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Flow-topography interactions in shallow-water turbulenceHanna Płotkaa; David G. Dritschela

a School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, UK

First published on: 06 February 2011

To cite this Article Płotka, Hanna and Dritschel, David G.(2011) 'Flow-topography interactions in shallow-waterturbulence', Geophysical & Astrophysical Fluid Dynamics,, First published on: 06 February 2011 (iFirst)To link to this Article: DOI: 10.1080/03091929.2010.537264URL: http://dx.doi.org/10.1080/03091929.2010.537264

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Geophysical and Astrophysical Fluid Dynamics2011, 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 30 October 2009; in final form 14 September 2010)

Topography and external time-varying flows such as tides can have a profound influence onatmospheric and oceanic fluid motion. The effects of topography on turbulence have been littlestudied, and until recently those of tides have been overlooked. Our lack of understanding ofthese effects is a serious impediment to predicting flows both in the atmosphere and in theoceans. As a first step toward modelling realistic tidal oscillations, this study focuses on how aninitially balanced flow containing a distribution of potential vorticity anomalies develops andbecomes unbalanced (by emitting gravity waves) as a result of interactions with topography inthe presence of an oscillating flow, here in the form of pure inertial oscillations (at the Coriolisfrequency). We investigate a large parameter space spanned by the Burger number, themaximum amplitude of topography, and the amplitude of the flow oscillations. Comparisonsare made with a time-independent and spatially-homogeneous steady flow in order to betterunderstand the role played by flow oscillations. The most persistent large-amplitude gravitywave activity occurs for large-amplitude flow oscillations and for large topography. The latterproduces 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, largepotential vorticity and depth anomalies are advected away from topographic features and sogravity 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.

Viudez and Dritschel 2006, McIntyre 2008 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. Email: [email protected]

Geophysical and Astrophysical Fluid Dynamics

ISSN 0309-1929 print/ISSN 1029-0419 online � 2011 Taylor & Francis

DOI: 10.1080/03091929.2010.537264

Downloaded By: [Plotka, Hanna] At: 10:33 7 February 2011

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 whichmay be relevant to the dynamics of the ocean. Specifically, we concentrate on how PVanomalies are locked to topography. The external flow considered here, consisting of

pure inertial oscillations, may be regarded as a first step toward considering realistictidal oscillations (Webster 1968). Inertial oscillations are the simplest way in which tidesover topography can be modelled in the doubly-periodic domain considered here, andare 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 spaceand cannot be handled straightforwardly (they would require explicit forcing tomaintain).

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 generatedas a result of flows interacting with it (see, e.g. Schar and Smith 1993, Aebischer andSchar 1998). Baines (1995, p. 2) points out that our lack of understanding of howtopography affects atmospheric flows is a significant impediment to improvements in

weather prediction. He also notes that surface drag contributes up to 50% of the totaldrag in the atmosphere. A similar thing can be said about the oceans (Johnson andVilenski 2005). While numerous studies have examined the effects of topography on the

propagation and evolution of eddies, this has been done mainly for isolated topographyor infinitely long ridges (e.g. Holloway 1987, McDonald 1998, Aelbrecht et al. 1999,Dewar 2002, Hinds et al. 2007), and without tidal or inertial oscillations. In fact, themajority of work on the flow–topography interaction problem examines not the locking

and development of eddies, but rather the mechanisms for the generation of internalwaves (e.g. Gerkema and Zimmerman 1995, Llewellyn Smith and Young 2002, Zarrouget al. 2010).

Here we focus on how an initially balanced flow evolves through time and becomesunbalanced 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 withthose obtained for a spatially-homogeneous and time-independent steady flow. Thisarticle is organised as follows. Details of the model, both mathematical and numerical,

are presented next in section 2. Following this, an extensive range of numericalsimulations of flow over complex topography is illustrated and analysed in section 3.Conclusions and ideas for further study are offered in section 4.

2. Model formulation

2.1. The shallow-water model

In this study, an unbounded inviscid rotating shallow-water (SW) fluid layer ofconstant density �0 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 basicprocesses in the oceans and atmosphere (Vallis 2008, p. 123), and it is the simplestmodel containing both vortical and GW motions for which PV-inversion is non-trivial.

2 H. Piotka and D. G. Dritschel


The fluid motion is entirely determined from the (vertically integrated) momentum and

continuity equations (Vallis 2008, p. 123), collectively called the shallow-water

equations, which may be written in the form

Du

Dt� fv ¼ �c2

@ ðhþ bÞ

@x,

Dv

Dtþ fu ¼ �c2

@ ðhþ bÞ

@y, ð1a;bÞ

@h

@tþ J � ðuhÞ ¼ 0 ð1cÞ

(Gill 1982, p. 191), where u¼ (u(x, y, t), v(x, y, t)) is the (horizontal) velocity, f is theCoriolis parameter (twice the background rotation rate �), h(x, y, t) is the fluid depth

(or height) scaled on the mean fluid depth H, b(x, y) is the height of the topography

scaled on H and c ¼ffiffiffiffiffiffiffigHp

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 p¼ pa� �0gzþ �0c2(hþ b),

with pa 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 2008, 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

where �¼ @v/@x� @u/@y is the vertical component of vorticity. That is, q remainsunchanged following fluid particles. The same holds for any perturbation q0 to the PV

relative to a constant background value, e.g. q0 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

Figure 1. Shallow-water environment with topography. Here, s¼ bþ h relative to a reference depth.

Flow-topography interactions in shallow water 3


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, 2001, 2008).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 2001), 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.

2000). 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.

2.2. 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 2009).

2.2.1. A variable transformation. Smith and Dritschel (2006) showed that whenformulating 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 � isthe 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 (1a,b):

@ ð1aÞ

@xþ@ ð1bÞ

@y)@�

@t¼ S0

� � c2r2b, ð4aÞ



where S0� is the usual divergence tendency without topography,

S0� ¼ �

0 þ 2@u

@x

@v

@y�@u

@y

@v

@x

� �� J � ð�uÞ, ð4bÞ

and �0¼ f�� c2r2h is the value of � without topography. The only new term introducedis �c2r2b. Defining Sb

� ¼ �c2r2b, we have

@�

@t¼ S0

� þ Sb� : ð4cÞ

Next, we examine the changes to �. From equations (1a,b), we obtain

Du

Dt¼ a ¼ fv� c2

@ ðhþ bÞ

@x, � fu� c2

@ ðhþ bÞ

@y

� �, ð5aÞ

and so

� ¼ J � a ¼ f � � c2r2ðhþ bÞ ¼ �0 þ Sb� : ð5bÞ

Turning to the evolution of �, we find

@�

@t¼ f

@�

@t� c2r2 @h

@t¼ c2r2 ðJ � ðhuÞ

�� fJ � ðuð� þ f ÞÞ, ð5cÞ

so topography has no effect on it, which is expected since topography is constant intime. 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 standardHelmholtz decomposition

u ¼ J�þ �@

@y,@

@x

� �, ð6Þ

where � is the divergence potential, while is the (non-divergent) streamfunction.Then,

r2� ¼ J � u ¼ �, r2 ¼@v

@x�@u

@y¼ �,

� þ f

h¼ q ð7a;b;cÞ

(Dritschel et al. 1999), while h is obtained from the definition of �0 and PV:

c2r2h� fqh ¼ ��0 � f 2: ð8Þ

Hence there is no direct effect of topography on the inversion.

2.2.2. Parameter dependencies. Three fundamental parameters control the flowevolution:

(1) The maximum amplitude of topography bm within the domain;(2) The Froude number F �U0/c, where c ¼

ffiffiffiffiffiffiffigHp

is the short-scale GW speedand U0 is the oscillating flow (or steady flow) amplitude (see section 2.3);

(3) The Burger number B � R2=F 2 ¼ L2D=L

2, where LD¼ c/f is the Rossbydeformation length, L is the typical horizontal scale of topography and



R¼U0/fL is the Rossby number. Here, L¼ 1/k0, where k0 is the wavenumbercharacterising the random topography b. k0¼ 4 is used throughout.

All three of these are varied to examine a range of possible values characterising

oceanic flows (Carton 2001). 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

B ¼ 0:25, B ¼ 1, B ¼ 4,

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

F ¼ 0:2, F ¼ 0:3, F ¼ 0:4:

Any choice larger than this would require the topography to be small in order toprevent 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 bM is found by running

simulations with increasing amplitudes of topography bm 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 bm were investigated: 0.1bM, 0.5bM and bM. The scaling is performed with

accuracy to two significant figures. For convenience, we refer to 0.1bM as minimum

topography, and bM as maximum topography. The values of the maximum topography

bOM and bSM (the superscript O refers to the oscillating flow, while S refers to the steady

flow) are listed in table 1. Notably, bOM 6¼ bSM; apart from when B¼ 4 and F ¼ 0.2, 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/bm is taken to be identical, as shown in

figure 2. The random topography is generated from the variance spectrum

Ak=ðk2 þ k20Þ2 , where k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2x þ k2y

q, and we have taken k0¼ 4. The amplitude A is

chosen to give a specified maximum amplitude of topography bm. The random

topography is generated by choosing random phases for the Fourier coefficients

Table 1. Maximum permissible amplitude of topography bM for eachFroude number F and for the range of Burger numbers considered.

B F bOM bSM

0.25 0.2 0.40 0.410.3 0.33 0.360.4 0.27 0.30

1 0.2 0.31 0.320.3 0.23 0.240.4 0.19 0.20

4 0.2 0.28 0.260.3 0.21 0.210.4 0.16 0.17



bðkx, kyÞ in each wavenumber shell n� k5nþ 1, where n is an integer. (Note jbj2

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

20Teddy, where Teddy� 4�/Rf is the ‘‘eddy-turnaround time’’ and R¼B1/2F .

2.3. The external flow and the initial conditions

We consider an external flow, in a doubly-periodic domain, that has the form

U ¼ �Uþ C cos ftþ S sin ft, V ¼ �Vþ 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 inertialfrequency (C cos ftþS 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 ¼ ð �Vx� �UyÞ f=c2.We examine either a steady unidirectional mean flow (referred to as the steady flow)

in the x direction ( �U¼U0; C ¼ S ¼ �V ¼ 0) or an oscillating flow initially in the negative

x direction ( �U ¼ �V ¼ S ¼ 0; C¼�U0):

U ¼ �U0 cos ft, V ¼ U0 sin ft: ð10a;bÞ

The initial state is at rest relative to the external flow (u0 ¼ v0 ¼ 0) with a flatfree surface, which implies that the (dimensionless) depth anomaly h0 ¼ h� 1¼�b.

Thus, the initial PV is

q0 ¼f

1þ h0¼

f

1� b: ð11Þ

Without loss of generality, we take f¼ 4�, so that our unit of time is a nominal ‘‘day’’.

Figure 2. Form of the topography b/bm used in all simulations. (a) A linear grey scale is used, with darkregions having negative values and light regions having positive ones. (b) Contours are used with a contourinterval of 0.1.



3. Results

The numerical simulation results are examined in three ways. First, the evolving spatialstructure of the depth, PV and divergence fields characterising balance and imbalanceare illustrated in several examples in section 3.1. Second, in section 3.2, the presence ofGW activity is quantified. And third, the degree of locking of the initial depth and PVanomalies to topography is examined in section 3.3.

3.1. Spatial structure of the PV and divergence fields

3.1.1. Example 1 – Evolution under an oscillating flow. First, the effect of theoscillating flow is shown for the largest Burger number and Froude number (B¼ 4,F ¼ 0.4), contrasting weak topography bm¼ 0.02 in figure 3(a) with strong topographybm¼ 0.16 in figure 3(b) (figure 4 illustrates the same flow as in figure 3(a) except at earlytimes, showing that the PV anomalies rotate counter-clockwise in small circles abouttopographic features while hardly changing in form). Here we see that the PV anomaliesdeform 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 theproximity of their initial positions. This is not so in the steady flow case, discussed insection 3.1.2.

We next examine the evolution of the depth anomaly h0 ¼ h� 1 and the freesurface anomaly s0 ¼ s� 1 (where s¼ hþ b) for B¼ 4, F ¼ 0.4, and for maximum

Figure 3. Case: oscillating flow, B¼ 4, F ¼ 0.4. Times: t/Teddy¼ 0 (left), t/Teddy¼ 10 (middle), t/Teddy¼ 20(right). Negative anomalies are dark and positive anomalies are light in this and subsequent figures. (a) Stagesin the evolution of the PV anomaly field q0. Minimum topography (bm¼ 0.02). Here, |q0|� 0.02. (b) Stages inthe evolution of the PV anomaly field q0. Maximum topography (bm¼ 0.16). Here, |q0|� 0.18.



topography bm¼ 0.16. Comparing the depth anomaly in figure 5(a) with the form ofthe topography in figure 2 shows that there is a close resemblance between h0 and b(their correlation is quantified in section 3.3). On the other hand, the freesurface anomaly s0 in figure 5(b) is much less tied to topography and is much moredynamically active, exhibiting GW features also seen in the divergence field(discussed in section 3.1.3). It may be noticed that at times t/Teddy¼ 10 and 20 thefree surface has a very similar form. At these times, the oscillating flow is at thesame stage, and so GWs are produced in a similar fashion over the topography.The oscillating flow is at a different stage at time t/Teddy¼ 1 and so the free surfacehas a different form. A final thing to note is that bm¼ 0.16 is smaller than the maximumdeformation of the free surface js0jmax¼ 0.39. Topography can produce substantial freesurface deformations.

Figure 4. The effect of the oscillating flow on the PV anomaly field q0. Here, jq0j � 0.02. Case: oscillatingflow, B¼ 4, F ¼ 0.4, minimum topography bm¼ 0.02. Times: t/Teddy¼ 0 (a), t/Teddy¼ 2 (b) and t/Teddy¼ 4 (c).

Figure 5. Case: oscillating flow, B¼ 4, F ¼ 0.4, maximum topography bm¼ 0.16. Times: t/Teddy¼ 0 (left),t/Teddy¼ 1 (middle left), t/Teddy¼ 10 (middle right), t/Teddy¼ 20 (right). (a) Stages in the evolution of thedepth anomaly h0 ¼ h� 1. Here, |h0|� 0.47. (b) Stages in the evolution of the free surface anomaly s0 ¼ s� 1.Here, |s0|� 0.39.



3.1.2. Example 2 – Evolution in a steady flow. This, like the previous example, is alsoshown for B¼ 4 and, for ease of comparison, F ¼ 0.4 and maximum topographybm¼ 0.16. As can be seen from figure 6, like in the case shown in figure 3(b), the PVanomalies become deformed. However, now they do not remain near their originalpositions but are swept away by the steady flow, despite the large topography.

3.1.3. Example 3 – Presence of divergent motions. The emergence of unbalancedmotions – GWs – from the geostrophically balanced initial state is illustrated for theoscillating flow case with B¼ 4, F ¼ 0.4 and maximum topography bm¼ 0.16 infigure 7. Here, divergent motions develop rapidly through topographic interactions andexhibit characteristic GW features in the form of wave trains (animations stronglysuggest that these are GWs). Further confirmation that these are GWs comes from thespace-time Hovmoller diagram in figure 8. The criss-crossing patterns present in boththe surface and divergence fields indicate pairs of counter-propagating GWs. Amongother places, at roughly y¼ 0 and t/Teddy¼ 39.14, some prominent and clearlydistinguishable wave trains are visible in the divergence � field. The nearly vertical linescorrespond to slow motions presumably associated with balanced motions not capturedby the leading-order geostrophic balance (in other words, part of the divergence iscontrolled by PV). Note that the strongest divergence is generated near the strongesttopographic features, as can be seen when figure 2 is compared with figure 7, and the

Figure 7. Stages in the evolution of the divergence field �. Here, j�j � 2.27. Case: oscillating flow, B¼ 4,F ¼ 0.4, maximum topography bm¼ 0.16. Times: t/Teddy¼ 0 (a), t/Teddy¼ 0.25 (b), and t/Teddy¼ 4 (c).

Figure 6. Stages in the evolution of PV anomaly field q0. Here, jq0j � 0.20. Case: steady flow, B¼ 4, F ¼ 0.4,maximum topography bm¼ 0.16. Times: t/Teddy¼ 0 (a), t/Teddy¼ 4 (b), t/Teddy¼ 13 (c), t/Teddy¼ 20 (d).



cross-section of topography at x¼ 0 is examined along with the Hovmoller diagram infigure 8. The level of divergence is comparable to the level of vorticity �. This is specialto oscillating flows over topography – much weaker values occur without these effects(Mohebalhojeh and Dritschel 2001).

3.2. 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 flowwhich develops over time can be used to quantify (within the limitations of geostrophicbalance) the presence of imbalance – i.e. GWs (as illustrated in section 3.1.3). Note thatfrom section 2.2.1, the divergence of acceleration � could also be used; however, � ispreferable as � is often dominated by (second-order) cyclostrophic balance in theuxvy� uyvx term. We first estimate the scaling of divergence to be able to make

π0

y

−π20.6

t/T

ed

dy

41.2

π0

y

−π20.6

t/T

ed

dy

41.2

(a) (b)

Figure 8. Hovmoller diagram of the surface anomaly s0 ¼ s� 1 (a) and the divergence � (b) at x¼ 0. Case:oscillating flow, B¼ 4, F ¼ 0.4, and maximum topography bm¼ 0.16. Here, time is on the vertical axis andranges from t/Teddy¼ 20.6 to 41.2 while the latitude is on the horizontal axis and ranges from y¼�� to �.The two top panels show a cross-section of the topography at x¼ 0.



predictions about its expected level. We then quantify the level of imbalance by ther.m.s. divergence in both oscillating and steady flows over topography.

3.2.1. Scaling of divergence in the presence of external flows and topography. We nextperform a scale analysis of the various terms in the evolution equation for � (section2.2.1) to estimate how much divergence is produced as a function of the fundamentalparameters controlling the flow evolution. With bm the maximum topographicamplitude and F ¼U0/c the Froude number (where U0 is the maximum external flowspeed), we assume both bm� 1 and F � 1. Recall that the Burger number is given byB¼ (c/fL)2, where L is the scale of topography.

From the definition of the PV anomaly q0 ¼ q� f, we have

q0

fþqh0

f¼�

f: ð12Þ

At t¼ 0, we know q0/f¼ 1/(1� b)� 1� b when b� 1, and thereafter q0 is materiallyconserved; hence, q0 remains O(b) for all time t. Assuming the typical scale (ormagnitude) of h0 is bm, it follows that the scale of �/f is also bm. This is confirmed infigure 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 ¼ ðc2r2 � f 2Þ�þ c2r2 ðJ � ðh0uÞ

�� fJ � ð�uÞ, ð13Þ

written using h0 ¼ h� 1, the (dimensionless) depth anomaly. Let us assume that �t is atmost comparable to f 2�, which is sensible given that the divergence equation is �t¼ �þnonlinear terms (section 2.2.1), and that @/@t is at most f in magnitude. Then, for thepurpose of determining the typical scale of �, we can ignore �t in (13). We can thenrestrict attention to the remaining terms in the equation.

Let us first consider J � (h0u)¼ u �Jh0 þ �h0. Since we expect that � arises from theexternal flow sweeping over topography, we estimate u in the above by the flow speedU0. Since h0 is similar to the topography b, Jh0 scales as bm/L. Hence, u �Jh0 scales asU0bm=L ¼ bmFc=L ¼ f bmF

ffiffiffiffiBp

. We will show a posteriori that �h0 is negligible incomparison.

0

0.5r.m

.s.h

t

Topography bm 0.5 Topography bm 0.50r.m

.s.ζ

t/f

¯

0.5

B√bmF 3.00

r.m

.s.δ

t/f

¯

0.3

Figure 9. Time average r.m.s. depth anomaly, vorticity and divergence, respectively �h0t, ��t=f, and ��t=f, withappropriate scales, for B¼ 0.25 (i points), B¼ 1 (h points), B¼ 4 (� points); steady flow cases are indicatedwith a þ. The solid line is a best-fit line for the data, having slopes m¼ 0.416 ( �h0t), m¼ 0.333 ( ��t=f), andm¼ 1.73 ( ��t=f ).



Similarly, J � (�u)¼ u �J�þ �� is dominated by the first term u �J�, which scales asf 2bmF

ffiffiffiffiBp

, assuming that �/f is also similar in structure to the topography b (then J�scales as fbm/L).

Now suppose B41. In this case, c2r2 f 2B4f 2 in magnitude, so (c2r2� f 2)� scalesas f 2BD, where D is a typical scale of the divergence. Also, when B41, the termc2r2(J � (h0u)) exceeds f J � (�u). But c2r2(J � (h0u)) scales as f 2B f bmF

ffiffiffiffiBp

, and sodividing out the common f 2B factor, we find

D

f¼ bmF

ffiffiffiffiBp

: ð14Þ

Now suppose B51. In this case, c2r2 f 2B5f 2 in magnitude, so (c2r2� f 2)� scalesas f 2D. Also, when B51, the term f J � (�u) exceeds c2r2(J � (h0u)). But f J � (�u) scales asf 3bmF

ffiffiffiffiBp

, and so dividing out a factor of f 3, we recover (14) again. Hence (14) appliesfor all B. Note that (14) can also be written as D/f¼ bmR. The scaling found isconfirmed in figure 9.

Note, the scaling in (14) implies that � itself scales as bmU0/L, independent of both theGW speed c and the Coriolis frequency f. Nevertheless, in the scale analysis we requiredU0/c� 1 and �/f� 1 (the latter we showed scales as bm in figure 9). Evidently, thisscaling for � is a consequence of the disparate scales of U0 and c, and of � and f.

We can also now justify neglecting �h0 compared to u �Jh0 in the above analysis (thesame argument can be used for h0 replaced by � here). From (14), �h0 has the scalef b2mF

ffiffiffiffiBp

, whereas u �Jh0 has the scale f bmFffiffiffiffiBp

. The ratio of these terms is bm, whichis assumed �1, so we can neglect �h0.

3.2.2. Effect of varying the amplitude of topography bm and the Froude number F . Aswe have seen, regardless of the Burger number B, the higher the Froude number F , themore divergence is produced. This makes sense, as larger flow oscillations (or ‘‘IOs’’ forinertial 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 thefact that the definition of PV (equation 2) tells us that the smaller the topography b (thebigger 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 B¼ 1 and for the different values of bmand F , in figure 10. This figure shows that the smaller the topography, the smaller thedifferences in the divergence produced by different Froude numbers (this also holds forother Burger numbers). In fact, in the case of the smallest amplitude topography 0:1bOM,changes in the amplitude of the IOs have practically no effect on the divergenceproduced. The minimal divergence differences between them are slightly larger forlarger Burger numbers.

An interesting feature of figure 10 is that the presence of IOs is most visible forFroude number F ¼ 0.3. This is true not only for B¼ 1, but also for B¼ 0.25. This canbe explained by the fact that weak IOs are not able to cause noticeable interactions withthe 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 theF ¼ 0.3 case. This does not carry over to B¼ 4, which has some random fluctuations inthe amount of divergence for all amplitudes of the IOs and topography, the strongest



being for large values of both. This, again, can be attributed to the fact that topographiclocking is not as strong as in the other cases, and so the main source of divergence isfrom the anomalies (both depth and PV) interacting with the topography itself.

3.2.3. Effect of varying the Burger number B. It has been found that larger Burgernumbers produce more divergence, at least initially. As time progresses, the amount ofdivergence falls in the larger Burger number cases, and the differences between flowshaving different values of B become less significant, particularly for the smallest Burgernumbers.

These differences are more significant for larger topography. For the smallest topo-graphy, there are nomajor differences in the amount of divergence produced between thedifferent Burger numbers or Froude numbers. Furthermore, the larger the Burgernumber, the bigger the differences become between flows with different IO amplitudesover the same topography. An explanation for these properties is provided in section 3.3.

3.2.4. Comparison with the steady flow. The presence of an oscillating flow has aprofound influence on GW activity, which can be most clearly seen when flows forcedby 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 Froudenumbers, with maximum topography bM (note that the cases with bOM and bSM arecompared, even though bOM 6¼ bSM). The divergence created by the steady flow is more‘‘constant’’ than in the case of IOs, where strong fluctuations occur. An exception tothis is in the beginning, when the steady flow rips the PV anomalies off the topographyand in doing so produces sudden and short-lived GW activity.

When oscillating flows are compared to the equivalent steady flow cases, it can beseen that there are bigger differences in the divergence between the two extreme Froudenumbers for IOs, this being most visible for B¼ 1 in figure 11. However, this does notmean that the oscillating flow cases with lower F have less divergence than the steadyflow cases. In fact, the situation is just the reverse. Even the case of the oscillating flowwhich produces the least amount of divergence (i.e. having the lowest F ) is still moredivergent than the case of the steady flow producing the most divergence (i.e. having thehighest F ). The only exception to this occurs in the case B¼ 4, which exhibits a ‘‘burst’’

00

0.28

(a) (b) (c)r.

m.s

. δ

t/Teddy 20

Figure 10. r.m.s. divergence � for B¼ 1 at Froude number F ¼ 0.2 (a), F ¼ 0.3 (b), and F ¼ 0.4 (c) for thedifferent topography amplitudes: bOM (thin line), 0:5bOM (þ line), and 0:1bOM ( line).



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.

3.2.5. Kinetic energy due to divergent flow. We quantify next the amount of energyscattered into propagating waves by measuring the kinetic energy associated with the

divergent component of the flow, namely

Ed ¼1

2hhðu2d þ v2dÞi, ð15Þ

where h is the total depth field, (ud,vd) is the divergent velocity, and h� � �i indicates aspatial average. Ed is then normalised by the total energy E to give the dimensionless

measure ~Ed ¼ Ed=E. The complement of ~Ed then gives the amount of energy stored in thenon-divergent, balanced, vortical component of the flow. As seen in figure 12, for an

oscillating flow ~Ed is approximately independent of the Burger number and is

proportional to the amplitude of topography squared b2m times the Froude number F .

Indeed, for large and small values of the Burger number, ~Ed=Fb2m collapses for all

Froude numbers to a value between 2 and 4. Ed itself, however, is never more than 6.7%

of the total energy. Even less energy is scattered into divergent motions for a steady

flow: Ed is never more than 4.0%. Moreover, Ed appears to scale with F 2b2m, as seenin figure 13.

3.3. Topographic locking of depth and PV anomalies

To measure the impact of topography on the flow evolution, and what role the lockingof 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 linearrelationship exists between them. It gives not only the ‘‘strength’’ of this relationship,

00

0.40

(a) (b) (c)r.m

.s. δ

t/Teddy 20

Figure 11. r.m.s. divergence � for B¼ 0.25 (a), B¼ 1 (b), and B¼ 4 (c) for maximum topography withFroude number F ¼ 0.2 (thin line – oscillating flow, dashed line – steady flow) and F ¼ 0.4 (þ line –oscillating flow, line – steady flow).



but also the direction, and is defined to be

Cðx, yÞ ¼hxyi

hx2i1=2hy2i1=2: ð16Þ

Both the correlations of topography b with depth anomalies h0 ¼ h� 1 and with PVanomalies q0 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

00

6E d

/(F

b2 m

)

t/Teddy 20

00

6

E d/(F

b2 m

)

t/Teddy 20

00

6

E d/( F

b2 m

)

t/Teddy 20

Figure 12. Normalised kinetic energy of the divergent flow ~Ed scaled by the amplitude of topographysquared b2m and Froude number F for oscillating flows when B¼ 0.25 (column 1), B¼ 1 (column 2), and B¼ 4(column 3) at Froude number F ¼ 0.2 (row 1), F ¼ 0.3 (row 2), and F ¼ 0.4 (row 3) for the differenttopography amplitudes: bOM (thin line), 0:5bOM (þ line), and 0:1bOM ( line).

yWhen ‘‘low’’ or ‘‘high’’ values of correlation are referred to, absolute values are meant. ‘‘Low’’ implies aweak correlation, while ‘‘high’’ implies a strong one.



3.3.1. Effect of varying the amplitude of topography bm and the Froude number F . Fora fixed Burger number, we find that increasing the Froude number decreases thecorrelation of the topography with both the depth and PV anomalies. There do,however, exist differences between the topography-depth correlation C(b, h0) and thetopography-PV correlation C(b, q0). It has been found that for all Burger numbers, thecorrelation C(b, h0) for the two smaller amplitudes of topography is much more similar(and has lower values) than that for the maximum topography. In the cases withFroude number F ¼ 0.2, all values of topography have similar correlations. This seemsto imply that there is some critical amplitude of topography, after which making it anysmaller does not make any difference to the correlation C(b, h0).

Furthermore, and also holding for all Burger numbers, we find that as the Froudenumber of the flow increases, the fluctuations in the correlation C(b, q0) have a largeramplitude, a higher frequency and last longer. As can be seen from figure 14, these aremost visible for small topography. In fact, the smaller the topography and the biggerthe amplitude of the IOs, the larger the fluctuations become. As the IOs swell, the PVanomalies are almost ripped off the topography (and the correlation falls), and as theIOs fall they return (the correlation rises). For B¼ 0.25 and B¼ 1, as time passes, these

00.30

1

(a) (b) (c)

C(b

,q„)

t/Teddy 20

Figure 14. Correlation C(b, q0) for B¼ 4 for Froude numbers F ¼ 0.2 (a), F ¼ 0.3 (b), and F ¼ 0.4 (c) forthree different amplitudes of topography: bOM (thin line), 0:5bOM (þ line), and 0:1bOM ( line).

0

0

3.75

(a) (b) (c)E d

/(F

2b2 m

)

t/Teddy 20

Figure 13. Normalised kinetic energy of the divergent flow ~Ed scaled by the amplitude of topographysquared b2m and Froude number squared F 2 for the steady flow when B¼ 0.25 (a), B¼ 1 (b), and B¼ 4 (c) formaximum topography with Froude number F ¼ 0.2 (thin line), F ¼ 0.3 (þ line), and F ¼ 0.4 ( line).



fluctuations subside, and stay roughly at the same value. Fluctuations for B¼ 4 alsohave a decreasing trend, but the simulation is not long enough for them to damp. As theBurger number increases, the damping of the fluctuations takes longer, as the initial PVanomaly has a small scale compared to LD (recall L/LD¼B

�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 ofcorrelation occur. However, the PV anomalies are not ripped off the topography; ratherthey are gradually dispersed, and thus the general trend in C(b, q0) as time passes isdownward. Recall that, in all simulations, the topography is identical (apart for themaximum permissible amplitude bM, which diminishes with Burger number), and thatthe depth and PV anomalies weaken with increasing Burger number. This explains thefact that the correlation C(b, h0) is higher for smaller Burger numbers than for largerones, especially when C(b, h0) is compared to C(b, q0). Another explanation for the factthat C(b, h0) is higher than C(b, q0) is that the PV is dispersed away from its origin, andthus the correlation falls, yet there will always be some kind of correlation betweentopography and depth anomalies in places where topography is bigger (for an example,see section 3.1.1).

Another link between the correlations C(b, h0) and C(b, q0) is that as the PV graduallyescapes the topography (characterised by the decreasing trend in fluctuations ofC(b, q0)), fluctuations in C(b, h0) begin. These are more regular for B¼ 0.25 and B¼ 1for the two smaller Froude numbers F , whereas they are more irregular for F ¼ 0.4. Inthe case when B¼ 4, the fluctuations seem to be more erratic than in the other twocases, but most noticeably so for F ¼ 0.4.

Like with changes in the Froude number, changes in the amplitude of the topographyalso have an influence on both the correlations C(b, h0) and C(b, q0). The larger thetopography, the higher the correlation C(b, h0), and the lower C(b, q0) (which becomeslow 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 largeand thus is able to escape more quickly. Furthermore, for large topography, there is astrong link between it and the depth anomaly.

When the effects on the correlations due to changes in the amplitude of topographybm or Froude number F are compared, we find that changes in F have a much biggerinfluence on C(b, h0) than changes in bm (as can be seen from figure 15(a)), regardless ofthe Burger number.

3.3.2. Effect of varying the Burger number B. As noted above, differences incorrelations occur between different Burger numbers. Although increasing the Burgernumber does not necessarily change the average value of C(b, h0), it does affect itssmoothness, as is clear from figure 15(a). There are much stronger fluctuations at largerBurger numbers, and this can be explained by the fact that the depth anomalies areswung back and forth by the IOs much more.

The correlation C(b, q0) is much more affected by changes in the Burger number thanis C(b, h0), as can be seen by comparing figures 15(a) and (b). Changes in the Burgernumber thus affect C(b, q0) more than changes in the amplitudes of either the IOs or thetopography. All flow simulations for B¼ 0.25 have strongly topographically lockeddepth and PV anomalies, regardless of the amplitude of the IOs or topography. As theBurger number grows, the same flows are less correlated, and there are bigger



differences in the correlations between flows with different Froude numbers over thesame topography.

A similarity between the cases with different Burger numbers is that they all followthe same pattern for all topography and Froude numbers. Only the time and magnitudescales vary. For smaller Burger numbers, the processes are quicker.

3.3.3. Comparison with the steady flow. To better understand the effects of oscillatingflows on the locking of depth and PV anomalies, the above simulations are nextcompared to the equivalent ones carried out with a steady flow.We find that in the steadyflow case, regardless of the Burger number, the correlations C(b, q0) have similar values.This is not the case for the correlations C(b, h0), which vary with the Burger number.

For B¼ 0.25, the correlation C(b, h0) is much higher for the IOs. This is because thedepth anomalies are not able to escape in this case, while in the steady flow case they areimmediately ripped off. Also, since the depth anomalies are spatially large, as they aredragged along by the steady flow there is no visible correlation – hence the lowvalues of C(b, h0).

For B¼ 1, the correlations C(b, h0) for the steady flow and the IOs are similar, withthe steady flow case fluctuating much more. Similarities for this value of the Burgernumber arise because the depth and PV anomalies are able to gradually escape thetopography in both cases. However, the fluctuations are not as pronounced with IOsbecause both types of anomalies are not advected very far by the flow, but remain in theproximity of their origin.

0–1

–0.70(a)

(b)

C(b

,h„)

C(b

,q„)

t/Teddy

t/Teddy

20Correlation C(b, h„)

Correlation C(b, q„)0

0.30

1

20

Figure 15. Correlations for B¼ 0.25 (A), B¼ 1 (b), and B¼ 4 (c) for the maximum topography with F beingmaximum (dashed line) and minimum (thin line), and minimum topography with F being maximum ( line)and minimum (þ line). Case: oscillating flow.



For B¼ 4, for the first time there is a higher correlation C(b, h0) for the steady flowthan for the IOs. Since the initial depth and PV anomalies are of a scale small comparedto LD, it takes a long time for them to disperse, even as the steady flow advects themaway. This leads to a stronger correlation than previously.

4. Conclusions

In this study, we have carried out a series of numerical simulations assessing the effectsof oscillating flows (which in certain cases may be viewed as idealised tides) andtopography on shallow-water turbulence. The simulations covered a broad range ofparameter space in the amplitude of the topography bm, the strength of the flowoscillations U0, and the Rossby deformation length LD. Our key findings aresummarised next.

First, regarding the generation of gravity waves, the strongest generation (measuredhere by the divergence �) is produced for large Burger numbers and for flows withstrong flow oscillations and for large topographic amplitudes. However, this isexpected. We showed, by scale analysis, that the amount of divergence produced for afixed Burger number B¼ (LD/L)

2 is roughly proportional to the amplitudes of theoscillating flow and of the topography (indeed �=f bmF

ffiffiffiffiBp

where F ¼U0/c). Theeffects of the two are linked – flow oscillations have less effect on flows over smallamplitude topography. In fact, it has been shown that � bmU0/L, which means thatmost divergence is produced for a strong oscillating flow over steep topography (bm/Lcan be seen as the gradient of topography). Oscillating flows also generate much moredivergence than steady flows. Even the oscillating flow case producing the leastdivergence (i.e. having the lowest Froude number) produces more or similar divergencethan the steady flow case producing the most divergence (i.e. having the highest Froudenumber). Only when the Burger number is large, the divergence is nearly the same.Furthermore, the amount of energy scattered into propagating waves is approximatelyindependent of the Burger number and proportional to the square of the amplitude oftopography times the Froude number Fb2m (oscillating flow case) and to the square ofthe amplitude of topography times the Froude number squared F 2b2m (steadyflow case).

Second, concerning the extent to which the initial depth and PV anomalies are lockedto the topography, the topography-depth anomaly correlation C(b, h0) is a betterindicator of the locking of the anomalies by topography than the topography-PVanomaly correlation C(b, q0). With increasing Burger and Froude numbers, we find thattopographic locking weakens. However, the effects associated with weak topographyare not as clear. Weak topography leads to slightly lower values of C(b, q0) than strongtopography at late times, but it appears to take much longer to reach these lowcorrelations – the release of the depth and PV anomalies is slow and gradual. It can besaid that the escape of the depth and PV anomalies in the oscillating flow case is not assudden or as strong as when they are ripped off the topography by a steady flow havingthe same Froude number F . Nevertheless, increasing the Burger number makes thetopography less able to trap the depth and PV anomalies.

Linking the two analyses, we conclude that the greatest gravity wave generation isproduced as the depth and PV anomalies escape from topographic locking. Indeed, as



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 0 and hþ b¼ 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 0 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.

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