First published on: 06 February 2011

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1 This article was downloaded by: [Plotka, Hanna] On: 7 February 211 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Geophysical & Astrophysical Fluid Dynamics Publication details, including instructions for authors and subscription information: Flow-topography interactions in shallow-water turbulence Hanna Płotka a ; David G. Dritschel a a School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, UK First published on: 6 February 211 To cite this Article Płotka, Hanna and Dritschel, David G.(211) 'Flow-topography interactions in shallow-water turbulence', Geophysical & Astrophysical Fluid Dynamics,, First published on: 6 February 211 (ifirst) To link to this Article: DOI: 1.18/ URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Geophysical and Astrophysical Fluid Dynamics 211, 1 22, ifirst Flow topography interactions in shallow-water turbulence HANNA PLOTKA* and DAVID G. DRITSCHEL School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, KY16 9SS, UK (Received 3 October 29; in final form 14 September 21) Topography and external time-varying flows such as tides can have a profound influence on atmospheric and oceanic fluid motion. The effects of topography on turbulence have been little studied, and until recently those of tides have been overlooked. Our lack of understanding of these effects is a serious impediment to predicting flows both in the atmosphere and in the oceans. As a first step toward modelling realistic tidal oscillations, this study focuses on how an initially balanced flow containing a distribution of potential vorticity anomalies develops and becomes unbalanced (by emitting gravity waves) as a result of interactions with topography in the presence of an oscillating flow, here in the form of pure inertial oscillations (at the Coriolis frequency). We investigate a large parameter space spanned by the Burger number, the maximum amplitude of topography, and the amplitude of the flow oscillations. Comparisons are made with a time-independent and spatially-homogeneous steady flow in order to better understand the role played by flow oscillations. The most persistent large-amplitude gravity wave activity occurs for large-amplitude flow oscillations and for large topography. The latter produces large potential vorticity and depth anomalies which remain locked to the topography, thereby allowing gravity waves to be continually generated. By contrast, for a steady flow, large potential vorticity and depth anomalies are advected away from topographic features and so gravity waves are only generated at early times. Keywords: Inertial oscillations; Topography; Tides; Potential vorticity; Balance 1. Introduction Two main types of fluid motions occur in the Earth s oceans and atmosphere. The first, dominant, balanced component is associated with slowly-spinning vortices or eddies, while the second, weaker, unbalanced component is associated with higher-frequency inertia gravity waves. The dominant balanced component is entirely controlled by a quasi-materially advected scalar, potential vorticity (PV), from which all other dynamical fields may be deduced through a process called PV-inversion (see, e.g. Viu dez and Dritschel 26, McIntyre 28 and references therein). The remaining unbalanced component may be said to be gravity waves. The nonlinearity of the governing mathematical equations makes this wave vortex decomposition difficult *Corresponding author. hanna@mcs.st-and.ac.uk Geophysical and Astrophysical Fluid Dynamics ISSN print/issn online ß 211 Taylor & Francis DOI: 1.18/

3 2 H. Piotka and D. G. Dritschel to define precisely, but accurate practical procedures have been developed (see the above references). In this study we examine unbalanced gravity waves (here, pure inertial oscillations) and balanced vortical motions in part generated by topography, an interaction which may be relevant to the dynamics of the ocean. Specifically, we concentrate on how PV anomalies are locked to topography. The external flow considered here, consisting of pure inertial oscillations, may be regarded as a first step toward considering realistic tidal oscillations (Webster 1968). Inertial oscillations are the simplest way in which tides over topography can be modelled in the doubly-periodic domain considered here, and are the only modes which can be decoupled from the spatially-varying part of the flow. While real tides typically have frequencies higher than f, these are non-uniform in space and cannot be handled straightforwardly (they would require explicit forcing to maintain). Topography also plays an important role in the large-scale dynamics of the Earth s oceans and atmosphere. Both PV anomalies and gravity waves (GWs) can be generated as a result of flows interacting with it (see, e.g. Scha r and Smith 1993, Aebischer and Scha r 1998). Baines (1995, p. 2) points out that our lack of understanding of how topography affects atmospheric flows is a significant impediment to improvements in weather prediction. He also notes that surface drag contributes up to 5% of the total drag in the atmosphere. A similar thing can be said about the oceans (Johnson and Vilenski 25). While numerous studies have examined the effects of topography on the propagation and evolution of eddies, this has been done mainly for isolated topography or infinitely long ridges (e.g. Holloway 1987, McDonald 1998, Aelbrecht et al. 1999, Dewar 22, Hinds et al. 27), and without tidal or inertial oscillations. In fact, the majority of work on the flow topography interaction problem examines not the locking and development of eddies, but rather the mechanisms for the generation of internal waves (e.g. Gerkema and Zimmerman 1995, Llewellyn Smith and Young 22, Zarroug et al. 21). Here we focus on how an initially balanced flow evolves through time and becomes unbalanced as a result of interactions between topography and an oscillating flow. To better understand the role played by the flow oscillations, we contrast these results with those obtained for a spatially-homogeneous and time-independent steady flow. This article is organised as follows. Details of the model, both mathematical and numerical, are presented next in section 2. Following this, an extensive range of numerical simulations of flow over complex topography is illustrated and analysed in section 3. Conclusions and ideas for further study are offered in section Model formulation 2.1. The shallow-water model In this study, an unbounded inviscid rotating shallow-water (SW) fluid layer of constant density held down by gravity g is considered on the (uniformly rotating) f-plane, as shown in figure 1. The SW model is appropriate for the study of many basic processes in the oceans and atmosphere (Vallis 28, p. 123), and it is the simplest model containing both vortical and GW motions for which PV-inversion is non-trivial.

4 Flow-topography interactions in shallow water 3 Figure 1. Shallow-water environment with topography. Here, s ¼ b þ h relative to a reference depth. The fluid motion is entirely determined from the (vertically integrated) momentum and continuity equations (Vallis 28, p. 123), collectively called the shallow-water equations, which may be written in the form Du ðh þ bþ fv ¼ Dv ðh þ bþ þ fu ¼ þ J ðuhþ ð1cþ (Gill 1982, p. 191), where u ¼ (u(x, y, t), v(x, y, t)) is the (horizontal) velocity, f is the Coriolis parameter (twice the background rotation rate ), h(x, y, t) is the fluid depth (or height) scaled on p the mean fluid depth H, b(x, y) is the height of the topography scaled on H and c ¼ ffiffiffiffiffiffi gh is the short-scale mean gravity wave speed. Notably the SW equations make use of hydrostatic balance, by replacing the pressure in the momentum equations by its hydrostatic part, here p ¼ p a gz þ c 2 (h þ b), with p a the (constant) atmospheric pressure. The potential imbalance in the SW equations comes from not additionally imposing geostrophic balance, as is done in the quasi-geostrophic model (Vallis 28, p. 144). Nevertheless, geostrophic balance gives (at least) a good qualitative estimate for the flow field, so long as the acceleration terms in equations (1a,b) are relatively small i.e. so long as the Rossby number R¼U/( fl) F 1, where L is a characteristic horizontal length scale and U is the typical horizontal velocity. A key feature of the SW equations, implicit in (1a c), is the material conservation of potential vorticity (PV), q ¼ þ f h, ð2þ is the vertical component of vorticity. That is, q remains unchanged following fluid particles. The same holds for any perturbation q to the PV relative to a constant background value, e.g. q ¼ q f, called the PV anomaly. PV is intimately related to the balanced component of the flow, and is important because in many situations it largely controls the dynamical evolution of the fluid through what is known as the invertibility principle (Hoskins et al. 1985). This principle states that information about the PV field can be used to determine (within a certain approximation) everything about the other fields (such as velocity, mass) at any

5 4 H. Piotka and D. G. Dritschel instant of time. PV inversion excludes gravity waves, the unbalanced component of the flow, but these waves rarely contribute significantly to the overall flow evolution, particularly at intermediate to large scales (McIntyre 1993, 21, 28). PV-inversion is performed by reducing the shallow-water equations to a set of balance relations, while retaining a single evolution equation for PV. Typically two time derivatives are removed from the equations, thereby filtering the gravity waves (GWs). There are many ways to do this (Mohebalhojeh and Dritschel 21), leaving some ambiguity in the definition of balance, and therefore of GWs. That is, one cannot unambiguously separate balanced, vortical motions and GWs in general (Ford et al. 2). Instead, the choice of balance relations is dictated by practicality and the particular application. In the present study, topography results in significant, clearly identifiable GW generation, and due to their large amplitude, it is more than adequate to use quasi-geostrophic balance to separate balanced and unbalanced motions to leading order The numerical model The Contour-Advective Semi-Lagrangian (CASL) algorithm is employed to solve the SW equations in this study. The algorithm is able to simulate complex SW flows accurately and with unprecedented efficiency. It uses material PV contours explicitly thereby allowing access to scales well below the inversion grid-scale used to represent u, v and h (Dritschel and Ambaum 1997, Dritschel et al. 1999, Dritschel and Scott 29) A variable transformation. Smith and Dritschel (26) showed that when formulating the CASL algorithm, in order to obtain a more accurate representation of both the balanced vortical flow and the unbalanced gravity waves, it is beneficial to use a variable transformation from the primitive variables (u, v, h) to(,, q), defined as ¼ J u, ¼ J a ¼ J Du Dt, q ¼ þ f h : ð3a;b;cþ Here is the divergence of the horizontal velocity u, q by definition is the PV and is the divergence of the acceleration Du/Dt. Note that f 1 is the ageostrophic part of the vorticity. The three variables (,, q) are merely auxiliary variables used to cast the problem in a way convenient for numerical evaluation. The primitive variables (u, v, h), needed for instance to compute the tendencies of (,, q), may be obtained by a straightforward inversion of equations (3a c). This inversion process is modified in the present study to include the effects of topography, as detailed below. We examine the changes to the equations for, and q brought about by including topography. First, to find the divergence tendency, we take the divergence of equations @t ¼ S c2 r 2 b, ð4aþ

6 Flow-topography interactions in shallow water 5 where S is the usual divergence tendency without topography, S ¼ J and ¼ f c 2 r 2 h is the value of without topography. The only new term introduced is c 2 r 2 b. Defining S b ¼ c2 r 2 b, ¼ S þ Sb : Next, we examine the changes to. From equations (1a,b), we obtain and so Du Dt ¼ a ¼ Turning to the evolution of, we find ðh þ bþ ðh þ bþ fv c2, fu ¼ J a ¼ f c 2 r 2 ðh þ bþ ¼ þ S c2 ¼ c2 r 2 ðj ðhuþ f J ðuð þ f ÞÞ, ð5cþ so topography has no effect on it, which is expected since topography is constant in time. Therefore, the only change we need to make is in itself not in its evolution. Finally, we examine the effect topography has on the inversion of (,, q) to find (u, v, h). This is done in the CASL algorithm as follows. First, we start with the standard Helmholtz decomposition u ¼ @, where is the divergence potential, while is the (non-divergent) streamfunction. Then, r 2 ¼ J u ¼, r ¼, þ f h (Dritschel et al. 1999), while h is obtained from the definition of and PV: ¼ q ð5bþ ð7a;b;cþ c 2 r 2 h fqh ¼ f 2 : Hence there is no direct effect of topography on the inversion. ð8þ Parameter dependencies. Three fundamental parameters control the flow evolution: (1) The maximum amplitude of topography b m p within the domain; (2) The Froude number FU /c, where c ¼ ffiffiffiffiffiffi gh is the short-scale GW speed and U is the oscillating flow (or steady flow) amplitude (see section 2.3); (3) The Burger number BR 2 =F 2 ¼ L 2 D =L2, where L D ¼ c/f is the Rossby deformation length, L is the typical horizontal scale of topography and

7 6 H. Piotka and D. G. Dritschel R¼U /fl is the Rossby number. Here, L ¼ 1/k, where k is the wavenumber characterising the random topography b. k ¼ 4 is used throughout. All three of these are varied to examine a range of possible values characterising oceanic flows (Carton 21). In both the oscillating and steady flow cases, the same values of the Burger number are used, ranging from small (with the free surface being easily deformed) to large (a stiff free surface), with a midpoint value. We consider B¼:25, B¼1, B¼4, where each value is four times the previous one. Three Froude numbers, through which the maximum amplitude of the external flow is implicitly defined will be used for each of the Burger numbers, namely F¼:2, F¼:3, F¼:4: Any choice larger than this would require the topography to be small in order to prevent the flow from forming shocks or bottoming out, and any smaller choice would make the oscillations increasingly difficult to discern. For each of these Froude numbers F, the maximum permissible amplitude of topography b M is found by running simulations with increasing amplitudes of topography b m until the iteration method used to find h from (8) fails to converge. To be able to fully examine the effect topography has on the flow, in the case of the oscillating flow three values of topography b m were investigated:.1b M,.5b M and b M. The scaling is performed with accuracy to two significant figures. For convenience, we refer to.1b M as minimum topography, and b M as maximum topography. The values of the maximum topography b O M and bs M (the superscript O refers to the oscillating flow, while S refers to the steady flow) are listed in table 1. Notably, b O M 6¼ bs M ; apart from when B¼4 and F¼.2, the maximum amplitude of topography allowed by the steady flow is larger than that for the equivalent oscillating flow case. In all simulations, the form of topography b/b m is taken to be identical, as shown in figure 2. The random topography qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is generated from the variance spectrum Ak=ðk 2 þ k 2 Þ2, where k ¼ k 2 x þ k2 y, and we have taken k ¼ 4. The amplitude A is chosen to give a specified maximum amplitude of topography b m. The random topography is generated by choosing random phases for the Fourier coefficients Table 1. Maximum permissible amplitude of topography b M for each Froude number F and for the range of Burger numbers considered. B F b O M b S M

8 Flow-topography interactions in shallow water 7 Figure 2. Form of the topography b/b m used in all simulations. (a) A linear grey scale is used, with dark regions having negative values and light regions having positive ones. (b) Contours are used with a contour interval of.1. ^bðk x, k y Þ in each wavenumber shell n k5n þ 1, where n is an integer. (Note j ^bj 2 summed over each shell equals the spectrum.) To sum up, for each of the three Burger numbers, we consider three different Froude numbers, and in turn have either one (for the steady flow) or three (for the oscillating flow) different topography amplitudes in total 36 cases. Each simulation is run for 2T eddy, where T eddy 4/Rf is the eddy-turnaround time and R¼B 1/2 F The external flow and the initial conditions We consider an external flow, in a doubly-periodic domain, that has the form U ¼ U þ C cos ft þ S sin ft, V ¼ V þ S cos ft C sin ft, ð9a;bþ which is the sum of a constant mean flow ( U, V) and an oscillating flow at the inertial frequency (C cos ft þ S sin ft, S cos ft C sin ft). This flow can be compensated for by a fixed (time independent) sloping topography of the form Bðx, yþ ¼ðVx UyÞ f=c 2. We examine either a steady unidirectional mean flow (referred to as the steady flow) in the x direction ( U ¼ U ; C ¼ S ¼ V ¼ ) or an oscillating flow initially in the negative x direction ( U ¼ V ¼ S ¼ ; C ¼ U ): U ¼ U cos ft, V ¼ U sin ft: ð1a;bþ The initial state is at rest relative to the external flow (u ¼ v ¼ ) with a flat free surface, which implies that the (dimensionless) depth anomaly h ¼ h 1 ¼ b. Thus, the initial PV is q ¼ f 1 þ h ¼ f 1 b : ð11þ Without loss of generality, we take f ¼ 4, so that our unit of time is a nominal day.

9 8 H. Piotka and D. G. Dritschel Figure 3. Case: oscillating flow, B¼4, F¼.4. Times: t/t eddy ¼ (left), t/t eddy ¼ 1 (middle), t/t eddy ¼ 2 (right). Negative anomalies are dark and positive anomalies are light in this and subsequent figures. (a) Stages in the evolution of the PV anomaly field q. Minimum topography (b m ¼.2). Here, q.2. (b) Stages in the evolution of the PV anomaly field q. Maximum topography (b m ¼.16). Here, q Results The numerical simulation results are examined in three ways. First, the evolving spatial structure of the depth, PV and divergence fields characterising balance and imbalance are illustrated in several examples in section 3.1. Second, in section 3.2, the presence of GW activity is quantified. And third, the degree of locking of the initial depth and PV anomalies to topography is examined in section Spatial structure of the PV and divergence fields Example 1 Evolution under an oscillating flow. First, the effect of the oscillating flow is shown for the largest Burger number and Froude number (B¼4, F¼.4), contrasting weak topography b m ¼.2 in figure 3(a) with strong topography b m ¼.16 in figure 3(b) (figure 4 illustrates the same flow as in figure 3(a) except at early times, showing that the PV anomalies rotate counter-clockwise in small circles about topographic features while hardly changing in form). Here we see that the PV anomalies deform in both cases, but they are much finer-scaled in the case of strong topography. Despite the fact that the PV anomalies become noticeably deformed, they remain in the proximity of their initial positions. This is not so in the steady flow case, discussed in section We next examine the evolution of the depth anomaly h ¼ h 1 and the free surface anomaly s ¼ s 1 (where s ¼ h þ b) for B¼4, F¼.4, and for maximum

10 Flow-topography interactions in shallow water 9 Figure 4. The effect of the oscillating flow on the PV anomaly field q. Here, jq j.2. Case: oscillating flow, B¼4, F¼.4, minimum topography b m ¼.2. Times: t/t eddy ¼ (a), t/t eddy ¼ 2 (b) and t/t eddy ¼ 4 (c). Figure 5. Case: oscillating flow, B¼4, F¼.4, maximum topography b m ¼.16. Times: t/t eddy ¼ (left), t/t eddy ¼ 1 (middle left), t/t eddy ¼ 1 (middle right), t/t eddy ¼ 2 (right). (a) Stages in the evolution of the depth anomaly h ¼ h 1. Here, h.47. (b) Stages in the evolution of the free surface anomaly s ¼ s 1. Here, s.39. topography b m ¼.16. Comparing the depth anomaly in figure 5(a) with the form of the topography in figure 2 shows that there is a close resemblance between h and b (their correlation is quantified in section 3.3). On the other hand, the free surface anomaly s in figure 5(b) is much less tied to topography and is much more dynamically active, exhibiting GW features also seen in the divergence field (discussed in section 3.1.3). It may be noticed that at times t/t eddy ¼ 1 and 2 the free surface has a very similar form. At these times, the oscillating flow is at the same stage, and so GWs are produced in a similar fashion over the topography. The oscillating flow is at a different stage at time t/t eddy ¼ 1 and so the free surface has a different form. A final thing to note is that b m ¼.16 is smaller than the maximum deformation of the free surface js j max ¼.39. Topography can produce substantial free surface deformations.

11 1 H. Piotka and D. G. Dritschel Figure 6. Stages in the evolution of PV anomaly field q. Here, jq j.2. Case: steady flow, B¼4, F¼.4, maximum topography b m ¼.16. Times: t/t eddy ¼ (a), t/t eddy ¼ 4 (b), t/t eddy ¼ 13 (c), t/t eddy ¼ 2 (d). Figure 7. Stages in the evolution of the divergence field. Here, jj2.27. Case: oscillating flow, B¼4, F¼.4, maximum topography b m ¼.16. Times: t/t eddy ¼ (a), t/t eddy ¼.25 (b), and t/t eddy ¼ 4 (c) Example 2 Evolution in a steady flow. This, like the previous example, is also shown for B¼4 and, for ease of comparison, F¼.4 and maximum topography b m ¼.16. As can be seen from figure 6, like in the case shown in figure 3(b), the PV anomalies become deformed. However, now they do not remain near their original positions but are swept away by the steady flow, despite the large topography Example 3 Presence of divergent motions. The emergence of unbalanced motions GWs from the geostrophically balanced initial state is illustrated for the oscillating flow case with B¼4, F¼.4 and maximum topography b m ¼.16 in figure 7. Here, divergent motions develop rapidly through topographic interactions and exhibit characteristic GW features in the form of wave trains (animations strongly suggest that these are GWs). Further confirmation that these are GWs comes from the space-time Hovmo ller diagram in figure 8. The criss-crossing patterns present in both the surface and divergence fields indicate pairs of counter-propagating GWs. Among other places, at roughly y ¼ and t/t eddy ¼ 39.14, some prominent and clearly distinguishable wave trains are visible in the divergence field. The nearly vertical lines correspond to slow motions presumably associated with balanced motions not captured by the leading-order geostrophic balance (in other words, part of the divergence is controlled by PV). Note that the strongest divergence is generated near the strongest topographic features, as can be seen when figure 2 is compared with figure 7, and the

12 Flow-topography interactions in shallow water 11 (a) 41.2 (b) 41.2 t/t eddy t/t eddy 2.6 π y 2.6 π π Figure 8. Hovmo ller diagram of the surface anomaly s ¼ s 1 (a) and the divergence (b) at x ¼. Case: oscillating flow, B¼4, F¼.4, and maximum topography b m ¼.16. Here, time is on the vertical axis and ranges from t/t eddy ¼ 2.6 to 41.2 while the latitude is on the horizontal axis and ranges from y ¼ to. The two top panels show a cross-section of the topography at x ¼. y π cross-section of topography at x ¼ is examined along with the Hovmo ller diagram in figure 8. The level of divergence is comparable to the level of vorticity. This is special to oscillating flows over topography much weaker values occur without these effects (Mohebalhojeh and Dritschel 21) Gravity wave activity The way in which the flow is set up ensures that it is initially geostrophically balanced. Balanced motions are here non-divergent, and therefore the divergence of the flow which develops over time can be used to quantify (within the limitations of geostrophic balance) the presence of imbalance i.e. GWs (as illustrated in section 3.1.3). Note that from section 2.2.1, the divergence of acceleration could also be used; however, is preferable as is often dominated by (second-order) cyclostrophic balance in the u x v y u y v x term. We first estimate the scaling of divergence to be able to make

13 12 H. Piotka and D. G. Dritschel r.m.s.h t r.m.s.ζ t /f r.m.s.δ t /f Topography b m.5 Topography b m.5 b m F B.3 Figure 9. Time average r.m.s. depth anomaly, vorticity and divergence, respectively h t, t =f, and t =f, with appropriate scales, for B¼.25 (i points), B¼1(hpoints), B¼4(points); steady flow cases are indicated with a þ. The solid line is a best-fit line for the data, having slopes m ¼.416 ( h t ), m ¼.333 ( t =f), and m ¼ 1.73 ( t =f ). predictions about its expected level. We then quantify the level of imbalance by the r.m.s. divergence in both oscillating and steady flows over topography Scaling of divergence in the presence of external flows and topography. We next perform a scale analysis of the various terms in the evolution equation for (section 2.2.1) to estimate how much divergence is produced as a function of the fundamental parameters controlling the flow evolution. With b m the maximum topographic amplitude and F¼U /c the Froude number (where U is the maximum external flow speed), we assume both b m 1 and F1. Recall that the Burger number is given by B¼(c/fL) 2, where L is the scale of topography. From the definition of the PV anomaly q ¼ q f, we have q f þ qh ¼ f f : ð12þ At t ¼, we know q /f ¼ 1/(1 b) 1 b when b 1, and thereafter q is materially conserved; hence, q remains O(b) for all time t. Assuming the typical scale (or magnitude) of h is b m, it follows that the scale of /f is also b m. This is confirmed in figure 9, which shows that the values used in our simulations do scale accordingly. We now examine the various terms in the evolution equation for (cf. section 2.2.1): t ¼ðc 2 r 2 f 2 Þ þ c 2 r 2 ðj ðh uþ f J ðuþ, ð13þ written using h ¼ h 1, the (dimensionless) depth anomaly. Let us assume that t is at most comparable to f 2, which is sensible given that the divergence equation is t ¼ þ nonlinear terms (section 2.2.1), and is at most f in magnitude. Then, for the purpose of determining the typical scale of, we can ignore t in (13). We can then restrict attention to the remaining terms in the equation. Let us first consider J (h u) ¼ u Jh þ h. Since we expect that arises from the external flow sweeping over topography, we estimate u in the above by the flow speed U. Since h is similar to thep topography b, Jh scales as b m /L. Hence, u Jh scales as U b m =L ¼ b m Fc=L ¼ fb m F ffiffiffi B. We will show a posteriori that h is negligible in comparison.

14 Flow-topography interactions in shallow water 13 Similarly, p J (u) ¼ u J þ is dominated by the first term u J, which scales as f 2 b m F ffiffiffi B, assuming that /f is also similar in structure to the topography b (then J scales as fb m /L). Now suppose B41. In this case, c 2 r 2 f 2 B4f 2 in magnitude, so (c 2 r 2 f 2 ) scales as f 2 BD, where D is a typical scale of the divergence. Also, when B41, p the term c 2 r 2 (J (h u)) exceeds f J (u). But c 2 r 2 (J (h u)) scales as f 2 B fb m F ffiffiffi B, and so dividing out the common f 2 B factor, we find D f ¼ b p mf ffiffiffi B : ð14þ Now suppose B51. In this case, c 2 r 2 f 2 B5f 2 in magnitude, so (c 2 r 2 f 2 ) scales as f 2 D. p Also, when B51, the term f J (u) exceeds c 2 r 2 (J (h u)). But f J (u) scales as f 3 b m F ffiffiffi B, and so dividing out a factor of f 3, we recover (14) again. Hence (14) applies for all B. Note that (14) can also be written as D/f ¼ b m R. The scaling found is confirmed in figure 9. Note, the scaling in (14) implies that itself scales as b m U /L, independent of both the GW speed c and the Coriolis frequency f. Nevertheless, in the scale analysis we required U /c 1 and /f 1 (the latter we showed scales as b m in figure 9). Evidently, this scaling for is a consequence of the disparate scales of U and c, and of and f. We can also now justify neglecting h compared to u Jh in the above analysis (the same argument can be used for h replaced by here). From (14), h has the scale fb 2 m F p ffiffiffi p B, whereas u Jh has the scale fb m F ffiffiffi B. The ratio of these terms is bm, which is assumed 1, so we can neglect h Effect of varying the amplitude of topography b m and the Froude number F. As we have seen, regardless of the Burger number B, the higher the Froude number F, the more divergence is produced. This makes sense, as larger flow oscillations (or IOs for inertial oscillations) have more energy to perturb the fluid. Similarly, regardless of B, the smaller the topography, the smaller the divergence is. This can be explained by the fact that the definition of PV (equation 2) tells us that the smaller the topography b (the bigger the fluid depth h, equal to 1 b initially), the smaller the PV anomaly q f. A smaller PV anomaly, combined with smaller topography, produces less turbulence. These results are consistent with the analysis performed in the previous subsection, and are illustrated in detail for Burger number B¼1 and for the different values of b m and F, in figure 1. This figure shows that the smaller the topography, the smaller the differences in the divergence produced by different Froude numbers (this also holds for other Burger numbers). In fact, in the case of the smallest amplitude topography :1b O M, changes in the amplitude of the IOs have practically no effect on the divergence produced. The minimal divergence differences between them are slightly larger for larger Burger numbers. An interesting feature of figure 1 is that the presence of IOs is most visible for Froude number F¼.3. This is true not only for B¼1, but also for B¼.25. This can be explained by the fact that weak IOs are not able to cause noticeable interactions with the topography, while for strong ones the PV is no longer topographically locked (section 3.3); hence, although the IOs play a role, it is not as directly visible as in the F¼.3 case. This does not carry over to B¼4, which has some random fluctuations in the amount of divergence for all amplitudes of the IOs and topography, the strongest

15 14 H. Piotka and D. G. Dritschel (a) (b) (c).28 r.m.s. δ t/t eddy 2 Figure 1. r.m.s. divergence for B¼1 at Froude number F¼.2 (a), F¼.3 (b), and F¼.4 (c) for the different topography amplitudes: b O M (thin line), :5bO M (þ line), and :1bO M ( line). being for large values of both. This, again, can be attributed to the fact that topographic locking is not as strong as in the other cases, and so the main source of divergence is from the anomalies (both depth and PV) interacting with the topography itself Effect of varying the Burger number B. It has been found that larger Burger numbers produce more divergence, at least initially. As time progresses, the amount of divergence falls in the larger Burger number cases, and the differences between flows having different values of B become less significant, particularly for the smallest Burger numbers. These differences are more significant for larger topography. For the smallest topography, there are no major differences in the amount of divergence produced between the different Burger numbers or Froude numbers. Furthermore, the larger the Burger number, the bigger the differences become between flows with different IO amplitudes over the same topography. An explanation for these properties is provided in section Comparison with the steady flow. The presence of an oscillating flow has a profound influence on GW activity, which can be most clearly seen when flows forced by IOs are compared to similar flows forced by a steady flow, as is done in figure 11. Here, similar or equivalent flows are those having the same Burger and Froude numbers, with maximum topography b M (note that the cases with b O M and bs M are compared, even though b O M 6¼ bs M ). The divergence created by the steady flow is more constant than in the case of IOs, where strong fluctuations occur. An exception to this is in the beginning, when the steady flow rips the PV anomalies off the topography and in doing so produces sudden and short-lived GW activity. When oscillating flows are compared to the equivalent steady flow cases, it can be seen that there are bigger differences in the divergence between the two extreme Froude numbers for IOs, this being most visible for B¼1 in figure 11. However, this does not mean that the oscillating flow cases with lower F have less divergence than the steady flow cases. In fact, the situation is just the reverse. Even the case of the oscillating flow which produces the least amount of divergence (i.e. having the lowest F ) is still more divergent than the case of the steady flow producing the most divergence (i.e. having the highest F). The only exception to this occurs in the case B¼4, which exhibits a burst

16 .4 Flow-topography interactions in shallow water 15 (a) (b) (c) r.m.s. δ t/t eddy 2 Figure 11. r.m.s. divergence for B¼.25 (a), B¼1 (b), and B¼4 (c) for maximum topography with Froude number F¼.2 (thin line oscillating flow, dashed line steady flow) and F¼.4 (þ line oscillating flow, line steady flow). of GW activity as the PV anomalies are ripped off the topography. Additionally, the divergence produced is only minimally greater than the maximum amount of divergence created by the IOs at later times in the simulation Kinetic energy due to divergent flow. We quantify next the amount of energy scattered into propagating waves by measuring the kinetic energy associated with the divergent component of the flow, namely E d ¼ 1 2 hhðu2 d þ v2 d Þi, where h is the total depth field, (u d,v d ) is the divergent velocity, and h i indicates a spatial average. E d is then normalised by the total energy E to give the dimensionless measure E ~ d ¼E d =E. The complement of E ~ d then gives the amount of energy stored in the non-divergent, balanced, vortical component of the flow. As seen in figure 12, for an oscillating flow E ~ d is approximately independent of the Burger number and is proportional to the amplitude of topography squared b 2 m times the Froude number F. Indeed, for large and small values of the Burger number, Ed ~ =Fb 2 m collapses for all Froude numbers to a value between 2 and 4. E d itself, however, is never more than 6.7% of the total energy. Even less energy is scattered into divergent motions for a steady flow: E d is never more than 4.%. Moreover, E d appears to scale with F 2 b 2 m, as seen in figure 13. ð15þ 3.3. Topographic locking of depth and PV anomalies To measure the impact of topography on the flow evolution, and what role the locking of depth and PV anomalies plays, correlations between the topography and depth anomalies, and between the topography and PV anomalies are examined next. The correlation C(x, y) between two variables x and y indicates the degree to which a linear relationship exists between them. It gives not only the strength of this relationship,

17 16 H. Piotka and D. G. Dritschel 6 Ẽ d / (Fb 2 m ) 6 t/t eddy 2 Ẽ d / (Fb 2 m ) Ẽ d / (Fb 2 m ) 6 t/t eddy 2 t/t eddy 2 Figure 12. Normalised kinetic energy of the divergent flow E ~ d scaled by the amplitude of topography squared b 2 m and Froude number F for oscillating flows when B¼.25 (column 1), B¼1 (column 2), and B¼4 (column 3) at Froude number F¼.2 (row 1), F¼.3 (row 2), and F¼.4 (row 3) for the different topography amplitudes: b O M (thin line), :5bO M (þ line), and :1bO M ( line). but also the direction, and is defined to be hxyi Cðx, yþ ¼ hx 2 i 1=2 hy 2 i 1=2 : ð16þ Both the correlations of topography b with depth anomalies h ¼ h 1 and with PV anomalies q ¼ q f are examined. From geostrophic balance we know that if there is an anomaly in PV, then there also exists one in depth. The two are linked. A strong correlation indicates that the anomalies are dependent on the topography i.e. that they are locked to it. On the other hand, values closer to zero indicate that the anomalies move independently of the location of the topography.y ywhen low or high values of correlation are referred to, absolute values are meant. Low implies a weak correlation, while high implies a strong one.

18 Flow-topography interactions in shallow water 17 (a) (b) (c) 3.75 Ẽ d / (F 2 b 2 m ) t/t eddy 2 Figure 13. Normalised kinetic energy of the divergent flow ~ E d scaled by the amplitude of topography squared b 2 m and Froude number squared F 2 for the steady flow when B¼.25 (a), B¼1 (b), and B¼4 (c) for maximum topography with Froude number F ¼.2 (thin line), F ¼.3 (þ line), and F ¼.4 ( line). (a) (b) (c) 1 C(b,q ).3 t/t eddy 2 Figure 14. Correlation C(b, q ) for B¼4 for Froude numbers F¼.2 (a), F¼.3 (b), and F¼.4 (c) for three different amplitudes of topography: b O M (thin line), :5bO M (þ line), and :1bO M ( line) Effect of varying the amplitude of topography b m and the Froude number F. For a fixed Burger number, we find that increasing the Froude number decreases the correlation of the topography with both the depth and PV anomalies. There do, however, exist differences between the topography-depth correlation C(b, h ) and the topography-pv correlation C(b, q ). It has been found that for all Burger numbers, the correlation C(b, h ) for the two smaller amplitudes of topography is much more similar (and has lower values) than that for the maximum topography. In the cases with Froude number F¼.2, all values of topography have similar correlations. This seems to imply that there is some critical amplitude of topography, after which making it any smaller does not make any difference to the correlation C(b, h ). Furthermore, and also holding for all Burger numbers, we find that as the Froude number of the flow increases, the fluctuations in the correlation C(b, q ) have a larger amplitude, a higher frequency and last longer. As can be seen from figure 14, these are most visible for small topography. In fact, the smaller the topography and the bigger the amplitude of the IOs, the larger the fluctuations become. As the IOs swell, the PV anomalies are almost ripped off the topography (and the correlation falls), and as the IOs fall they return (the correlation rises). For B¼.25 and B¼1, as time passes, these

19 18 H. Piotka and D. G. Dritschel fluctuations subside, and stay roughly at the same value. Fluctuations for B¼4 also have a decreasing trend, but the simulation is not long enough for them to damp. As the Burger number increases, the damping of the fluctuations takes longer, as the initial PV anomaly has a small scale compared to L D (recall L/L D ¼B 1/2 ). For all Burger numbers, PV anomalies are least locked to the topography for large Froude numbers and for small amplitudes of topography, as then the lowest values of correlation occur. However, the PV anomalies are not ripped off the topography; rather they are gradually dispersed, and thus the general trend in C(b, q ) as time passes is downward. Recall that, in all simulations, the topography is identical (apart for the maximum permissible amplitude b M, which diminishes with Burger number), and that the depth and PV anomalies weaken with increasing Burger number. This explains the fact that the correlation C(b, h ) is higher for smaller Burger numbers than for larger ones, especially when C(b, h ) is compared to C(b, q ). Another explanation for the fact that C(b, h ) is higher than C(b, q ) is that the PV is dispersed away from its origin, and thus the correlation falls, yet there will always be some kind of correlation between topography and depth anomalies in places where topography is bigger (for an example, see section 3.1.1). Another link between the correlations C(b, h ) and C(b, q ) is that as the PV gradually escapes the topography (characterised by the decreasing trend in fluctuations of C(b, q )), fluctuations in C(b, h ) begin. These are more regular for B¼.25 and B¼1 for the two smaller Froude numbers F, whereas they are more irregular for F¼.4. In the case when B¼4, the fluctuations seem to be more erratic than in the other two cases, but most noticeably so for F¼.4. Like with changes in the Froude number, changes in the amplitude of the topography also have an influence on both the correlations C(b, h ) and C(b, q ). The larger the topography, the higher the correlation C(b, h ), and the lower C(b, q ) (which becomes low quickly with few fluctuations, unlike the small topography case). This is because for (relatively) large topography (and hence small fluid depth h), the PV anomaly is large and thus is able to escape more quickly. Furthermore, for large topography, there is a strong link between it and the depth anomaly. When the effects on the correlations due to changes in the amplitude of topography b m or Froude number F are compared, we find that changes in F have a much bigger influence on C(b, h ) than changes in b m (as can be seen from figure 15(a)), regardless of the Burger number Effect of varying the Burger number B. As noted above, differences in correlations occur between different Burger numbers. Although increasing the Burger number does not necessarily change the average value of C(b, h ), it does affect its smoothness, as is clear from figure 15(a). There are much stronger fluctuations at larger Burger numbers, and this can be explained by the fact that the depth anomalies are swung back and forth by the IOs much more. The correlation C(b, q ) is much more affected by changes in the Burger number than is C(b, h ), as can be seen by comparing figures 15(a) and (b). Changes in the Burger number thus affect C(b, q ) more than changes in the amplitudes of either the IOs or the topography. All flow simulations for B¼.25 have strongly topographically locked depth and PV anomalies, regardless of the amplitude of the IOs or topography. As the Burger number grows, the same flows are less correlated, and there are bigger

20 Flow-topography interactions in shallow water 19 (a).7 C(b,h ) 1 t/t eddy 2 Correlation C(b, h ) (b) 1 C(b,q ).3 t/t eddy 2 Correlation C(b, q ) Figure 15. Correlations for B¼.25 (A), B¼1 (b), and B¼4 (c) for the maximum topography with F being maximum (dashed line) and minimum (thin line), and minimum topography with F being maximum ( line) and minimum (þ line). Case: oscillating flow. differences in the correlations between flows with different Froude numbers over the same topography. A similarity between the cases with different Burger numbers is that they all follow the same pattern for all topography and Froude numbers. Only the time and magnitude scales vary. For smaller Burger numbers, the processes are quicker Comparison with the steady flow. To better understand the effects of oscillating flows on the locking of depth and PV anomalies, the above simulations are next compared to the equivalent ones carried out with a steady flow. We find that in the steady flow case, regardless of the Burger number, the correlations C(b, q ) have similar values. This is not the case for the correlations C(b, h ), which vary with the Burger number. For B¼.25, the correlation C(b, h ) is much higher for the IOs. This is because the depth anomalies are not able to escape in this case, while in the steady flow case they are immediately ripped off. Also, since the depth anomalies are spatially large, as they are dragged along by the steady flow there is no visible correlation hence the low values of C(b, h ). For B¼1, the correlations C(b, h ) for the steady flow and the IOs are similar, with the steady flow case fluctuating much more. Similarities for this value of the Burger number arise because the depth and PV anomalies are able to gradually escape the topography in both cases. However, the fluctuations are not as pronounced with IOs because both types of anomalies are not advected very far by the flow, but remain in the proximity of their origin.

21 2 H. Piotka and D. G. Dritschel For B¼4, for the first time there is a higher correlation C(b, h ) for the steady flow than for the IOs. Since the initial depth and PV anomalies are of a scale small compared to L D, it takes a long time for them to disperse, even as the steady flow advects them away. This leads to a stronger correlation than previously. 4. Conclusions In this study, we have carried out a series of numerical simulations assessing the effects of oscillating flows (which in certain cases may be viewed as idealised tides) and topography on shallow-water turbulence. The simulations covered a broad range of parameter space in the amplitude of the topography b m, the strength of the flow oscillations U, and the Rossby deformation length L D. Our key findings are summarised next. First, regarding the generation of gravity waves, the strongest generation (measured here by the divergence ) is produced for large Burger numbers and for flows with strong flow oscillations and for large topographic amplitudes. However, this is expected. We showed, by scale analysis, that the amount of divergence produced for a fixed Burger number B¼(L D /L) 2 is roughly proportionalpto the amplitudes of the oscillating flow and of the topography (indeed =f b m F ffiffiffi B where F¼U /c). The effects of the two are linked flow oscillations have less effect on flows over small amplitude topography. In fact, it has been shown that b m U /L, which means that most divergence is produced for a strong oscillating flow over steep topography (b m /L can be seen as the gradient of topography). Oscillating flows also generate much more divergence than steady flows. Even the oscillating flow case producing the least divergence (i.e. having the lowest Froude number) produces more or similar divergence than the steady flow case producing the most divergence (i.e. having the highest Froude number). Only when the Burger number is large, the divergence is nearly the same. Furthermore, the amount of energy scattered into propagating waves is approximately independent of the Burger number and proportional to the square of the amplitude of topography times the Froude number Fb 2 m (oscillating flow case) and to the square of the amplitude of topography times the Froude number squared F 2 b 2 m (steady flow case). Second, concerning the extent to which the initial depth and PV anomalies are locked to the topography, the topography-depth anomaly correlation C(b, h ) is a better indicator of the locking of the anomalies by topography than the topography-pv anomaly correlation C(b, q ). With increasing Burger and Froude numbers, we find that topographic locking weakens. However, the effects associated with weak topography are not as clear. Weak topography leads to slightly lower values of C(b, q ) than strong topography at late times, but it appears to take much longer to reach these low correlations the release of the depth and PV anomalies is slow and gradual. It can be said that the escape of the depth and PV anomalies in the oscillating flow case is not as sudden or as strong as when they are ripped off the topography by a steady flow having the same Froude number F. Nevertheless, increasing the Burger number makes the topography less able to trap the depth and PV anomalies. Linking the two analyses, we conclude that the greatest gravity wave generation is produced as the depth and PV anomalies escape from topographic locking. Indeed, as

22 Flow-topography interactions in shallow water 21 can be seen from figures 12 and 14, there is a gain in the kinetic energy due to the divergent flow (corresponding to energy being scattered into gravity waves) as the correlation between topography and the PV anomaly drops. In fact, the highest instantaneous burst of divergence occurs for the largest Burger number when a steady flow rips anomalies off the topography. However, as the steady flow advects the anomalies, they become dispersed instead of remaining spatially localised, and gravity wave amplitudes fall. To have significant and persistent divergence, locking of the anomalies by topography is required in order to prevent dispersion, and to cause a struggle as the anomalies try to escape. Therefore, the larger the topography, the more locking occurs, and the more interaction. Furthermore, high amplitude flow oscillations are required. In summary, the most long-lived gravity wave activity is produced for large (in space) depth and PV anomalies, which are strongly locked to large topography, and are under the influence of flow oscillations. The most instantaneous GW activity is produced for spatially large anomalies, which are ripped off large topography by a steady flow with a high Froude number. Several avenues for future research have been identified. As realistic tidal oscillations cannot be represented in the f-plane shallow-water model used here (see section 1), a global shallow-water model is necessary. This would allow simulating a whole range of different tidal frequencies which occur in both the oceans and atmosphere. Also, in this study, the initial geostrophically balanced flow is set up directly from the definition of the shallow-water potential vorticity (see equation (2)) with ¼ and h þ b ¼ 1 (flat free surface). This means that decreasing the topography b also decreases the PV anomaly q f. A further extension to this study would be to examine 6¼ initially, i.e. a random initial vorticity field (in e.g. geostrophic balance), so that there is a pre-existing turbulent flow interacting with the oscillating flow and topography. References Aebischer, U. and Scha r, C., Low-level potential vorticity and cyclogenesis to the lee of the Alps. J. Atmos. Sci. 1998, 55, Aelbrecht, D., d Hie` res, G. and Renouard, D., A 3-D coastal tidal rectification process: observations, theory and experiments. Cont. Shelf Res. 1999, 19, Baines, P., Topographic Effects in Stratified Flows, Cambridge Monographs on Mechanics, (Cambridge: Cambridge University Press). Carton, X., Hydrodynamical modeling of oceanic vortices. Surv. Geophys. 21, 22, Dewar, W., Baroclinic eddy interaction with isolated topography. J. Phys. Oceanogr. 22, 32, Dritschel, D. and Ambaum, M., A contour-advective semi-lagrangian algorithm for the simulation of finescale conservative dynamical fields. Quart. J. Roy. Meteorol. Soc. 1997, 123, Dritschel, D., Polvani, L. and Mohebalhojeh, A., The contour-advective semi-lagrangian algorithm for the shallow water equations. Mon. Weath. Rev. 1999, 127, Dritschel, D. and Scott, R., On the simulation of nearly inviscid two-dimensional turbulence. J. Comput. Phys 29, 228, Ford, R., McIntyre, M. and Norton, W., Balance and the slow quasimanifold: some explicit results. J. Atmos. Sci. 2, 57, Gerkema, T. and Zimmerman, J., Generation of nonlinear internal tides and solitary waves. J. Phys. Oceanogr. 1995, 25, Gill, A.E., Atmosphere Ocean Dynamics, (London: Academic Press). Hinds, A., Johnson, E. and McDonald, N., Vortex scattering by step topography. J. Fluid Mech. 27, 571,