An energy and potential enstrophy conserving numerical scheme for the multi-layer shallow water equations with complete Coriolis force

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1 An energy an potential enstrophy conserving numerical scheme for the multi-layer shallow water equations with complete Coriolis force Anrew L. Stewart a, Paul J. Dellar b a Department of Atmospheric an Oceanic Sciences, University of California, Los Angeles, CA 90095, USA b OCIAM, Mathematical Institute, Anrew Wiles Builing, Racliffe Observatory Quarter, Woostock Roa, Oxfor OX2 6GG, UK Abstract We present an energy- an potential enstrophy-conserving scheme for the non-traitional shallow water equations that inclue the complete Coriolis force an topography. These integral conservation properties follow from material conservation of potential vorticity in the continuous shallow water equations. The latter property cannot be preserve by a iscretisation on a fixe Eulerian gri, but exact conservation of a iscrete energy an a iscrete potential enstrophy seems to be an effective substitute that prevents any istortion of the forwar an inverse cascaes in quasi-two imensional turbulence through spurious sources an sinks of energy an potential enstrophy, an also increases the robustness of the scheme against nonlinear instabilities. We exploit the existing Arakawa Lamb scheme for the traitional shallow water equations, reformulate by Salmon as a iscretisation of the Hamiltonian an Poisson bracket for this system. The non-rotating, traitional, an our non-traitional shallow water equations all share the same continuous Hamiltonian structure an Poisson bracket, provie one istinguishes between the particle velocity an the canonical momentum per unit mass. We have etermine a suitable iscretisation of the non-traitional canonical momentum, which inclues aitional coupling between the layer thickness an velocity fiels, an moifie the iscrete kinetic energy to suppress an internal symmetric computational instability that otherwise arises for multiple layers. The resulting scheme exhibits the expecte secon-orer convergence uner spatial gri refinement. We also show that the rifts in the iscrete total energy an potential enstrophy ue to temporal truncation error may be reuce to machine precision uner suitable refinement of the timestep using the thir-orer Aams Bashforth or fourth-orer Runge Kutta integration schemes. Keywors: geophysical flui ynamics, energy conserving schemes, Hamiltonian structures Receive 2 August 205, accepte 8 December 205, J. Comput. Phys oi:0.06/j.jcp Introuction Geophysical flui ynamics is concerne with the large-scale motions of stratifie flui on a rotating, nearspherical planet. In principle such motions are governe by the compressible Navier Stokes equations as moifie to inclue rotation, but simplifie an approximate forms of these equations are wiely use in both theoretical an computational moels [, 2]. At large scales viscous effects are relatively unimportant, an usually neglecte. The resulting invisci equations possess various conservation laws, for instance of energy, momentum, an potential vorticity [2]. The last property is perhaps the most important. Energy an momentum may be transporte over large istances by waves, but potential vorticity is tie to flui elements [3]. It is esirable for the simplifie an approximate equations to share these conservation properties, an as far as possible for any numerical scheme to preserve them too. Planetary scale flui flow is typically ominate by the Coriolis force. However, almost all moels of the ocean an atmosphere employ the so-calle traitional approximation [4]. This retains only the contribution to the Coriolis force from the component of the planetary rotation vector that is locally normal to geopotential surfaces. The traitional approximation is typically justifie on the basis that the vertical lengthscales of rotationally-ominate geophysical flows are much smaller than their horizontal lengthscales [5]. However, the non-traitional component of the Coriolis force is ynamically important in a wie range of geophysical flows [6]. In the ocean, nontraitional effects can substantially alter the structure an propagation properties of gyroscopic an internal waves [7 2], the evelopment an growth rate of inertial instabilities [3, 4], the instability of Ekman layers [5], high-latitue eep convection [6], the static stability of the water column [7], an the flow of abyssal waters close to the equator [8 20]. A parallel line of evelopment has le to eep atmosphere moels that inclue non-traitional effects [2 26]. c 206 This manuscript version is mae available uner the CC-BY-NC-ND 4.0 license

2 The focus of this article is a set of shallow water equations that inclue the full Coriolis force [27, 28]. Shallow water moels approximate the ocean s stratification using a stack of layers of constant-ensity flui. This is equivalent to a Lagrangian vertical iscretization of the water column using ensity as the vertical coorinate. Density or isopycnal coorinates are a natural choice for ocean moels because mixing an transport across isopycnal surfaces in the ocean interior is weak, so large-scale flows are largely constraine to flow along ensity surfaces [29]. Moels base on vertical coorinates aligne with surfaces of constant geopotential ten to prouce spuriously large iabatic mixing ue to the slight misalignment between gri cells an isopycnal surfaces [30]. Shallow water an isopycnal coorinate moels are thus wiely use to stuy the ocean circulation, both in conceptual moelling stuies an comprehensive ocean moels [3 33]. This isopycnal moelling framework has recently been extene to solve the non-hyrostatic three-imensional stratifie equations with the non-traitional components of the Coriolis force [34]. In this paper we erive an energy- an potential enstrophy-conserving numerical scheme for our extene shallow water equations that inclue the complete Coriolis force [27, 28]. There are three principal motivations for exactly preserving these conservation laws in our scheme:. Ieally any numerical scheme shoul materially conserve potential vorticity, as this is a crucial ynamical variable in large-scale geophysical flows [3]. Material conservation of potential vorticity, Dq/Dt = 0, implies conservation of all the integral moments of q, t x hcq = 0, for any function cq. These spatial integrals are much more amenable to approximation on a fixe gri. The most important is conservation of potential enstrophy, for which cq = q 2. However, exact material conservation of potential vorticity is only possible via specialise Lagrangian schemes such as the particle-mesh metho [35] or the contour-avective semi-lagrangian metho [36]. Thompson [37] showe that a vorticity evolution equation for an incompressible flui that conserves energy an potential enstrophy, an satisfies reasonable properties such as spatial locality for the streamfunction an invariance uner rotations, is inevitably of material conservation form. Although erive for partial ifferential equations in continuous space, this result suggests that conservation of energy an potential enstrophy on a fixe Eulerian gri provies a suitable substitute for material conservation of potential vorticity. 2. Conservation of energy an potential enstrophy is crucial for the evelopment of the simultaneous inverse cascae of energy an forwar cascae of potential enstrophy that characterises quasi-two-imensional turbulence [2]. Long-term integrations using non-conservative finite-ifference schemes are prone to a spurious cascae of energy towars the scale of the gri spacing, where the spatial truncation error is largest [38]. Conserving the iscrete analogue of total energy alone oes not prevent the cascae of energy to small scales, an leas to an increase in potential enstrophy [39]. Thus long-time integrations require a scheme that conserves both energy an potential enstrophy, though even this is not sufficient to completely eliminate spurious spectral transfers of these quantities [40]. 3. Enforcing these conservation properties eliminates the possibility of nonlinear instabilities riven by spurious sources of energy or potential enstrophy at the gri scale. For example, Fornberg [4] showe that the only stable member of a class of semi-iscrete centre finite ifference schemes for the invisci Burgers equation is the scheme that conserves a iscrete energy. Arakawa [38] constructe a finite ifference scheme for the two-imensional incompressible Euler equations, now known as the Arakawa Jacobian, that exactly conserves a iscrete kinetic energy an a iscrete enstrophy when formulate in continuous time. Arakawa & Lamb [42] later extene this approach to construct a finite ifference scheme for the invisci shallow water equations that exactly conserves total mass, energy, an potential enstrophy. The spurious energy cascaes observe in non-conservative long-time integrations of the SWE see point 2 above are ue to errors introuce in the spatial, rather than temporal, iscretisation [39], as we shall confirm in 5. The Arakawa Lamb scheme therefore iscretises only the spatial erivatives of the traitional shallow water equations, yieling evolution equations in continuous time for the iscrete particle velocities an layer thicknesses at spatial gri points. This semi-iscrete finite ifference scheme has since been extene to multiple layers [43], to fourth-orer accuracy for the vorticity in the non-ivergent limit [44 47], to spherical geoesic gris [48], an to unstructure finite element meshes [49]. Arakawa & Lamb [42] began with a general finite ifference iscretisation of the shallow water equations on a staggere gri, known as the Arakawa C-gri, as sketche in figure. They wrote own a iscretisation that containe many unetermine coefficients, incluing some unphysical Coriolis terms that were parallel, not perpenicular, to the flui velocity. They obtaine a scheme that conserves consistent spatially iscrete approximations to the energy an potential enstrophy via irect algebraic manipulation of the iscrete equations to etermine suitable coefficients. In particular, the coefficients of the unphysical Coriolis terms become proportional to gri spacing square. That 2

3 a iscrete scheme shoul exist that conserves not quaratic invariants like the Arakawa Jacobian [38], but a cubic quantity energy an a quotient potential enstrophy was long mysterious. Salmon [45] interprete the Arakawa Lamb scheme as a particular iscretisation of a non-canonical Hamiltonian formulation of the traitional shallow water equations, in which the evolution is expresse using a Hamiltonian functional an a Poisson bracket. Symmetries of a Hamiltonian system correspon irectly to conservation laws via Noether s theorem, making this a natural starting point for the evelopment of conservative schemes. The equations of ieal flui mechanics, incluing the shallow water equations, can be expresse as non-canonical Hamiltonian systems using space-fixe Eulerian variables [2, 50 52]. To keep the canonical form of Hamilton s equations one must either use Lagrangian variables that follow flui elements, or introuce artificial Clebsch potentials. The non-rotating, traitional, an our non-traitional shallow water equations all share the same Hamiltonian structure an Poisson bracket, provie one istinguishes between the particle velocity an the canonical momentum per unit mass [27, 28]. This istinction is often glosse over in the traitional shallow water equations because the two iffer only by a function of space alone, so their time erivatives are ientical. We use this insight to construct an analog of the Arakawa Lamb scheme for our non-traitional shallow water equations by aapting Salmon s [45] Hamiltonian formulation. The main ifficulty we encounter is that the layer thickness fiel enters the canonical momentum through the non-traitional terms, so we nee to fin a suitable average of the layer thickness fiel that is compatible with the placement of variables on the Arakawa C gri. Like the Arakawa Jacobian an Arakawa Lamb schemes, we obtain a system of orinary ifferential equations ODEs for the evolution in time of fiel values at gri points. Discrete analogs of the energy E an potential enstrophy P are exactly conserve uner the exact evolution of this ODE system. To get a concrete numerical scheme we further iscretise in time using the 4th orer Runge Kutta an 3r orer Aams Bashforth schemes [53]. We investigate the time step that is necessary to reuce the temporal truncation errors in our iscrete approximations of E an P own to machine precision. Recently a similar Hamiltonian an Poisson bracket formulation has been employe to erive energy conserving schemes for the eep an shallow atmosphere equations [54], again using the canonical momenta to formulate the iscrete Poisson bracket an ensure antisymmetry. However, these authors were unable to construct a iscrete Poisson bracket that conserves potential enstrophy. A more general approach to constructing schemes that conserve energy an potential enstrophy formulates the shallow water equations as evolution equations for the potential vorticity, ivergence, an layer thickness fiels using iscrete Nambu brackets [55, 56]. This formulation requires an elliptic inversion to compute the velocity fiel, which is computationally expensive but allows for a straightforwar incorporation of bounary conitions. The structure of this paper is as follows. In 2 we briefly review the shallow water equations with complete Coriolis force an their Hamiltonian formulation [27, 28]. In 3 we review Salmon s Hamiltonian erivation of the Arakawa Lamb scheme for the traitional shallow water equations 3., an exten it to inclue the complete Coriolis force in a single layer 3.2 an in multiple layers 3.3. In 4 we present a series of numerical verification experiments, an in 5 we emonstrate that energy an potential enstrophy can be conserve to machine precision if the time-step is sufficiently short. Finally, in 6 we summarise our results an provie concluing remarks. 2. Shallow water equations with complete Coriolis force The erivation of our numerical scheme begins with the non-canonical Hamiltonian formulation of the traitional shallow water equations, henceforth abbreviate as SWE, as escribe in [50 52]. The single an multi-layer nontraitional shallow water equations henceforth NTSWE were recently erive in [27, 28]. For clarity we first present the non-canonical Hamiltonian formulation of the single-layer NTSWE. We begin by writing the single layer NTSWE [27, 28] in the vector invariant form [57], ũ Φ hqv + t x = 0, ṽ Φ + hqu + t y = 0, h t + x hu + hv = 0. y 2c 2a 2b These equations are written in pseuo-cartesian coorinates with x an y enoting horizontal istances within a constant geopotential surface [5, 58]. The effective gravity g acts in the z irection perpenicular to this surface. They escribe a layer of invisci flui flowing over bottom topography at z = h b x, y in a frame rotating with angular velocity vector Ω = Ω x, Ω y, Ω z. We allow for an arbitrary orientation of the x an y axes with respect to North, so Ω x an Ω y may both be non-zero, an may epen arbitrarily on x an y, but not z [28]. 3

4 The mass conservation equation 2c for the flui layer thickness hx, y, t contains the particle velocity components u an v in the x an y irections. These particle velocity components also appear in the Bernoulli potential Φ = 2 u2 + 2 v2 + gh b + h + h Ω x v Ω y u. 3 The vertical momentum equation for the unerlying three-imensional flui contains aitional non-traitional contributions proportional to the horizontal components Ω x an Ω y of the rotation vector. The three-imensional pressure whose epth-average appears in the shallow water equation thus satisfies a three-imensional quasihyrostatic balance [22], rather than the usual hyrostatic balance. However, the time erivatives in 2a an 2b are of the canonical velocity components ũ = u + 2 Ω y h b + 2 h, ṽ = v 2 Ω x h b + 2 h, 4 which together form the vector ũ = ũ, ṽ. These velocity components also appear in the potential vorticity q = 2 Ω z + ṽ h x ũ, 5 y where Ω z is the component of the rotation vector parallel to the z-axis. We call ũ the canonical velocity because it is relate to the canonical momentum per unit mass, or to the epthaverage of the particle velocity relative to an inertial frame [27, 28]. For spatially-uniform Ω x, Ω y, Ω z, i.e. making an f -plane approximation, the latter quantity is ũ = u Ω z y + 2 Ω y h b + 2 h, ṽ = v + Ω z x 2 Ω x h b + 2 h, 6 which may be erive from the epth average of the three-imensional vector fiel ũ = u + Ω x + xzω y yzω x = u + 2zΩ y yω z, xω z 2zΩ x, 0. 7 The first two terms u + Ω x give the three-imensional velocity relative to an inertial frame. The aitional graient of a gauge potential eliminates the vertical component of the vector potential for the Coriolis force, as require for a shallow water system formulate in terms of the epth-average horizontal velocity alone. The potential vorticity is q = ṽ h x ũ, 8 y with no explicit contribution from the rotation vector. The terms proportional to Ω z in 6 are inepenent of time, so they o not contribute to the time erivatives in 2a an 2b. Thus the istinction between the particle velocity an canonical velocity is commonly not mae in the traitional shallow water equations, in which the aitional terms proportional to both Ω x, Ω y an the time-epenent hx, y, t are absent. Instea, the contribution from Ω z is put explicitly in the efinition 5 of the potential vorticity. By using 4 instea of 6 we omit the time-inepenent terms proportional to Ω z, which are much larger than the other terms in the canonical momenta by a factor of the inverse Rossby number. This reuces the accumulation of floating point roun-off errors in transforming between the canonical an particle velocities. It is also then straightforwar to impose perioic bounary conitions, because the explicit x an y epenence in 6 is absent. Moreover, the non-traitional numerical scheme we erive below will reuce to the exact form of the Arakawa Lamb scheme when we set Ω x an Ω y to zero. Conversely, the aitional terms proportional to x an y in 6 will be ifferentiate exactly by a secon-orer accurate spatial iscretisation, so the equivalent scheme base on the full canonical velocity 6 instea of 4 woul give ientical results in exact arithmetic. By istinguishing between the particle velocity u an the canonical velocity ũ we have written the NTSWE in the same structural form 2a 2c as the SWE. However, in the following we treat the particle velocity as a known function of the canonical velocity an the layer thickness, as given by solving 4 for u = uũ, h. The Hamiltonian for the system 2a 2c is simply the total energy, which is inepenent of the Coriolis force when expresse in terms of the particle velocity components u an v, H = E = { x 2 h u 2 + v 2 + gh } h b + 2 h. 9 However, we treat H = H [uũ, h, h] as a functional of the canonical velocity components an layer thickness. We o not specify a omain of integration for this functional, but assume that the bounaries of the omain are perioic, or permit no normal fluxes, to allow us to integrate by parts as necessary below without acquiring bounary terms. 4

5 The non-canonical Hamiltonian formalism expresses the time evolution of any functional F in terms of a Poisson bracket as [2, 50 52] F = {F, H}. 0 t The Poisson bracket must be bilinear, antisymmetric, an satisfy the Jacobi ientity {F, {G, K}} + {G, {K, F }} + {K, {F, G}} = 0, for any three functionals F, G, K. These three properties provie a coorinate-inepenent abstraction of the essential geometrical properties of Hamilton s canonical Poisson bracket. The Poisson bracket for the shallow water equations may be written as {A, B} = { x q δa δb δũ δṽ δb δũ δa δa δṽ δũ δb δh + δb } δũ δa, 2 δh for any pair of functionals A an B. This bracket may be rewritten in several ifferent but equivalent forms using integrations by parts [5]. The functional erivatives in 2 are efine with respect to the canonical velocities ũ an h, so this bracket reuces to the traitional shallow water Poisson bracket when Ω x = Ω y = 0, since ũ then equals u. The evolution equations 2a 2c may be erive by setting in turn F = ũx 0, t, F = ṽx 0, t, an F = hx 0, t for a fixe location x 0 in 0. For example, setting F = h 0 = hx 0, t in 0 gives h 0 t = {h 0, H} = x { q δh0 δh δũ δṽ δh δũ δh 0 δṽ δh 0 δũ δh δh + δh δũ δh 0 δh For the purpose of taking variational erivatives, ũ an h are treate as inepenent variables so δh 0 /δũ = 0. The particle velocity u is treate as an explicit function of ũ an h, so the variational chain rule gives δh δũ = δh δh δũ + u δu u ũ, δh δu u δh = ṽ δu, δh = δh = hu. 3 δv δu The notation u inicates that u is hel constant while taking the variational erivative with respect to ũ. Now δh 0 /δh is nonzero only at x = x 0, so { h 0 δh = x t δu δhx } 0, t { } = x hu δx x 0, 4 δhx, t where δx x 0 is the Dirac elta function in two imensions. Finally, we integrate by parts an use the ivergence theorem with the perioic or no-flux bounary conitions to obtain { } t hx 0, t = x hu δx x 0 = hu, 5 x=x0 which is the mass conservation equation 2c. Symmetries in the variational principle for a Hamiltonian system imply conservation laws via Noether s theorem. The usual momentum an energy conservation laws arising from translation symmetries in space an time. Material conservation of potential vorticity arises from a more subtle relabelling symmetry in the particle-to-label map use to formulate the shallow water equations in Lagrangian coorinates [2, 50, 59, 60]. In the non-canonical Hamiltonian formalism, these conservation laws arise instea from properties of the Poisson bracket. Conservation of energy follows immeiately from antisymmetry of the Poisson bracket, E t = H t Non-canonical Poisson brackets typically support Casimir functionals C that satisfy = {H, H} = 0. 6 {C, F } = 0 7 for any functional F. Two salient Casimir functionals for the shallow water Poisson bracket in 2 are the total mass M, an the potential enstrophy P, { } { } M = x h an P = x 2 hq2. 8 }. 5

6 2 r r r r q q q r r r r r r r r q q q r r r r q q r r r r r r r r a q q q r r r r r r r r r r r r r r b q q r r r r r r r r r r Figure : Staggere layout of the variables on the Arakawa C gri. Panel a shows the inices of the computational gri an its alignment with the coorinate axes. Panel b shows the layout of the variables within a single computational cell. The latter is just one of an infinite family of Casimir functionals of the form { } C = x h cq, 9 where cq is an arbitrary function of the potential vorticity. This family of Casimir functionals thus expresses the material conservation of potential vorticity in an Eulerian setting. In the following sections we will show that iscretizing the non-canonical Hamiltonian formulation of the NTSWE allows us to enforce exact conservation of iscrete analogues of the total energy E, potential enstrophy P, an total mass M. 3. Spatial iscretisation In this section we erive an energy- an potential enstrophy-conserving spatial iscretisation for the NTSWE. We begin with a brief review of the Arakawa Lamb scheme [42] for the single-layer SWE an its Hamiltonian formulation [45], then exten the scheme to the NTSWE, an to multiple layers. 3.. Hamiltonian formulation of the Arakawa Lamb scheme The Arakawa Lamb [42] scheme for the traitional shallow water equations begins with a staggere placement of the u, v, h an q variables known as the Arakawa C gri, an illustrate in figure. The C gri may be thought of as a mesh of square cells of sie length. The layer thickness h is efine at cell centres, the velocity components are efine at cell faces, an the potential vorticity is efine at cell corners. This layout is optimise for computing the pressure graient terms in the evolution equations for u an v, an for computing the vorticity as the curl of a two-imensional velocity fiel. The velocity components u an v are also naturally place for computing mass fluxes across cell faces. The schemes presente here generalise straightforwarly to rectangular cells with ifferent sie lengths in the x an y irections. Salmon [45] showe that the Arakawa Lamb scheme is equivalent to a iscrete Hamiltonian system with a iscrete Hamiltonian that approximates 9 an a iscrete Poisson bracket that approximates 2. Salmon s iscrete Hamiltonian for the Arakawa Lamb scheme is [45] H = E = 2 h i+/2, j+/2 u 2x i+/2, j+/2 + v 2y i+/2, j+/2 + g h i+/2, j+/2 h b + 2 h i+/2, j+/2. 20 This sum ranges over all cells in the computational gri. We have omitte the factor of 2 that woul make 20 a iscrete approximation to a two-imensional integral x. This factor cancels with the same omission in our iscrete Poisson bracket, which is also a two-imensional integral x. The right han sie of 20 is a straightforwar iscretisation of the total energy 9 on the Arakawa C gri. The kinetic energy terms u 2x an v 2y enote the averages 6

7 of the contributions from the two u an v points either sie of the h point at the centre of each computational cell, as shown in figure b, u 2x i+/2, j+/2 = 2 u 2 +/2 + u 2 i+, j+/2, v 2 y i+/2, j+/2 = 2 v 2 i+/2, j + v 2 i+/2, j+. 2 The evolution of any function F of the iscrete variables u +/2, v i+/2, j, h i+/2, j+/2 is given by the iscrete version of 0 F = {F, H}. 22 t We use non-calligraphic symbols for functions of iscrete quantities, which approximate the corresponing functionals of continuous fiel variables written in calligraphic font. Antisymmetry of the iscrete Poisson bracket in 22 automatically implies conservation of the iscrete energy by the iscrete analog of 6 E t = H t = {H, H} = The Arakawa Lamb scheme also conserves the iscrete potential enstrophy an iscrete total mass efine by P = h xy q 2, M = h i+/2, j+/2. 24 The iscrete potential enstrophy involves the iscrete potential vorticity which contains the natural iscrete relative vorticity on the Arakawa C gri, ζ = q = 2Ω z h xy + ζ, 25 vi+/2, j v i /2, j u +/2 + u /2, 26 an the natural nearest-neighbour average of h on each q gripoint, h xy = 4 hi+/2, j+/2 + h i /2, j+/2 + h i+/2, j /2 + h i /2, j /2. 27 As in the continuous theory, conservation of P an M is guarantee if they are Casimirs of the iscrete Poisson bracket, i.e. {F, P} = {F, M} = 0 for any Fu +/2, v i+/2, j, h i+/2, j+/2. 28 Here Fu +/2, v i+/2, j, h i+/2, j+/2 enotes an arbitrary function of all the u, v an h gripoints, not just those from one particular gri cell. By the chain rule, this is equivalent to requiring { u+/2, P } = { v i+/2, j, P } = { h i+/2, j+/2, P } = 0 for all, 29a { u+/2, M } = { v i+/2, j, M } = { h i+/2, j+/2, M } = 0 for all. 29b Thus the erivation of an energy-, potential enstrophy-, an mass-conserving scheme for the SWE amounts to fining a iscretisation of 2 that satisfies 29a an 29b. Salmon [45] split the iscrete Poisson bracket into a potential vorticity epenent part an a remainer, {A, B} = {A, B} Q + {A, B} R, 30 where A an B are algebraic functions of the u +/2, v i+/2, j, an h i+/2, j+/2 for all i an j. The potential vorticity epenent part is {A, B} Q = q α A, B u +/2+n, v i+/2, j+m A, B + β A, B + γ u +/2+n, u +/2+m v i+/2, j+n, v i+/2, j+m, 3 7

8 where A, B u, v = A B u v B A u v is the Jacobian of A an B with respect to u an v. Again, we omit the factor of 2 that woul make 3 an approximation of the integral in 2. The variational erivatives of the continuous Poisson bracket 2 have become partial erivatives, as in Hamiltonian particle mechanics. The inices n = n x, n y an m = m x, m y are pairs of integers, an we suppose that the coefficients α m,n, β m,n, an γ m,n are only nonzero within a stencil that is 2M + gri points wie, i.e. for n x, n y, m x, m y M. The first term proportional to α m,n approximates to the term proportional to q in the continuous Poisson bracket 2. The aitional terms proportional to β an γ introuce the unphysical Coriolis terms in the Arakawa Lamb scheme that are neee to conserve potential enstrophy [45]. They introuce only an O 2 error, which is consistent with the overall secon-orer accuracy of the scheme. The remaining part of the bracket is {A, B} R = { B A A h i+/2, j+/2 u +/2 + B v i+/2, j h i /2, j+/2 A h i+/2, j+/2 A h i+/2, j /2 32 } A B, 33 where A B inicates that all the preceing terms in 33 shoul be repeate with A an B exchange to enforce the anti-symmetry. We have iscretise the graient in the continuous Poisson bracket 2 using secon-orer central finite ifferences. The coefficients α, β an γ in 3 must now be chosen to satisfy 29a an 29b, so that P an M are Casimirs. The mass M satisfies conition 29b for any α, β an γ because M/ u +/2 = M/ v i+/2, j = 0 an M/ h i+/2, j+/2 =. There are infinitely many suitable combinations of α, β, an γ that conserve potential enstrophy P [45], but for the purpose of illustration we restrict our attention to the coefficients that give the original Arakawa Lamb scheme [42]. These are liste in the appenix. Combining 20, 22, an 30 recovers the iscrete evolution equations for u +/2 t, v i+/2, j t, an h i+/2, j+/2 t erive by Arakawa & Lamb [42], t h i+/2, j+/2 = t u +/2 = + k,l k,l t v i+/2, j = + k,l u i+, j+/2 u +/2 + v i+/2, j+ v i+/2, j, 34a v k+/2,l u k,l+/2 k,l δ k,l+m n α q k,l m δ k,l+m n β β m,n q k,l m Φi+/2, j+/2 Φ i /2, j+/2, 34b u k,l+/2 v k+/2,l δ k,l+n m α q k,l n δ k,l+m n γ γ m,n q k,l m Φi+/2, j+/2 Φ i+/2, j /2, 34c where δ is the Kronecker elta. The iscrete Bernoulli potential is H Φ i+/2, j+/2 = = x h 2 u2 i+/2, j+/2 + 2 v2y i+/2, j+/2 + gh b + h i+/2, j+/2, 35 i+/2, j+/2 using the averages u 2x i+/2, j+/2 an v 2y i+/2, j+/2 efine in 2. The mass fluxes across cell bounaries are u +/2 = H = u +/2 2 v i+/2, j = H = v i+/2, j 2 h i /2, j+/2 + h i+/2, j+/2 u +/2, h i+/2, j /2 + h i+/2, j+/2 v i+/2, j, 36a 36b which involve the natural averages of the h values either sie of the u or v point Extension to the non-traitional shallow water equations The Hamiltonian an Poisson bracket for the NTSWE are formulate using the canonical velocity ũ an layer thickness h as inepenent variables. The particle velocity u is treate as a known function of ũ an h via 4. The 8

9 first step in constructing a iscretisation of the NTSWE is therefore to efine a iscrete canonical velocity. We assign the iscrete canonical an particle velocity components to the same gri points, so u +/2 an v i+/2, j are collocate with ũ +/2 an ṽ i+/2, j respectively see figure. We then efine the particle velocity components u +/2 an v i+/2, j using two arbitrarily-weighte averages over the h points, [ u +/2 = ũ +/2 µ r,s 2Ω y h b + 2 h], 37a i+/2+r, j+/2+s r,s [ v i+/2, j = ṽ i+/2, j + ξ r,s 2Ω x h b + 2 h]. i+/2+r, j+/2+s r,s The summations are taken over all integers r an s. The weights µ r,s an ξ r,s must be chosen so that 37a 37a are consistent with the continuous relations 4 to at least secon orer in. We take µ r,s = ξ r,s = 0 for r, s > M, where M efines our stencil size as before. As in our continuous Hamiltonian formulation of the NTSWE, we efine the iscrete potential an relative vorticities using the canonical velocity components as q = 2Ω z h xy + ζ, ζ = ṽi+/2, j ṽ i /2, j ũ +/2 + ũ /2 It is implicit in 37a 37b an 38 that the horizontal components Ω x an Ω y of the rotation vector are efine on the h points, while the vertical component Ω z is efine on the q points. As iscusse in 2, the total energy is not change by the Coriolis force, whether traitional or non-traitional. We therefore use exactly the same iscrete Hamiltonian 20 for the non-traitional equations, but now treat u +/2 an v i+/2, j as explicit functions of ũ +/2, ṽ i+/2, j, h i+/2, j+/2 via 37a 37b. We pose a iscretisation of the nontraitional Poisson bracket 2 that is analogous to 30 3 for the traitional Poisson bracket, {A, B} Q = q α A, B ũ +/2+n, ṽ i+/2, j+m {A, B} R = ṽ i+/2, j+n, ṽ i+/2, j+m, + β A, B ũ +/2+n, ũ +/2+m + γ A, B { B ũ +/2 + A A h i+/2, j+/2 B A h i+/2, j+/2 ṽ i+/2, j h i /2, j+/2 A h i+/2, j /2 37b 38 39a } A B. 39b The only ifference is the replacement of u an v by ũ an ṽ. The iscrete ynamics are still given by 22, so it follows from the antisymmetry of the Poisson bracket 39a 39b that 23 still hols, an so the iscrete energy E is conserve. The total potential enstrophy P an mass M are given by 24, but now with the potential vorticity q expresse in terms of ũ an ṽ through 38. As in the traitional case, P an M are Casimirs of the non-traitional bracket 39a 39b if they satisfy {ũ+/2, P } = { ṽ i+/2, j, P } = { h i+/2, j+/2, P } = 0 for all, 40a {ũ+/2, M } = { ṽ i+/2, j, M } = { h i+/2, j+/2, M } = 0 for all. 40b The conitions 40a 40b an Poisson bracket 39a 39b are exactly isomorphic to the traitional conitions 29a 29b an Poisson bracket 3 an 33 after replacing u an v by ũ an ṽ. Any choice of α, β, an γ that satisfies 29a 29b also satisfies 40a 40b. For the purpose of illustration we use the coefficients corresponing to the original Arakawa Lamb scheme [42], as given in the appenix. It now only remains to fin explicit evolution equations for ũ +/2, ṽ i+/2, j an h i+/2, j+/2 from 22 using the iscrete Hamiltonian 20 an Poisson bracket 39a 39b. For example, t ũk,l+/2 = { ũ k,l+/2, H } = { ũ k,l+/2, H } Q + { ũ k,l+/2, H } R, t ṽk+/2,l = { ṽ k+/2,l, H } = { ṽ k+/2,, H } Q + { ṽ k+/2,l, H } R, t h k+/2,l+/2 = { h k+/2,l+/2, H } = { h k+/2,l+/2, H } Q + { h k+/2,l+/2, H } R, 4a 4b 4c 9

10 for integers k an l. The Hamiltonian H epens on ũ only through u, so by the chain rule H ũ k,l+/2 = H ṽ k+/2,l = H u k,l+/2 u k,l+/2 ũ k,l+/2 = H v k+/2,l v k+/2,l ṽ k+/2,l = H u k,l+/2 = u k,l+/2, H v k+/2,l = v k+/2,l. 42a 42b Thus the contribution of { ũ k,l+/2, H } Q to the non-traitional iscrete equations is ientical to the contribution of { uk,l+/2, H } Q to the traitional iscrete equations. The same is true for { ṽ k+/2,l, H } Q, { h k+/2,l+/2, H} Q, an { h k+/2,l+/2, H} R. However, the remaining parts of the iscrete evolution equations 4a 4c, { ũ k,l+/2, H } R an { ṽ k+/2,l, H } R, introuce aitional terms via the epenence of u an v on h, H H { = H u +/2 + + H } v i+/2, j. 43 ũ u h k+/2,l+/2 h k+/2,l+/2 h k+/2,l+/2 u +/2 v i+/2, j h k+/2,l+/2 Here the subscript u inicates that all the u an v values at gri points shoul be hel constant in the erivative. Equation 43 efines the iscrete Bernoulli potential Φ k+/2,l+/2 = H/ h k+/2,l+/2. Evaluating the erivatives of u with respect to h in 43 using 37a 37b yiels u +/2 h k+/2,l+/2 = Ω y i+/2, j+/2 µ i k, j l, v i+/2, j h k+/2,l+/2 = Ω x i+/2, j+/2 ξ i k, j l. 44 The iscrete Bernoulli potential may therefore be written as Φ i+/2, j+/2 = 2 u2x i+/2, j+/2 + 2 v2y i+/2, j+/2 + gh b + h i+/2, j+/2 Ω y i+/2, j+/2 µ r, s u i+r, j+/2+s + Ωx i+/2, j+/2 ξ r, s v i+/2+r, j+s, 45 r,s which contains iscrete non-traitional terms proportional to weighte averages of the mass fluxes u an v over the nearby ũ an ṽ points. Comparison with 37a 37b shows that this is symmetric to the weighte average over h gri points in the efinition of the iscrete particle velocities u +/2, v i+/2, j. Thus 45 is guarantee to be a secon-orer accurate approximation to the continuous Bernoulli potential 3, provie the iscrete particle velocities 37a 37b give at least a secon-orer accurate approximation to 4. The choice of the weights µ r,s an ξ r,s oes not impact the conservation properties of the numerical scheme. For simplicity, we use a secon-orer two-gripoint average in 37a 37b, which correspons to µ 0,0 = µ,0 = 2, ξ 0,0 = ξ 0, = 2, an µ r,s = ξ r,s = 0 for all other r an s. The iscrete particle velocities are then efine as r,s u +/2 = ũ +/2 2Ω y h b + 2 hx +/2, v i+/2, j = ṽ i+/2, j + 2Ω x h b + 2 hy i+/2, j, 46a 46b an the iscrete Bernoulli potential becomes Φ i+/2, j+/2 = 2 u2x i+/2, j+/2 + 2 v2y i+/2, j+/2 + gh b + h i+/2, j+/2 + Ω x i+/2, j+/2 v y i+/2, j+/2 Ω y i+/2, j+/2 u x i+/2, j+/2. 47 Note that Ω x an Ω y nee not be locate on h-gripoints, as implie throughout this section. Locating Ω x on v- gripoints i.e. Ω x i+/2, j an Ωy on u-gripoints i.e. Ω y +/2 is natural for calculating the iscrete three-imensional ivergence of Ω at the q-points where Ω z is locate. The three-imensional ivergence of the rotation vector must vanish for potential vorticity to be materially conserve in the continuously stratifie flui equations [58]. The only effect on our scheme from this change woul be to move Ω y an Ω x outsie of the averaging operators in 46a 46b, an to move them within the averaging operators in 47. For example, the rightmost term in 46a woul become 2Ω y +/2 hb + 2 hx, an the rightmost term in 47 woul become +/2 Ωy u x i+/2, j+/2, 3.3. Extension to multiple layers The multi-layer shallow water equations escribe a stack of layers of immiscible fluis whose ensities ρ k ecrease upwars. We use a superscript k to inicate the velocity, thickness, an ensity of the k th layer, with k = 0

11 corresponing to the top layer an k = K to the bottom. We also efine the vertical positions of the interfaces between layers as l=k η K+ = h b, η k = h b + h l. 48 The multi-layer analogue of the Hamiltonian 9 is [28] H = E = K ρ k k= l=k { [ x 2 hk u k 2 ] + v k 2 + gh η k k+ + 2 } hk. 49 The upper surface η k+ of the layer below thus appears as an effective bottom topography in the contribution to the Hamiltonian from layer k. The multi-layer Poisson bracket is just a sum of single-layer brackets weighte by inverse ensity, {A, B} = K k= ρ k { δa x q k δũ k The canonical velocity components in each layer are δb δb δa δṽ k δũ k δṽ k δa δũ δb δb δa + k δh k δũ k δh k }. 50 ũ k = u k + 2Ω y η k+ + 2 hk, ṽ k = v k 2Ω x η k+ + 2 hk, 5 an the potential vorticity in each layer is q k = 2Ω z + ṽk h k x ũk. 52 y The multi-layer NTSWE may be erive from 49 an 50 by substituting F = ũ k, F = ṽ k, an F = h k in turn into 0, as outline in 3.2. The multi-layer shallow water equations conserve the total energy E over all layers, an the mass M k an potential enstrophy P k in each layer, { P k = ρ k x 2 hk q k } } 2, M k = ρ k x {h k. 53 As in the single-layer case, we erive a numerical scheme for the multi-layer NTSWE via a iscretisation of the Hamiltonian an Poisson bracket that conserves iscrete analogues of the total mass, energy an potential enstrophy. We iscretise the Hamiltonian 49 as H = K ρ k k= [ 2 k 2 hk i+/2, j+/2 u 3 2xyy i+/2, j+/2 + v 2xxy k ] i+/2, j+/2 + u 3 2x i+/2, j+/2 + v 2y i+/2, j+/2 + gh k i+/2, j+/2 ηk + 2 hk i+/2, j+/2. 54 The only qualitative change from the single-layer case lies in the averaging of the square particle velocity components use to approximate the kinetic energy in each layer. These averages must be taken over a larger stencil in the multilayer case to avoi an internal symmetric computational instability [43]. For example, the superscripts xyy an xxy correspon to the averages u 2xyy i+/2, j+/2 = 4 u 2 i+, j+/2 + u 2 +/2 + 8 u 2 i+, j+3/2 + u 2 +3/2 + u2 i+, j /2 + u2 /2, 55a v 2xxy i+/2, j+/2 = 4 v 2 i+/2, j+ + v 2 i+/2, j + 8 v 2 i+3/2, j+ + v 2 i+3/2, j + v2 i /2, j+ + v2 i /2, j. 55b

12 We pose the iscrete Poisson bracket as a sum of a potential vorticity epenent part an a remainer in each layer, {A, B} = K {A, B} k = k= {A, B} k Q = ρ k + β k {A, B} k R = ρ k q k K k= {A, B} k Q αk + {A, B}k R, 56a A, B ũ k +/2+n, ṽk i+/2, j+m A, B ũ k +/2+n, + γ k ũk +/2+m B ũ k +/2 + B ṽ k i+/2, j A h k i+/2, j+/2 A h k i+/2, j+/2 A, B ṽ k i+/2, j+n, ṽk i+/2, j+m A h k i /2, j+/2 A h k i+/2, j /2, A B, with coefficients α k, β k an γ k to be etermine. The iscrete potential vorticity in each layer is efine as 56b 56c q k = 2Ω z h kxy + ζ k, where ζ k = ṽk i+/2, j ṽk i /2, j ũk +/2 + ũk /2. 57 The iscrete potential enstrophy an mass in each layer are P k = ρ k { 2 hkxy q k 2 }, M k = ρ k { } h k i+/2, j+/2. 58 As in the single-layer case, the ynamics of the iscrete system are given by 22, where H is the multi-layer Hamiltonian 54. Conservation of energy follows trivially from the antisymmetry of the Poisson bracket 56b 56c, so 23 hols for the multi-layer Hamiltonian H. To conserve potential enstrophy P k within each layer, it is sufficient to ensure that the Poisson bracket satisfies {ũl, P k} = { ṽ l, P k} = { h l, P k} = 0, 59 for all l =,..., K. This requirement may be simplifie by recognising that P k is a function only of ũ k, ṽ k an h k, an that only the k th term of the Poisson bracket 56a contains erivatives with respect to these variables. Thus 59 reuces to {ũk, P k} k = {ṽk, P k} k = { h k, P k} k = Requirement 60 for the iscrete potential enstrophy 58 an Poisson bracket 56b 56c is exactly isomorphic to requirement 29a for the single-layer potential enstrophy 24 an Poisson bracket 3 an 33. Therefore, we may ensure conservation of potential enstrophy using exactly the same coefficients α k = α, β k = β an γ k = γ in the multi-layer case as in the single-layer case. For the purpose of illustration we use the coefficients corresponing to the original Arakawa Lamb scheme, as given in the appenix. Finally, conservation of mass within each layer again follows from M k being inepenent of all velocity variables, an M k / h l i+/2, j+/2 = δ kl being a Kronecker elta. Substituting F = h k i+/2, j+/2, F = ũk +/2, an F = ṽk i+/2, j in turn into 22 with the iscrete Hamiltonian 54 an 2

13 Poisson bracket 56a gives the following scheme for the multi-layer NTSWE: t hk i+/2, j+/2 = u k i+, j+/2 uk +/2 + vk i+/2, j+ vk i+/2, j, 6a t ũk +/2 = v k r+/2,s δ r,s+m n α q k r,s m + r,s r,s u k r,s+/2 t ṽk i+/2, j = + r,s r,s v k r+/2,s δ r,s+m n β β m,n q k r,s m u k r,s+/2 δ r,s+n m α q k r,s n δ r,s+m n γ γ m,n q k r,s m Φ k i+/2, j+/2 Φk i /2, j+/2, 6b Φ k i+/2, j+/2 Φk i+/2, j /2. 6c The larger averaging stencil in the Hamiltonian 54 etermines the mass fluxes u k +/2 = x uk +/2 3 hk +/ hkxyy +/2, 62a v k i+/2, j = x vk i+/2, j i+/2, j hkxyy i+/2, j 62b in terms of the particle velocity components 3 hk u k +/2 = ũk +/2 2Ωy η k+ + 2 hkx +/2, 63a v k i+/2, j = ṽk i+/2, j + 2Ωx η k+ + 2 hky i+/2, j. 63b The iscrete Bernoulli potential for each layer is u Φ k i+/2, j+/2 =2 k 2 xyy i+/2, j+/2 3 + v k 2 xxy i+/2, j+/2 + u k 2 x i+/2, j+/2 3 + v k 2 y i+/2, j+/2 l=k+ + g ηk i+/2, j+/2 + ρ l hl ρ k i+/2, j+/ Integration in time l= + Ω x i+/2, j+/2 vl y i+/2, j+/2 Ω y i+/2, j+/2 ul x i+/2, j+/2 + l=k+ l= 2 ρl Ω x ρ k i+/2, j+/2 vl y i+/2, j+/2 Ω y i+/2, j+/2 ul x i+/2, j+/2. 64 For the numerical experiments below, we integrate the iscretise NTSWE or multi-layer NTSWE scheme forwar in time using the stanar 4 th orer Runge Kutta scheme with a constant timestep chosen to satisfy the Courant Frierichs Lewy CFL stability conition throughout the integration. 4. Numerical experiments We first nonimensionalise our equations by writing x = R ˆx, u k = cû k, h k = Hĥ k, h b = Hĥ b, Ω x, Ω y, Ω z = Ω ˆΩ x, ˆΩ y, ˆΩ z, 65 where a hat ˆ enotes a imensionless variable. We use the layer thickness scale H, gravitational acceleration g, an planetary rotation rate Ω to construct the gravity wave spee c, Rossby eformation raius R, an non-traitional parameter δ [8, 22, 27] c = gh, R = c 2Ω, δ = H/R = 2ΩH/c. 66 3

14 The last of these is the aspect ratio base on the eformation raius an thickness scale. It is also the ratio between the change 2ΩH in zonal velocity create by isplacing a particle at the equator by a vertical istance H while conserving angular momentum an the gravity wave spee c. The imensionless components of the rotation vector at latitue ϕ are ˆΩ x = 0, ˆΩ y = cos ϕ, ˆΩ z = sin ϕ, 67 where we have aligne our x-axis with East to make ˆΩ x vanish. We take ϕ = π/4, which correspons to a latitue of 45 N. However, we have base our eformation raius on Ω rather than Ω z = Ω sin ϕ. Uner this nonimensionalisation, the single-layer shallow water equations become ˆũ ˆt + ˆqĥ ẑ û + ˆ ˆΦ = 0, where the imensionless canonical velocities are ĥ ˆt + ˆ ĥû = 0, 68 ˆũ = û + δ ˆΩ y ĥ b + 2ĥ, ˆṽ = ˆv δ ˆΩ x ĥ b + 2ĥ. 69 The imensionless potential vorticity an Bernoulli potential are ˆq = ˆΩ z + ˆṽ ĥ ˆx ˆũ, ˆΦ = ŷ 2û2 + 2 ˆv2 + ĥ b + ĥ + 2 δĥ ˆΩ xˆv ˆΩ y û. 70 We will henceforth rop the hat ˆ notation for imensionless variables. We choose H = 000 m as a characteristic vertical lengthscale for the ocean, an the Earth s rotation rate is Ω ra s. We take g = 0 3 m s 2 to be a reuce gravity for which our shallow water equations give a 2-layer or equivalent barotropic escription of a relatively thin active layer separate from a much thicker quiescent layer by a small ensity contrast [7, 8, 20]. These parameters yiel a gravity wave spee c = m s, a eformation raius R 6.88 km, an a non-traitional parameter δ Single-layer geostrophic ajustment We first focus on the propagation of inertia-gravity waves, which are substantially moifie by the non-traitional component of the Coriolis force. Their group velocity an structure epen on their irection of propagation relative to the horizontal component of the rotation vector [2]. We prescribe an initially motionless layer with an upwar bulge in the centre of a square oubly-perioic omain with a sie length of 0 eformation raii: u t=0 = 0, v t=0 = 0, h t=0 = + 2 4x 2 exp 5 2 4y 5. 7 Figures 2a illustrate the generation an propagation of inertia-gravity waves from the collapse of the axisymmetric initial peak. Nonlinear interactions subsequently generate shorter an shorter waves that continue to propagate aroun the omain for as long as we have been able to evolve them at least until t = 000 because the total energy is conserve. Figures 2e,f shows the convergence of the numerical solution as we increase the number of gripoints N along both the x- an y-axes. We quantify the convergence using the iscrete l 2 norms of the ifferences in the potential vorticity ϵ q an layer thickness ϵ h relative to a reference solution compute using a Fourier pseuospectral metho with collocation points, an the Hou Li spectral filter to provie issipation at the finest resolve scales [6]. We use the average h xy efine in 27 to construct a layer thickness fiel on the same gri points as the potential vorticity see figure. Figure 2e shows the expecte secon-orer convergence of the potential vorticity fiel at the four times t {5, 0, 5, 20}. However, the convergence rate for the layer thickness fiel shown in figure 2f eteriorates at later times ue to the presence of small-amplitue shock waves [62]. These are create by the nonlinear steepening of the original inertia-gravity waves, an form increasingly more complicate patterns as shocks collie with their perioic images. The asymmetry along the x-axis in figures 2c, is ue to the East/West asymmetry of the gravity wave ispersion relation create by non-traitional effects [2]. The horizontal component of the rotation vector Ω creates a preferre horizontal irection. To explore the consequences of this asymmetry we continue the integration of both the SWE an the NTSWE up to t = 200 on a gri. To istinguish the geostrophically ajuste state from the fluctuations associate with the inertia-gravity waves, which are trappe by the perioic bounary conitions, we plot the average of each quantity over the time interval from t = 00 to t = 200. Figure 3 compares the mean layer thickness h an the layer thickness variance h 2 = h 2 h 2 between the traitional an non-traitional SWE. 4

15 a b c t = 20 t = 5 t = 0 t = 5 ε h t = 20 t = 5 t = 0 t = 5 ε q N N 2 e N f N Figure 2: Panels a show snapshots of the layer thickness h from the single-layer geostrophic ajustment test case escribe in 4. on a gri. Convergence in the l 2 norm of e the potential vorticity q an f the layer thickness h uner refinement of the N N gri. 5

16 5 Traitional SWE mean height h.5 5 SWE with full Coriolis mean height h a b 5 Traitional SWE height variance h 2 x SWE with full Coriolis height variance h 2 x c Figure 3: Time-average iagnostics from our single-layer geostrophic ajustment test case between t = 00 an t = 200, compute on a gri. Panels a an b compare the time-average layer thickness h in the SWE an NTSWE solutions respectively. Panels c an compare the layer thickness variance h 2. Figures 3a,b show that the geostrophically ajuste mean states are almost inistinguishable. However, their variances show significant ifferences. Figure 3c shows that the fluctuations in the traitional shallow water equations are focuse at the site of the initial surface isplacement an its antipoe, an show near-perfect 4-fol rotational symmetry. The slight rotational asymmetry visible may be attribute to the imperfect cyclone/anticyclone symmetry in the traitional shallow water equations [63]; a simulation in which Ω z has the opposite sign prouces the opposite asymmetry. Figure 3 shows a much greater loss of 4-fol rotational symmetry ue to the non-traitional component of the Coriolis force that spreas the fluctuations into zonal bans Single-layer shear instability The inertia-gravity waves prouce in our first test involve a horizontally ivergent flow with relatively large vertical velocities. Our secon test instea emphasises the vorticity ynamics of our moel. We consier the roll-up of an unstable shear layer that is initially in geostrophic balance, referre to in a geophysical context as a barotropic instability []. A velocity fiel u g is in geostrophic balance with the layer thickness fiel if the traitional component of the Coriolis force balances the pressure graient, Ω z z u g = h. 72 This traitional geostrophic relation hols when non-traitional effects are weak δ, the imensionless velocity is small u, an eviations of the layer thickness are small h. The latter two conitions are equivalent 6

17 0 potential vorticity at t = 0 0 potential vorticity at t = a b ε q t = 00 t = 75 t = 50 t = t = 00 t = 75 t = 50 t = ε h 0 6 N c N N N Figure 4: Snapshots of the potential vorticity q from the single-layer shear instability test case escribe in 4.2 compute on a gri at a t = 0 an b t = 75. Convergence of the l 2 norm of c the potential vorticity q an the layer thickness h uner refinement of the N N spatial gri. to smallness of the Rossby an Froue numbers respectively. The geostrophic velocity given by 72 has no horizontal ivergence. The vertical velocity component must thus be inepenent of epth, an hence vanishes when the flow is over a horizontal lower bounary. We prescribe an initial layer thickness that varies sinusoially with amplitue h = 0.2 in the y-irection, an compute the corresponing velocity components from 72, { 2π h t=0 = + h sin L u t=0 = 2π h { 2π Ω z L cos L v t=0 = 4π2 h y Ω z L 2 cos [ y y sin [ y y sin { 2π L 2πx L 2πx L 2πx L [ y y sin ]}, 73a ]}, 73b ]} cos 2πx L. 73c These initial conitions escribe an alternating patterns of geostrophic jets in the x-irection whose paths are perturbe by a sinusoial meaner of amplitue y = 0.5 in the y-irection. The moel parameters are otherwise the same as in our first test case in 4.. As before, we take the imensionless omain length to be L = 0, an the reuce gravity to be g = 0 3 m s 2, so the non-traitional parameter is δ 0.45 Figures 4a,b illustrate the roll-up of the potential vorticity from its initial state until the shear evelops filaments on the orer of the gri scale. Beyon this point the solution begins to evelop spurious gri-scale potential vorticity anomalies, but remains numerically stable in the sense that it oes not blow up ue to the exact conservation of 7