of the role of barotropicstanding waves in setting the momentum balance and transport in the Southern Ocean.In a study of channel spin-up, Ward & Hogg (2011) showed that a suddenly imposedwind stress is balanced by topographic form stress within two or three weeks. This fastbalance is achieved by barotropic pressure gradients associated with sea-surface height.Interior equilibration, involving transmission of momentum by interfacial form stresses andbaroclinic instability (Johnson & Bryden 1989), takes about 10 years to establish. Windstress on the Southern Ocean is never steady and thus fast barotropic eddy saturationmay be as important as slow baroclinic eddy saturation.The authors acknowledge fruitful discussions with N. A. Bakas, P. Cessi, T. D. Drivas,B. F. Farrell, G. R. Flierl, A. McC. Hogg, P. J. Ioannou and A. F. Thompson. Commentsfrom three anonymous reviewers greatly improved the paper. NCC was supported bythe NOAA Climate & Global Change Postdoctoral Fellowship Program, administered byUCAR’s Cooperative Programs for the Advancement of Earth System Sciences. WRY wassupported by the National Science Foundation under OCE-1357047. The Matlab codeused in this paper is available at the github repository: https://github.com/navidcy/QG_ACC_1layer . Appendix A. Energy and enstrophy power integrals and balances
In this appendix we derive energy and enstrophy power integrals as well as the time-averaged energy and enstrophy balances for each of the four flow components: ¯ U , U (cid:48) , ¯ ψ and ψ (cid:48) .The energy and enstrophy of the flow are defined in (2.5). From (1.1) and (2.3) we findthat: d E d t = F U − µU − (cid:10) µ | ∇ ψ | + νζ (cid:11) , (A 1 a )d Q d t = F β − (cid:104) η D ζ (cid:105) − µβU − (cid:10) µζ + ν | ∇ ζ | (cid:11) . (A 1 b )The rate of working by the wind stress, F U , appears on the right of (A 1 a ): because F isconstant the energy injection varies directly with the large-scale mean flow U ( t ). On the8 Constantinou and Young other hand, the main enstrophy injection rate on the right of (A 1 b ) is fixed and equalto F β . The subsidiary enstrophy source (cid:104) η D ζ (cid:105) becomes important if β is small relativeto the gradients of the topographic PV; in the special case β = 0, (cid:104) η D ζ (cid:105) is the onlyenstrophy source.Following (2.6), we represent all flow fields as a time-mean plus a transient; note that¯ q = ¯ ζ + η and q (cid:48) = ζ (cid:48) . Equations (1.1) and (2.3) decompose into:J( ¯ ψ − ¯ U y, ¯ q + βy ) + ∇ · E + D¯ ζ = 0 , (A 2 a ) q (cid:48) t + J( ψ (cid:48) − U (cid:48) y, ¯ q + βy ) + J( ¯ ψ − ¯ U y, q (cid:48) ) + ∇ · ( E (cid:48)(cid:48) − E ) + D ζ (cid:48) = 0 , (A 2 b ) F − µ ¯ U − (cid:104) ¯ ψη x (cid:105) = 0 , (A 2 c ) U (cid:48) t = − µU (cid:48) − (cid:104) ψ (cid:48) η x (cid:105) , (A 2 d )where the eddy PV fluxes are E (cid:48)(cid:48) def = U (cid:48) q (cid:48) and E def = U (cid:48) q (cid:48) .A.1. Energy and enstrophy balances
Following the definitions in (2.5), the energy of each flow component is: E ¯ U = ¯ U , E U (cid:48) = U (cid:48) , E ¯ ψ = (cid:104)| ∇ ¯ ψ | (cid:105) and E ψ (cid:48) = (cid:104)| ∇ ψ (cid:48) | (cid:105) . (A 3)Thus the total energy of the large-scale flow and of the eddies is: E U = E ¯ U + E U (cid:48) + ¯ U U (cid:48) and E ψ = E ¯ ψ + E ψ (cid:48) + (cid:104) ∇ ¯ ψ · ∇ ψ (cid:48) (cid:105) . (A 4)The cross-terms above are removed by time-averaging.The time-mean energy balances for each flow component are obtained by manipulationsof (A 2) as follows: E ¯ ψ : (cid:104)− ¯ ψ × (A 2 a ) (cid:105) ⇒ U (cid:104) ¯ ψη x (cid:105) + (cid:104) ¯ ψ ∇ · E (cid:105) + (cid:104) ¯ ψ D¯ ζ (cid:105) , (A 5 a ) E ψ (cid:48) : (cid:104)− ψ (cid:48) × (A 2 b ) (cid:105) ⇒ U (cid:48) (cid:104) ψ (cid:48) η x (cid:105) − (cid:104) ¯ ψ ∇ · E (cid:105) + (cid:104) ψ (cid:48) D ζ (cid:48) (cid:105) , (A 5 b ) E ¯ U : ¯ U × (A 2 c ) ⇒ F ¯ U − µ ¯ U − ¯ U (cid:104) ¯ ψη x (cid:105) , (A 5 c ) E U (cid:48) : U (cid:48) × (A 2 d ) ⇒ − µU (cid:48) − U (cid:48) (cid:104) ψ (cid:48) η x (cid:105) . (A 5 d )Summing (A 5 d ) and (A 5 c ) we obtain the energy power integral for the total (standingplus transient) large-scale flow. Summing (A 5 b ) and (A 5 a ) the conversion term (cid:104) ¯ ψ ∇ · E (cid:105) cancels and we obtain the energy power integral (2.7) for the total (standing plus transient)eddy field.The time-mean of the energy integral in (A 1 a ) is the sum of equations (A 5). Note thatfrom (A 5 d ) we have that U (cid:48) (cid:104) ψ (cid:48) η x (cid:105) < b ) we infer that (cid:104) ¯ ψ ∇ · E (cid:105) < ∇ · E as PV diffusion (8.2). The effective PV diffusivity κ eff can be diagnosed by requiring that the time-mean eddy energy balance (A 5 a ) andthe transient eddy flow energy balance (A 5 b ) are satisfied. according to this requirementgives: κ eff = (cid:0) ¯ U (cid:104) ¯ ψη x (cid:105) + (cid:104) ¯ ψ D¯ ζ (cid:105) (cid:1) (cid:46) (cid:10) ¯ ζ ¯ q (cid:11) (A 6 a )= − (cid:0) (cid:104) ψ (cid:48) D ζ (cid:48) (cid:105) + U (cid:48) (cid:104) ψ (cid:48) η x (cid:105) (cid:1) (cid:46) (cid:10) ¯ ζ ¯ q (cid:11) . (A 6 b )In (A 6 a ) the terms ¯ U (cid:104) ¯ ψη x (cid:105) and (cid:104) ¯ ψ D¯ ζ (cid:105) are of opposite sign; the magnitude of the formeris generally much larger than that of the latter. In (A 6 b ), the term U (cid:48) (cid:104) ψ (cid:48) η x (cid:105) is negligiblecompared to (cid:104) ψ (cid:48) D ζ (cid:48) (cid:105) . Neglecting the small term, and using D ζ (cid:48) = µζ (cid:48) , we simplify (A 6 b )to obtain the expression for κ eff in (8.3). eta-plane turbulence above monoscale topography U , the standing eddies ¯ ψ , and the corresponding transientcomponents U (cid:48) and ψ (cid:48) .The enstrophy of each flow component is: Q ¯ U = β ¯ U , Q U (cid:48) = βU (cid:48) , Q ¯ ψ = (cid:104) ¯ q (cid:105) and Q ψ (cid:48) = (cid:104) q (cid:48) (cid:105) , (A 7)The transient large-scale flow has by definition Q U (cid:48) = 0. The enstrophy power integralsfollow by manipulations similar to those in (A 5): Q ¯ ψ : (cid:104) ¯ q × (A 2 a ) (cid:105) ⇒ β (cid:104) ¯ ψη x (cid:105) − (cid:104) ¯ q ∇ · E (cid:105) − (cid:104) ¯ ζ D¯ ζ (cid:105) − (cid:104) η D¯ ζ (cid:105) , (A 8 a ) Q ψ (cid:48) : (cid:104) q (cid:48) × (A 2 b ) (cid:105) ⇒ (cid:104) ¯ q ∇ · E (cid:105) − (cid:104) ζ (cid:48) D ζ (cid:48) (cid:105) , (A 8 b ) Q ¯ U : (cid:104) β × (A 2 c ) (cid:105) ⇒ F β − µβ ¯ U − β (cid:104) ¯ ψη x (cid:105) . (A 8 c )The time-mean of the enstrophy integral in (A 1 b ) is the sum of equations (A 8).Equation (A 8 b ) implies that (cid:104) ¯ q ∇ · E (cid:105) >
0; the term (cid:104) η D¯ ζ (cid:105) in (A 8 a ) can have either sign.The enstrophy power integrals (A 8) are summarized in figure 21(b). Appendix B. Form stress for isotropic topography
For the case of isotropic topography analytic progress follows to the QL expression forthe form stress by converting the sum over k in (5.4) into an integral: (cid:104) ψη x (cid:105) = U (cid:90) µk x | k | | ˆ η ( k ) | µ | k | + k x ( β − | k | U ) d k , (B 1)where ˆ η ( k ) def = (cid:82) η ( x )e − i k · x d x . Now assume that the topography is isotropic, i.e. itspower spectral density S ( k ) is only a function of the total wavenumber k = | k | andas S ( k ) = 2 π k | ˆ η ( k ) | , so that η = (cid:82) S ( k ) d k . In this case we further simplify theintegral (B 1) using polar coordinates ( k x , k y ) = k (cos θ , sin θ ): (cid:104) ψη x (cid:105) = U π µ (cid:90) ∞ S ( k ) (cid:73) cos θ ξ cos θ d θ d k , (B 2)where ξ def = (cid:0) β − k U (cid:1) / ( µk ) >
0. The θ -integral above is evaluated analytically so that (cid:104) ψη x (cid:105) = µU (cid:90) ∞ k S ( k ) d kµ k + ( β − k U ) + µk (cid:113) µ k + ( β − k U ) . (B 3)For the special case of idealized monoscale topography: S ( k ) = η δ (cid:0) k − (cid:96) − η (cid:1) , the k -integral in (B 3) can be evaluated in closed form. In that case (B 3) reduces to (5.5).0 Constantinou and Young
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