On the Discrete Normal Modes of Quasigeostrophic Theory
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On the Discrete Normal Modes of Quasigeostrophic Theory
Houssam Yassin ∗ Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, NJ, USA
Stephen M. Griffies
NOAA/Geophysical Fluid Dynamics Laboratory andProgram in Atmospheric and Oceanic Sciences, Princeton University, Princeton, NJ, USA
ABSTRACTThe baroclinic modes of quasigeostrophic theory are incomplete and the incompleteness manifests as a loss of information in theprojection process. The incompleteness of the baroclinic modes is related to the presence of two previously unnoticed stationary step-wavesolutions of the Rossby wave problem with flat boundaries. These step-waves are the limit of surface quasigeostrophic waves as boundarybuoyancy gradients vanish. A complete normal mode basis for quasigeostrophic theory is obtained by considering the traditional Rossbywave problem with prescribed buoyancy gradients at the lower and upper boundaries. The presence of these boundary buoyancy gradientsactivates the previously inert boundary degrees of freedom. These Rossby waves have several novel properties such as the presence ofmultiple equivalent barotropic modes, a finite number of modes with negative norms, and their vertical structures form a basis capableof representing any quasigeostrophic state. Using this complete basis, we are able to obtain a series expansion to the potential vorticityof Bretherton (with Dirac delta contributions). We compare the convergence and differentiability properties of these complete modeswith various other modes in the literature. We also examine the quasigeostrophic vertical velocity modes and derive a complete basis forsuch modes as well. In the process, we introduce the concept of the quasigeostrophic phase space which we define to be the space of allpossible quasigeostrophic states. A natural application of these modes is the development of a weakly non-linear wave-interaction theoryof geostrophic turbulence that takes prescribed boundary buoyancy gradients into account.
1. Introduction a. Background
The vertical decomposition of quasigeostrophic motioninto normal modes plays an important role in bounded strat-ified geophysical fluids (e.g., Charney 1971; Flierl 1978; Fuand Flierl 1980; Wunsch 1997; Chelton et al. 1998; Smithand Vallis 2001; Tulloch and Smith 2009; Lapeyre 2009;Ferrari et al. 2010; Ferrari and Wunsch 2010; de La Lamaet al. 2016; LaCasce 2017; Brink and Pedlosky 2019).Most prevalent are the traditional baroclinic modes (e.g.,section 6.5.2 in Vallis 2017) that are the vertical structuresof Rossby waves in a quiescent ocean with no topographyor boundary buouyancy gradients. In a landmark contribu-tion, Wunsch (1997) partitions the ocean’s kinetic energyinto the baroclinic modes and finds that the zeroth and firstbaroclinic modes dominate over most of the extratropicalocean. Additionally, Wunsch (1997) concludes that thesurface signal primarily reflects the first baroclinic modeand, therefore, the motion of the thermocline.However, the use of baroclinic modes has come underincreasing scrutiny in recent years (Lapeyre 2009; Roulletet al. 2012; Scott and Furnival 2012; Smith and Vanneste2012). Lapeyre (2009) observes that the vertical shear of ∗ Corresponding author : Houssam Yassin, [email protected] the baroclinic modes vanishes at the boundaries, thus lead-ing to the concomitant vanishing of the boundary buoyancy.Consequently, Lapeyre (2009) proposes that the baroclinicmodes cannot be complete due to their inability to rep-resent boundary buoyancy. To supplement the baroclinicmodes, Lapeyre (2009) includes a boundary-trapped expo-nential surface quasigeostrophic solution (see Held et al.1995) and suggests that the surface signal primarily re-flects, not thermocline motion, but boundary-trapped sur-face quasigeostrophic dynamics (see also Lapeyre 2017).Appending additional functions to the collections ofnormal modes as in Lapeyre (2009) or Scott and Furni-val (2012) does not result in a set of normal modes sincethe appended functions are not orthogonal to the originalmodes. It is only with Smith and Vanneste (2012) that a setof normal modes capable of representing arbitrary surfacebuoyancy is derived.Yet it is not clear how the normal modes of Smith andVanneste (2012) differ from the baroclinic modes or whatthese modes correspond to in linear theory. Indeed, Rochaet al. (2015), noting that the baroclinic series expansion ofany sufficiently smooth function converges uniformly to thefunction itself, argues that the incompleteness of the baro- A collection of functions is said to be complete in some functionspace F if this collection forms a basis of F . Specifying the underlyingfunction space F turn out to be crucial, as we see in section 2d. a r X i v : . [ phy s i c s . a o - ph ] J a n AMS JOURNAL NAME clinic modes has been “overstated”. Moreover, de La Lamaet al. (2016) and LaCasce (2017), motivated by the obser-vation that the leading empirical orthogonal function ofWunsch (1997) vanishes near the ocean bottom, proposean alternate set of modes—the surface modes—that havea vanishing pressure at the bottom boundary.We thus have a variety of proposed normal modes andit is not clear how their properties differ. Are the baro-clinic modes actually incomplete? What about the surfacemodes? What does completeness mean in this context?What modes should be used to project model/observationaldata? The purpose of this paper is to answer these ques-tions. b. Normal modes and eigenfunctions
A normal mode is a linear motion in which all compo-nents of a system move coherently at a single frequency.Mathematically, a normal mode has the form Φ 𝑎 ( 𝑥, 𝑦, 𝑧 ) e − i 𝜔 𝑎 𝑡 (1)where Φ 𝑎 describes the spatial structure of the mode and 𝜔 𝑎 is its angular frequency. The function Φ 𝑎 is obtainedby solving a differential eigenvalue problem and henceis an eigenfunction. The collection of all eigenfunctions { Φ 𝑎 } 𝑎 forms a basis of some function space relevant to theproblem.By an abuse of terminology, the spatial structure Φ 𝑎 ,alone, is often called a normal mode (e.g., the term “Fouriermode” is often used for e i 𝑘𝑥 where 𝑘 is a wavenumber).In linear theory, this misnomer is often benign as each Φ 𝑎 corresponds to a frequency 𝜔 𝑎 . For example, given someinitial condition Ψ ( 𝑥, 𝑦, 𝑧 ) , we decompose Ψ as a sum ofmodes at 𝑡 = Ψ ( 𝑥, 𝑦, 𝑧 ) = ∑︁ 𝑎 𝑐 𝑎 Φ 𝑎 ( 𝑥, 𝑦, 𝑧 ) , (2)where the 𝑐 𝑎 are the Fourier coefficients, and the timeevolution is then given by ∑︁ 𝑎 𝑐 𝑎 Φ 𝑎 ( 𝑥, 𝑦, 𝑧 ) e − i 𝜔 𝑎 𝑡 . (3)However, in non-linear theory, this abuse of terminologycan be confusing. Given some spatial structure Ψ ( 𝑥, 𝑦, 𝑧 ) in a non-linear fluid, we can exploit the basis properties ofthe eigenfunctions Φ 𝑎 to decompose Ψ as in equation (2).Whereas in a linear fluid only wave motion of the form (1)is possible, a non-linear fluid permits a larger collection ofnon-linear solutions (e.g., non-linear waves and vortices)and so the linear wave solution (3) no longer follows fromthe decomposition (2).For this reason, we call the linear solution (1) a physical normal mode to distinguish it from the spatial structure Φ 𝑎 ,which is only an eigenfunction. Otherwise, we will use the terms “normal mode” and “eigenfunction” interchangeablyto refer to the spatial structure Φ 𝑎 , as is prevalent in theliterature.Our strategy here is then the following. We find the physical normal modes [of the form (1)] to various Rossbywave problems and examine the basis properties of theirconstituent eigenfunctions Φ 𝑎 . Our goal is to find a collec-tion of eigenfunctions (i.e., “normal modes” in the preva-lent terminology) capable of representing every possiblequasigeostrophic state. c. The contents of this article This article constitutes an examination of all collectionsof discrete (i.e., non-continuum ) quasigeostrophic nor-mal modes. We include the baroclinic modes, the surfacemodes of de La Lama et al. (2016) and LaCasce (2017),the surface-aware mode of Smith and Vanneste (2012), aswell as various generalizations. To study the complete-ness of a set of normal modes, one must first define theunderlying space in question. From general considera-tions, we introduce in section 2 the quasigeostrophic phasespace, defined as the space of all possible quasigeostrophicstates. Subsequently, in section 3 we use the general the-ory of differential eigenvalue problems with eigenvaluedependent boundary condition, developed in Yassin (inprep.), to study Rossby waves in an ocean with prescribedboundary buoyancy gradients (e.g., topography, see sec-tion 2a). Intriguingly, in an ocean with no topography, wefind that, in addition to the usual baroclinic modes, thereare two additional stationary step-mode solutions that havenot been noted before. The stationary step-modes are thelimits of boundary-trapped surface quasigeostrophic wavesas the boundary buoyancy gradient vanishes. Our study ofRossby waves then leads us examine all possible discretecollections of normal modes in section 4.As shown in section 4, the baroclinic modes are incom-plete, as argued by Lapeyre (2009), but in a rather subtlemanner. The baroclinic modes are incomplete in the sensethat one loses information after projecting a function ontothe baroclinic modes. In contrast, modes such as those sug-gested by Smith and Vanneste (2012) are complete in thequasigeostrophic phase space so that projecting a functiononto such modes provides an equivalent representation ofthe function.
2. Mathematics of the quasigeostrophic phase space a. The potential vorticity
Consider a three-dimensional region D of the form D = D × [ 𝑧 , 𝑧 ] , (4) Continuum modes appear once a sheared mean-flow is present, e.g.,Drazin et al. (1982), Balmforth and Morrison (1994, 1995), and Brinkand Pedlosky (2019).
2. Furthermore, 𝑔 𝑗 corresponds to a prescribed buoyancy 𝑏 𝑗 = 𝑁 𝑓 𝑔 𝑗 (9)at the 𝑗 th boundary [see equation (A6)].The gradient of the surface potential vorticity density, 𝑅 𝑗 , is proportional to the buoyancy gradient at the 𝑗 thboundary. A non-zero ∇ 𝑅 𝑗 signals outcrops or incrops ofbuoyancy on D . In quasigeostrophy, boundary buoyancycan be either prescribed, as in the first term of equation(8), or induced by a geostrophic flow, as in the second termof equation (8). An example prescribed buoyancy gradientarises in the following manner. Namely, quasigeostrophyformally has zero topography so that the lower and upperboundaries are rigid and flat. However, topography can beincluded by proxy through prescribing a surface potentialvorticity 𝑔 𝑗 = 𝑓 ℎ 𝑗 (10)where ℎ 𝑗 represents infinitesimal topography at the 𝑗 thboundary.We can interpret a surface potential vorticity density 𝑅 𝑗 as an infinitesimally thin volume potential vorticitydensity (Bretherton 1966). This interpretation motivatesthe definition of the potential vorticity density distribution 𝔔 = 𝑄 + ∑︁ 𝑗 = 𝑅 𝑗 𝛿 ( 𝑧 − 𝑧 𝑗 ) (11)where 𝛿 ( 𝑧 ) is the Dirac delta distribution with the dimen-sion of inverse length. The potential vorticity in the volume D is then given by the volume integral ∫ D 𝔔 d 𝑉 . (12)Knowledge of
𝑄, 𝑅 and 𝑅 determines the streamfunc-tion 𝜓 through equations (5) and (8). After determiningthe geostrophic horizontal velocity by u = ˆ z × ∇ 𝜓 , thequasigeostrophic state is evolved using 𝜕𝑄𝜕𝑡 + u · ∇ 𝑄 = 𝑧 ∈ ( 𝑧 , 𝑧 ) (13a) 𝜕𝑅 𝑗 𝜕𝑡 + u · ∇ 𝑅 𝑗 = 𝑧 = 𝑧 𝑗 . (13b)Thus, the dynamical system is entirely determined by thematerial conservation of the volume potential vorticity den-sity, 𝑄 , in the interior and material conservation of the sur-face potential vorticity densities, 𝑅 and 𝑅 , at the lowerand upper boundaries. b. Defining the quasigeostrophic phase space We define the quasigeostrophic phase space to be thespace of all possible quasigeostrophic states, with a quasi-geostrophic state determined by the potential vorticity den-sities,
𝑄, 𝑅 , and 𝑅 . Equivalently, a quasigeostrophic AMS JOURNAL NAME state is determined by a distribution 𝔔 of the form givenin equation (11). Note that the volume potential vorticitydensity, 𝑄 , is defined throughout the whole fluid region D , so that 𝑄 = 𝑄 ( 𝑥, 𝑦, 𝑧, 𝑡 ) . In contrast, the surface poten-tial vorticity densities, 𝑅 and 𝑅 , are only defined on thetwo-dimensional lower and upper boundary surfaces, D ,so that 𝑅 𝑗 = 𝑅 𝑗 ( 𝑥, 𝑦, 𝑡 ) .It will be useful to restate the previous paragraph withsome mathematical precision. For that purpose, let 𝐿 [D] be the space of square-integrable functions in the fluid vol-ume D , and let 𝐿 [D ] be the space of square-integrablefunctions on the boundary area D . Elements of 𝐿 [D] are functions of three spatial coordinates whereas elementsof 𝐿 [D ] are functions of two spatial coordinates. Hence, 𝑄 ∈ 𝐿 [D] and 𝑅 , 𝑅 ∈ 𝐿 [D ] .Define the space 𝒫 by 𝒫 = 𝐿 [D] ⊕ 𝐿 [D ] ⊕ 𝐿 [D ] , (14)where ⊕ is the direct sum. Equation (14) states that anyelement of 𝒫 is the sum of a function defined on the volume D and an element of 𝐿 [D] , plus two functions defined onthe area D that are elements of 𝐿 [D ] . By construction,the potential vorticity distribution 𝔔 is determined by afunction 𝑄 on the volume D and two functions 𝑅 and 𝑅 on the area D . We conclude that 𝔔 ∈ 𝒫 and that 𝒫 is thespace of all possible quasigeostrophic states. We thus call 𝒫 the quasigeostrophic phase space. c. The phase space in terms of the streamfunction The geostrophic streamfunction, 𝜓 , contains the samedynamical information as 𝔔 , and as such it offers an equiv-alent description of the dynamics. We can thus choose torepresent the phase space, 𝒫 , either in terms of 𝔔 orin terms of 𝜓 . Consequently, the space of all possiblegeostrophic streamfunctions is identical to the space ofall possible potential vorticity distributions 𝔔 . That is,both spaces are 𝒫 . Just as 𝑄 ∈ 𝐿 [D] is independent of 𝑅 𝑖 ∈ 𝐿 [D ] , then 𝜓 ( 𝑥, 𝑦, 𝑧, 𝑡 ) , for 𝑧 ∈ ( 𝑧 , 𝑧 ) , is indepen-dent of 𝜓 | 𝑧 = 𝑧 and 𝜓 | 𝑧 = 𝑧 .One may have expected the streamfunction, 𝜓 , as a func-tion defined on the volume D , to belong to 𝐿 [D] . But 𝜓 instead belongs to the larger space, 𝒫 , given in equation(14). To make this point explicit, write 𝜓 as the sum 𝜓 = 𝜓 int + 𝜓 low + 𝜓 upp . (15) The definition of 𝐿 [D] is more subtle than presented here.Namely, elements of 𝐿 [D] are not functions, but rather equivalenceclasses of functions leading to the unintuitive properties seen in thissection. See Yassin (in prep.) and citations within for more details. Streamfunctions in the phase space are defined up to a gauge trans-formation (see Schneider et al. 2003). We consider two streamfunctionsthat give rise to the same potential vorticity distribution as equivalent inthe phase space 𝒫 . 𝜓 int is the flow generated by the volume potential vorticitydensity, 𝑄 , in the fluid interior 𝑄 − 𝑓 = ∇ 𝜓 int + 𝜕𝜕𝑧 (cid:18) 𝑓 𝑁 𝜕𝜓 int 𝜕𝑧 (cid:19) for 𝑧 ∈ ( 𝑧 , 𝑧 ) = 𝑓 𝑁 𝜕𝜓 int 𝜕𝑧 for 𝑧 = 𝑧 , 𝑧 , (16)whereas 𝜓 low is the flow generated by the surface potentialvorticity density, 𝑅 , on the lower boundary0 = ∇ 𝜓 low + 𝜕𝜕𝑧 (cid:32) 𝑓 𝑁 𝜕𝜓 low 𝜕𝑧 (cid:33) for 𝑧 ∈ ( 𝑧 , 𝑧 ) 𝑅 − 𝑔 = 𝑓 𝑁 𝜕𝜓 low 𝜕𝑧 for 𝑧 = 𝑧 = 𝑓 𝑁 𝜕𝜓 low 𝜕𝑧 for 𝑧 = 𝑧 , (17)and 𝜓 upp is the flow generated by the surface potentialvorticity density, 𝑅 , on the upper boundary0 = ∇ 𝜓 upp + 𝜕𝜕𝑧 (cid:32) 𝑓 𝑁 𝜕𝜓 upp 𝜕𝑧 (cid:33) for 𝑧 ∈ ( 𝑧 , 𝑧 ) = 𝑓 𝑁 𝜕𝜓 upp 𝜕𝑧 for 𝑧 = 𝑧 𝑅 + 𝑔 = − 𝑓 𝑁 𝜕𝜓 upp 𝜕𝑧 for 𝑧 = 𝑧 . (18)We observe that 𝜓 consists of three independent com-ponents : 𝜓 int belongs to 𝐿 [D] , 𝜓 low | 𝑧 = 𝑧 belongs to 𝐿 [D ] , and 𝜓 upp | 𝑧 = 𝑧 belongs to 𝐿 [D ] . It follows that 𝜓 is an element of the phase space 𝒫 .Equations (16)–(18) motivate the definition of the rel-ative potential vorticity densities, 𝑞 = 𝑄 − 𝑓 and 𝑟 𝑗 = 𝑅 𝑗 − (− ) 𝑗 + 𝑔 𝑗 , which are the portions of the potentialvorticity providing a source for a streamfunction. Explic-itly, the relative potential vorticity densities are 𝑞 = ∇ 𝜓 + 𝜕𝜕𝑧 (cid:32) 𝑓 𝑁 𝜕𝜓𝜕𝑧 (cid:33) for 𝑧 ∈ ( 𝑧 , 𝑧 ) (19a) 𝑟 = 𝑓 𝑁 𝜕𝜓𝜕𝑧 for 𝑧 = 𝑧 (19b) 𝑟 = − 𝑓 𝑁 𝜕𝜓𝜕𝑧 for 𝑧 = 𝑧 . (19c)We then define a relative potential vorticity density distri-bution, 𝔮 , analogous to 𝔔 in equation (11) 𝔮 = 𝑞 + ∑︁ 𝑗 = 𝑟 𝑗 𝛿 ( 𝑧 − 𝑧 𝑗 ) . (20) We emphasize that 𝜓 low = 𝜓 low ( 𝑥, 𝑦, 𝑧, 𝑡 ) and 𝜓 upp = 𝜓 upp ( 𝑥, 𝑦, 𝑧, 𝑡 ) , that is, both are functions of three spatial coordinates.However, both 𝜓 low and 𝜓 upp are determined by the distribution ofsurface potential vorticity 𝑅 𝑗 − (− ) 𝑗 + 𝑔 𝑗 on the 𝑗 th boundary. d. The vertical structure phase space Since the fluid region, D , is separable, we can expandthe potential vorticity density distribution 𝔮 and the stream-function 𝜓 in terms of the eigenfunctions, 𝑒 k , of the hori-zontal Laplacian. For a horizontal domain D , the eigen-function 𝑒 k ( x ) satisfies −∇ 𝑒 k = 𝑘 𝑒 k . (21)where x = ( 𝑥, 𝑦 ) is the horizontal position vector, k = ( 𝑘 𝑥 , 𝑘 𝑦 ) is the horizontal wavevector, and 𝑘 = | k | is thehorizontal wavenumber. For example, in a horizontally pe-riodic domain the eigenfunctions 𝑒 k ( x ) are proportionalto complex exponentials, e i k · x . The horizontal eigenfunc-tions 𝑒 k obey the orthogonality relation 𝛿 a , b = 𝐴 ∫ D 𝑒 ∗ a ( x ) 𝑒 b ( x ) d 𝐴, (22)where 𝛿 a , b is the Kronecker delta, 𝐴 is the area of theregion D , and 𝑒 ∗ a is the complex conjugate of 𝑒 a .Projecting the relative potential vorticity density distri-bution, 𝔮 , onto the horizontal eigenfunctions, 𝑒 k , yields 𝔮 ( x , 𝑧, 𝑡 ) = ∑︁ k 𝔮 k ( 𝑧, 𝑡 ) 𝑒 k ( x ) . (23)Only the vertical dependence is retained in the verticalstructure, 𝔮 k , of the potential vorticity density distribution 𝔮 k ( 𝑧, 𝑡 ) = 𝑞 k ( 𝑧, 𝑡 ) + ∑︁ 𝑗 = 𝑟 𝑗 k ( 𝑡 ) 𝛿 ( 𝑧 − 𝑧 𝑗 ) (24)where 𝑞 k is a function of 𝑧 and 𝑟 k and 𝑟 k are independentof 𝑧 . Hence, 𝑞 k is an element of 𝐿 [( 𝑧 , 𝑧 )] whereas 𝑟 k and 𝑟 k are elements of the space of complex numbers , C .We conclude that the vertical structure of the potentialvorticity, 𝔮 k , is an element of (cid:98) 𝒫 = 𝐿 [( 𝑧 , 𝑧 )] ⊕ C ⊕ C , (25)so that the vertical structure, 𝔮 k , of the potential vorticitydistribution is determined by a function, 𝑞 k , in 𝐿 [( 𝑧 , 𝑧 )] and two 𝑧 -independent elements, 𝑟 k and 𝑟 k , of C . Simi-larly, the streamfunction can be represented as 𝜓 ( x , 𝑧, 𝑡 ) = ∑︁ k 𝜓 k ( 𝑧, 𝑡 ) 𝑒 k ( x ) , (26) Since all physical fields must be real, only a single degree of freedomis gained from C . Furthermore, when complex notation is used (e.g.,complex exponentials for the horizontal eigenfunctions 𝑒 k ) it is only thereal part of the fields that is physical. where 𝜓 k and 𝔮 k are related by 𝑞 k = − 𝑘 𝜓 k + 𝜕𝜕𝑧 (cid:32) 𝑓 𝑁 𝜕𝜓 k 𝜕𝑧 (cid:33) (27a) 𝑟 𝑗 k = (− ) 𝑗 + (cid:32) 𝑓 𝑁 𝜕𝜓 k 𝜕𝑧 (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 𝑧 = 𝑧 𝑗 . (27b)As before, knowledge of the vertical structure of thestreamfunction, 𝜓 k ( 𝑧 ) , is equivalent to knowing the verti-cal structure of the potential vorticity distribution, 𝔮 k ( 𝑧 ) .Thus, 𝜓 k must also belong to (cid:98) 𝒫 . As a consequence ofthe surface potential vorticity, 𝑟 𝑗 k , on the boundaries, theproperties of 𝜓 k ( 𝑧 𝑗 ) on the lower and upper boundaries areindependent of the interior values of the streamfunction.That 𝜓 k belongs to (cid:98) 𝒫 and not 𝐿 [( 𝑧 , 𝑧 )] underliesmuch of the confusion over baroclinic modes. Assertionsof completeness, based on Sturm-Liouville theory, assumethat 𝜓 is an element of 𝐿 [( 𝑧 , 𝑧 )] . However, as wehave shown, that is an incorrect assumption. A collectionof functions { 𝜙 𝑛 ( 𝑧 )} ∞ 𝑛 = is complete (in quasigeostrophictheory) only if this collection forms a basis of the func-tion space (cid:98) 𝒫 . In the context of quasigeostrophic theory,the space (cid:98) 𝒫 first appeared in Smith and Vanneste (2012).More generally, (cid:98) 𝒫 appears in the presence of non-trivialboundary dynamics (Yassin in prep.).We call (cid:98) 𝒫 the vertical structure phase space, and forconvenience we denote 𝐿 [( 𝑧 , 𝑧 )] by 𝐿 for the remain-der of the article. The vertical structure phase space (cid:98) 𝒫 isthen written as the direct sum (cid:98) 𝒫 = 𝐿 ⊕ C . (28) e. Representing the energy and potential enstrophy We find it convenient to represent several quadraticquantities in terms of the eigenfunctions of the horizon-tal Laplacian, 𝑒 k ( x ) . The energy per unit mass in thevolume D is given by 𝐸 = 𝑉 ∫ D (cid:34) | ∇ 𝜓 | + 𝑓 𝑁 (cid:12)(cid:12)(cid:12)(cid:12) 𝜕𝜓𝜕𝑧 (cid:12)(cid:12)(cid:12)(cid:12) (cid:35) d 𝐴 d 𝑧 = ∑︁ k 𝐸 k , (29)where the horizontal energy mode is given by the verticalintegral 𝐸 k = 𝐻 ∫ 𝑧 𝑧 (cid:34) 𝑘 | 𝜓 k | + 𝑓 𝑁 (cid:12)(cid:12)(cid:12)(cid:12) 𝜕𝜓 k 𝜕𝑧 (cid:12)(cid:12)(cid:12)(cid:12) (cid:35) d 𝑧, (30)with 𝑉 = 𝐴 𝐻 the domain volume and 𝐻 = 𝑧 − 𝑧 the do-main depth.Similarly, for the relative volume potential enstrophydensity, 𝑍 , we have 𝑍 = 𝑉 ∫ D | 𝑞 | d 𝐴 d 𝑧 = ∑︁ k 𝑍 k (31) AMS JOURNAL NAME
Fig. 2. Polar plots of the absolute value of the non-dimensional an-gular frequency | 𝜔 𝑛 |/( 𝛽𝐿 𝑑 ) of the first five modes of the traditionaleigenvalue problem (section 3a) as a function of the wave propagationdirection, k /| k | , for constant stratification. The outer most ellipse, withthe largest absolute angular frequency, represents the angular frequencyof the barotropic ( 𝑛 =
0) mode. The higher modes have smaller absolutefrequencies and are thus concentric and within the barotropic angularfrequency curve. Since the absolute value of the angular frequency of thebarotropic mode becomes infinitely large at small horizontal wavenum-bers 𝑘 , we have chosen a large wavenumber 𝑘 , given by 𝑘 𝐿 𝑑 =
7, so thatthe angular frequency of the first five modes can be plotted in the samefigure. We have chosen 𝑓 = − s − , 𝛽 = − m − s − , 𝑁 = − s − and 𝐻 = 𝐿 𝑑 = 𝑁 𝐻 / 𝑓 =
100 km.Numerical solutions to all eigenvalue problems in this article are ob-tained using
Dedalus (Burns et al. 2020). where 𝑍 k = 𝐻 ∫ 𝑧 𝑧 | 𝑞 k | d 𝑧. (32)Finally, analogous to 𝑍 , we have the relative surface po-tential enstrophy densities, 𝑌 𝑗 , on the area D 𝑌 𝑗 = 𝐴 ∫ D (cid:12)(cid:12) 𝑟 𝑗 (cid:12)(cid:12) d 𝐴 = ∑︁ k 𝑌 𝑗 k (33)where 𝑌 𝑗 k = (cid:12)(cid:12) 𝑟 𝑗 k (cid:12)(cid:12) . (34)
3. Rossby waves in a linear ocean
In this section, we study Rossby waves in an otherwisequiescent ocean; in other words, we examine the physi-cal normal modes of a quiescent ocean. Linearizing the equations of motion (13) about a resting ocean yields 𝜕𝑞𝜕𝑡 + 𝛽 𝑣 = 𝑧 ∈ ( 𝑧 , 𝑧 ) (35a) 𝜕𝑟 𝑗 𝜕𝑡 + u · ∇ (cid:2) (− ) 𝑗 + 𝑔 𝑗 (cid:3) = 𝑧 = 𝑧 𝑗 . (35b)We assume that the prescribed surface potential vorticitydensities at the lower and upper boundaries, 𝑔 and 𝑔 , arelinear, which ensures the resulting eigenvalue problem isseparable.The linear quasigeostrophic system (35) with no back-ground mean-flow is artificial since the ocean has a circu-lation while the above equations do not. A consequenceis that the linear system (35) has infinitely many Rossbywave modes and no advective continuum modes. However,once a mean-flow is present, then it is possible that only afinite number of Rossby waves exist (Killworth and Ander-son 1977) and a large number of degrees of freedom willgenerally be associated with advective continuum modeswhich the above model lacks (the situation is analogous toDrazin et al. 1982; Balmforth and Morrison 1994, 1995).The importance of the linear problem (35) is that itprovides all possible discrete Rossby wave normal modesin a quasigeostrophic fluid. Substituting a wave ansatz ofthe form [compare with equation (1) for physical normalmodes] 𝜓 ( x , 𝑧, 𝑡 ) = ˆ 𝜓 ( 𝑧 ) 𝑒 k ( x ) e − i 𝜔𝑡 (36)into the linear problem (35) renders (− i 𝜔 ) (cid:34) − 𝑘 ˆ 𝜓 + dd 𝑧 (cid:32) 𝑓 𝑁 d ˆ 𝜓 d 𝑧 (cid:33)(cid:35) + i 𝑘 𝑥 𝛽 ˆ 𝜓 = 𝑧 ∈ ( 𝑧 , 𝑧 ) , and (− i 𝜔 ) (cid:32) 𝑓 𝑁 d ˆ 𝜓 d 𝑧 (cid:33) + i ˆ z · (cid:0) k × ∇ 𝑔 𝑗 (cid:1) ˆ 𝜓 = 𝑧 = 𝑧 , 𝑧 . a. Traditional Rossby wave problem We first study the traditional case of linear fluctuations toa quiescent ocean with isentropic lower and upper bound-aries. Setting ∇ 𝑔 = ∇ 𝑔 = 𝜔 (cid:34) − 𝑘 𝜙 + dd 𝑧 (cid:32) 𝑓 𝑁 d 𝜙 d 𝑧 (cid:33)(cid:35) − 𝛽 𝑘 𝑥 𝜙 = 𝜔 (cid:32) 𝑓 𝑁 d 𝜙 d 𝑧 (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 𝑧 = 𝑧 𝑗 = , (39b)where ˆ 𝜓 ( 𝑧 ) = ˆ 𝜓 𝜙 ( 𝑧 ) and 𝜙 is a non-dimensional function.There are two cases to consider depending on whether 𝜔 vanishes. Fig. 3. The first six baroclinic modes in constant stratification for a propagation direction k /| k | with 𝑘 𝑥 ≠
0. Other parameters are as in figure 2.See section 3a for details.
1) Traditional baroclinic modesAssuming 𝜔 ≠ 𝐿 − dd 𝑧 (cid:32) 𝑓 𝑁 d 𝜙 d 𝑧 (cid:33) = 𝜆 𝜙 for 𝑧 ∈ ( 𝑧 , 𝑧 ) (40a) 𝑓 𝑁 d 𝜙 d 𝑧 = 𝑧 = 𝑧 , 𝑧 , (40b)where the eigenvalue 𝜆 is the square vertical wavenumber,given by 𝜔 = − 𝛽 𝑘 𝑥 𝑘 + 𝜆 . (41)See figure 2 for an illustration of the dependence of | 𝜔 | onthe wavevector k .From Sturm-Liouville theory, the eigenvalue problem(40) has infinitely many eigenfunctions 𝜙 , 𝜙 , 𝜙 , . . . withdistinct and ordered eigenvalues, 𝜆 𝑛 , satisfying0 = 𝜆 < 𝜆 < · · · → ∞ . (42)The 𝑛 th mode, 𝜙 𝑛 , has 𝑛 internal zeros in the interval ( 𝑧 , 𝑧 ) (see figure 3). The eigenfunctions are orthonormalwith respect to the inner product, (cid:104)· , ·(cid:105) , given by (cid:104) 𝜑, 𝜙 (cid:105) = 𝐻 ∫ 𝑧 𝑧 𝜑 𝜙 d 𝑧, (43)with orthonormality meaning that 𝛿 𝑚𝑛 = (cid:104) 𝜙 𝑚 , 𝜙 𝑛 (cid:105) (44)where 𝛿 𝑚𝑛 is the Kronecker delta. A powerful and com-monly used result of Sturm-Liouville theory is that the set { 𝜙 𝑛 } ∞ 𝑛 = forms an orthonormal basis of 𝐿 . 2) Stationary step-modesThere are two additional solutions to the Rossby waveeigenvalue problem (39) not previously noted in the liter-ature. If 𝜔 = 𝛽 𝑘 𝑥 𝜙 = 𝑧 ∈ ( 𝑧 , 𝑧 ) (45a)0 = 𝑧 = 𝑧 , 𝑧 . (45b)Consequently, if 𝑘 𝑥 ≠
0, then 𝜙 ( 𝑧 ) = 𝑧 ∈ ( 𝑧 , 𝑧 ) . Thatis, 𝜙 must vanish in the interior of the interval. However,since 𝜔 = 𝜓 can take arbitrary values atthe lower and upper boundaries. Thus two solutions are 𝜙 step 𝑗 ( 𝑧 ) = (cid:40) 𝑧 = 𝑧 𝑗 𝜙 step 𝑗 in terms of the baroclinic modes will fail andproduce a series that is identically zero.The two stationary step-modes, 𝜙 step1 and 𝜙 step2 , corre-spond to the two inert degrees of freedom in eigenvalueproblem (39). These two solutions are neglected in thetraditional eigenvalue problem (40) through the assump-tion that 𝜔 ≠
0. We will see that these two step-wavesare obtained as limits of boundary-trapped modes as theboundary buoyancy gradients 𝑁 ∇ 𝑔 𝑗 / 𝑓 become small.3) The general solutionFor a wavevector k with 𝑘 𝑥 ≠
0, the vertical structure ofthe streamfunction must be of the form 𝜓 k ( 𝑧, 𝑡 = ) = Ψ ( 𝑧 ) + ∑︁ 𝑗 = Ψ 𝑗 𝜙 step 𝑗 ( 𝑧 ) (47) AMS JOURNAL NAME where Ψ ( 𝑧 ) is a twice differentiable function satisfyingd Ψ ( 𝑧 𝑗 )/ d 𝑧 = 𝑗 = , Ψ , Ψ are arbitrary con-stants. We can represent Ψ according to the expansion Ψ = ∞ ∑︁ 𝑛 = (cid:104) Ψ , 𝜙 𝑛 (cid:105) 𝜙 𝑛 (48)and so the time-evolution is 𝜓 k ( 𝑧, 𝑡 ) = ∞ ∑︁ 𝑛 = (cid:104) Ψ , 𝜙 𝑛 (cid:105) 𝜙 𝑛 e − i 𝜔 𝑛 𝑡 + ∑︁ 𝑗 = Ψ 𝑗 𝜙 step 𝑗 . (49)It is this time-evolution expression, which is valid only inlinear theory for a quiescent ocean, that gives the baroclinicmodes a clear physical meaning. More precisely, equation(49) states that the vertical structure Ψ ( 𝑧 ) disperses intoits constituent Rossby waves with vertical structures 𝜙 𝑛 .Outside the linear theory of this section, baroclinic modesdo not have a physical interpretation, although they remaina mathematical basis for 𝐿 . b. The generalized Rhines problem We now study the case with constant buoyancy gradients ∇ 𝑔 and ∇ 𝑔 at the lower and upper boundaries. The spe-cial case of a "flat" upper boundary (i.e., isentropic upperboundary ∇ 𝑔 = 𝜆 -dependent boundaryconditions and obtains various completeness and expan-sions results as well as a qualitative theory for the modes.Below, we generalize these results, study the two limitingboundary conditions, and examine the phase speed of theresulting waves.1) The eigenvalue problemLet ˆ 𝜓 ( 𝑧 ) = ˆ 𝜓 𝜑 ( 𝑧 ) where 𝜑 is a non-dimensionalfunction. We then manipulate the eigenvalue problem(37)–(38) to obtain − dd 𝑧 (cid:32) 𝑓 𝑁 d 𝜑 d 𝑧 (cid:33) = 𝜆 𝜑 for 𝑧 ∈ ( 𝑧 , 𝑧 ) (50a) − 𝑘 𝜑 + (− ) 𝑗 𝛾 − 𝑗 (cid:32) 𝑓 𝑁 d 𝜑 d 𝑧 (cid:33) = 𝜆 𝜑 for 𝑧 = 𝑧 𝑗 , (50b)where the length-scale 𝛾 𝑗 is given by 𝛾 𝑗 = (− ) 𝑗 + ˆ z · (cid:0) k × ∇ 𝑔 𝑗 (cid:1) ˆ z · ( k × ∇ 𝑓 ) = (− ) 𝑗 + (cid:18) 𝛼 𝑗 𝑘𝛽 𝑘 𝑥 (cid:19) sin (cid:0) Δ 𝜃 𝑗 (cid:1) (51) where 𝛼 𝑗 = | ∇ 𝑔 𝑗 | and Δ 𝜃 𝑗 is the angle between thewavevector k and ∇ 𝑔 𝑗 measured counterclockwise from k . The parameter 𝛾 𝑗 depends only on the direction of thewavevector k and not its magnitude 𝑘 = | k | . If 𝛾 𝑗 = 𝑗 th boundary condition can be written as a 𝜆 -independent boundary condition. For now, we considerthe case 𝛾 𝑗 ≠ 𝜆 appears in both boundary con-ditions (50b), the eigenvalue problem (50) occurs in thefunction space 𝐿 ⊕ C and hence is not a Sturm-Liouvilleproblem. Instead, we must use the mathematical theorydeveloped in Yassin (in prep.) for eigenvalue problemswith 𝜆 -dependent boundary conditions .2) Characterizing the eigen-solutionsThe eigenvalue problem (50) has a countable infinityof eigenfunctions 𝜑 , 𝜑 , 𝜑 , . . . with ordered and distinctnon-zero eigenvalues 𝜆 𝑛 satisfying 𝜆 < 𝜆 < 𝜆 < · · · → ∞ . (52)The inner product (cid:104)· , ·(cid:105) induced by the eigenvalue problem(50) is (cid:104) 𝜑, 𝜙 (cid:105) = 𝐻 (cid:169)(cid:173)(cid:171)∫ 𝑧 𝑧 𝜑 𝜙 dz + ∑︁ 𝑗 = 𝛾 𝑗 𝜑 ( 𝑧 𝑗 ) 𝜙 ( 𝑧 𝑗 ) (cid:170)(cid:174)(cid:172) , (53)which depends on the direction of the horizontal wavevec-tor k through 𝛾 and 𝛾 . Furthermore, 𝛾 and 𝛾 arenot necessarily positive, with one consequence being thatsome functions 𝜑 may have a negative square (cid:104) 𝜑, 𝜑 (cid:105) < 𝜑 𝑛 then takes the form ± 𝛿 𝑚𝑛 = (cid:104) 𝜑 𝑚 , 𝜑 𝑛 (cid:105) , (54)where at most two modes 𝜑 𝑛 satisfy (cid:104) 𝜑 𝑛 , 𝜑 𝑛 (cid:105) = − (cid:16) 𝑘 + 𝜆 𝑛 (cid:17) (cid:104) 𝜑 𝑛 , 𝜑 𝑛 (cid:105) > 𝜑 𝑛 with (cid:104) 𝜑 𝑛 , 𝜑 𝑛 (cid:105) > 𝜑 𝑛 with (cid:104) 𝜑 𝑛 , 𝜑 𝑛 (cid:105) < 𝛽 > 𝛾 and 𝛾 and as depicted in figures 4, 5, and6. In the following, we assume 𝑘 ≠
0. [These results area consequence of the mathematical theory in Yassin (inprep.)]i. 𝛾 > 𝛾 >
0. All eigenvalues satisfy 𝜆 𝑛 > − 𝑘 ,all modes satisfy (cid:104) 𝜑 𝑛 , 𝜑 𝑛 (cid:105) >
0, and all waves prop-agate westward. The 𝑛 th mode, 𝜑 𝑛 , has 𝑛 internal To apply the theory in Yassin (in prep.), let ˜ 𝜆 = 𝜆 − 𝑘 be theeigenvalue in place of 𝜆 ; the resulting eigenvalue problem for ˜ 𝜆 willthen satisfy the left-definiteness conditions in Yassin (in prep.). Fig. 4. Polar plots of the absolute value of the non-dimensional angular frequency | 𝜔 𝑛 |/( 𝛽𝐿 𝑑 ) of the first five modes from section 3b asa function of the wave propagation direction k /| k | for a horizontal wavenumber given by 𝑘 𝐿 𝑑 = . 𝜔 (not visible in this figure, but see figure 5), the dashed line to 𝜔 , with these two modes becoming boundary trapped at largewavenumbers 𝑘 = | k | . The remaining modes, 𝜔 𝑛 for 𝑛 = , ,
4, are shown with solid lines. White regions are angles where 𝛾 > 𝛾 > 𝜔 𝑛 and so have a westward phase speed.Gray regions are angles where 𝛾 < 𝛾 <
0. The two gravest angular frequencies 𝜔 and 𝜔 are both positive while the remaining angularfrequencies 𝜔 𝑛 for 𝑛 > 𝜔 and 𝜔 each correspond to a Rossby waves with an eastward phasespeed whereas the remaining Rossby waves have westward phase speeds . Stippled regions are angles where 𝛾 > 𝛾 <
0. In the stippledregion, 𝜔 is positive and has an eastward phase speed. The remaining Rossby waves in the stippled region have negative angular frequenciesand have westward phase speeds. The lower boundary buoyancy gradient, proportional to ∇ 𝑔 , points towards 55 ◦ while the upper boundarybuoyancy gradient, proportional to ∇ 𝑔 , points towards 200 ◦ . These buoyancy gradients correspond to topographic gradients at the lower and upperboundaries through 𝑔 𝑗 = 𝑓 ℎ 𝑗 . The 𝑔 𝑗 are chosen so that | ∇ ℎ | = . × − and | ∇ ℎ | = − , thus leading to 𝛾 / 𝐻 = .
15 and 𝛾 / 𝐻 = . 𝜑 𝑛 , for angles 𝜃 = ◦ , ◦ , ◦ are shown in figure 6. zeros. See the regions in white in figures 4 and 5 andplots (a) and (b) in figure 6.ii. 𝛾 𝛾 <
0. There is one mode, 𝜑 , with a negativesquare, (cid:104) 𝜑 , 𝜑 (cid:105) <
0, corresponding to an eastwardpropagating wave. The associated eigenvalue, 𝜆 ,satisfies 𝜆 < − 𝑘 . The remaining modes, 𝜑 𝑛 for 𝑛 >
1, have positive squares, (cid:104) 𝜑 𝑛 , 𝜑 𝑛 (cid:105) >
0, correspondingto westward propagating waves and have eigenvalues 𝜆 𝑛 satisfying 𝜆 𝑛 > − 𝑘 . Both 𝜑 and 𝜑 have nointernal zeros whereas the remaining modes, 𝜑 𝑛 , have 𝑛 − 𝑛 >
1. See the stippled regionsin figures 4 and 5 and plots (c) and (d) in figure 6. iii. 𝛾 < 𝛾 <
0. There are two modes 𝜑 and 𝜑 with negative squares, (cid:104) 𝜑 𝑛 , 𝜑 𝑛 (cid:105) <
0, that propa-gate eastward and have eigenvalues, 𝜆 𝑛 , satisfying 𝜆 𝑛 < − 𝑘 for 𝑛 = ,
2. The remaining modes, 𝜑 𝑛 ,for 𝑛 > (cid:104) 𝜑 𝑛 , 𝜑 𝑛 (cid:105) >
0, prop-agate westward, and have eigenvalues, 𝜆 𝑛 , satisfying 𝜆 𝑛 > − 𝑘 . The zeroth mode, 𝜑 , has one internalzero, the first and second modes, 𝜑 and 𝜑 , haveno internal zeros, and the remaining modes, 𝜑 𝑛 , have 𝑛 − 𝑛 >
2. See the shaded regionsin figures 4 and 5 and plots (e) and (f) in figure 6.Figures 4 and 5 illustrate that the mode number, 𝑛 , nolonger refers to a unique physical mode. Instead, 𝑛 is0 AMS JOURNAL NAME
Fig. 5. As in figure 4 but with a larger horizontal wavenumber 𝑘 given by 𝑘 𝐿 𝑑 =
7. The smaller horizontal length-scale means that 𝜔 is smallenough to be visible in this figure, unlike in figure 4 where 𝜔 is large and thus sits outside of the figure. Plots of the first four modes, 𝜑 𝑛 , forangles 𝜃 = ◦ , ◦ , ◦ are shown in figure 6. simply a mathematical label. For instance, in the white re-gions in figure 5, the dotted line ( 𝑛 =
0) corresponds to thebottom-trapped mode. Following the dotted line into thestippled region reveals the dotted line becomes a dashedline indicating that the bottom-trapped mode now has 𝑛 = 𝑘 𝑦 → − 𝑘 𝑦 , 𝜕 𝑥 𝑔 𝑗 → − 𝜕 𝑥 𝑔 𝑗 symmetry in the eigenvalue problem (50).To elucidate the meaning of 𝜆 𝑛 < − 𝑘 in cases (i)–(iii)above, note that a pure surface quasigeostrophic mode has a vertical structure e 𝑘 ( 𝑧 − 𝑧 ) decaying from the upperboundary at 𝑧 = 𝑧 ; this exponential vertical structure im-plies a square vertical wavenumber 𝜆 = − 𝑘 (e.g., an e-folding scale of | 𝜆 | − / = 𝑘 − ). Thus 𝜆 𝑛 < − 𝑘 means thatat least one of the boundary-trapped modes decays awayfrom the boundary more rapidly than a pure surface quasi- A pure surface quasigeostrophic mode is the mode found after set-ting 𝛽 = 𝑧 = −∞ . geostrophic wave. Indeed, the limit of 𝜆 → −∞ yieldsone of the step-modes (46) of the previous subsection.The step-mode limit is obtained as 𝛾 𝑗 → − . This limitis found as either | ∇ 𝑔 𝑗 | → 𝛾 𝑗 < k becomes parallel or anti-parallel to ∇ 𝑔 𝑗 (whichever limit satisfies 𝛾 𝑗 → − ). In this limit, weobtain a step-mode exactly confined at the boundary (thee-folding scale is now | 𝜆 | − / =
0) with zero phase speed[see figure 7(a)]. The remaining modes then satisfy theisentropic boundary condition (cid:32) 𝑓 𝑁 d 𝜑 𝑛 d 𝑧 (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 𝑧 = 𝑧 𝑗 = . (56)The other limit is that of | 𝛾 𝑗 | → ∞ which is obtained asthe buoyancy gradient becomes large, | ∇ 𝑔 𝑗 | → ∞ . In thislimit, the eigenvalue 𝜆 𝑚 → − 𝑘 where 𝜆 𝑚 is the eigen-value of the mode 𝜑 𝑚 associated with the 𝑗 th bound-ary [see figure 7(b)]. Furthermore, the phase speedof the wave trapped to the 𝑗 th boundary becomes infi-nite, an indication that the quasigeostrophic approxima-tion breaks down. Indeed, the large buoyancy gradient1 Fig. 6. This figure illustrates the dependence of the vertical structure 𝜑 𝑛 of the streamfunction to the horizontal wavevector k as discussed insection 3b . Three propagation directions are shown 𝜃 = ◦ , ◦ , ◦ and correspond to the rows in the figure [e.g., the row containing (a) and(b) are the vertical structures of waves at 𝜃 = ◦ ]; two wavenumbers 𝑘 𝐿 𝑑 = . , 𝑘 = | k | ) and they correspond to the columnsin the above figure [e.g., (b), (d) and (f) are the vertical structure of waves with 𝑘 𝐿 𝑑 = 𝑛 = ,
1) are generally only boundary-trapped at small horizontalscales. At larger horizontal scales, we typically obtain a depth-independent mode along with another mode with large-scale features in the vertical.Second, note that for 𝛾 , 𝛾 >
0, as in panels (a) and (b), the 𝑛 th mode has 𝑛 internal zeros, as in Sturm-Liouville theory; for 𝛾 > , 𝛾 <
0, as inpanels (c) and (d), the first two modes ( 𝑛 = , ) have no internal zeros; and for 𝛾 , 𝛾 <
0, the zeroth mode 𝜑 has one internal zero, the first andsecond modes, 𝜑 and 𝜑 have no internal zeros, and the third mode 𝜑 has one internal zero. Note that the zero-crossing for the 𝑛 = 𝜑 is small near the zero-crossing. limit corresponds to steep topographic slopes and so weobtain the topographically-trapped internal gravity waveof Rhines (1970), which has an infinite phase speed inquasigeostrophic theory. The remaining modes then sat-isfy the vanishing pressure boundary condition 𝜑 ( 𝑧 𝑗 ) = k . However, as Straub (1994) observed, only the firstfew modes have a strong dependence on k whereas highermodes are generally insensitive to k . All modes, 𝜑 𝑛 with 𝑛 (cid:29)
1, satisfy the k -independent boundary condition (57)if 𝛾 𝑗 ≠
0. Alternatively, if 𝛾 𝑗 =
0, the 𝑛 (cid:29) 𝑐 𝑥 = 𝜔 / 𝑘 𝑥 is 𝑐 𝑥 = − 𝛽𝑘 + 𝜆 , (58) Fig. 7. The two limits of the boundary-trapped surface quasi-geostrophic waves, as discussed in section 3b. (a) Convergence to thestep mode given in equation (46) with 𝑗 = 𝛾 → − for three valuesof 𝛾 at a wavenumber 𝑘 = | k | given by 𝑘 𝐿 𝑑 =
1. The phase speedapproaches zero in the limit 𝛾 → − . (b) Here, 𝛾 / 𝐻 ≈
10 for the threevertical structures 𝜑 𝑛 shown. Consequently, the bottom trapped wavehas a square vertical wavenumber 𝜆 ≈ − 𝑘 and the phase speeds are large.The vertical structure, 𝜑 , for three values of 𝑘 𝐿 𝑑 are shown, illustratingthe dependence on 𝑘 of this mode, which behaves as a boundary-trappedexponential mode with an e-folding scale of | 𝜆 | − / = 𝑘 − . In both (a)and (b), the wave propagation direction 𝜃 = ◦ . All other parametersare identical to figure 4. AMS JOURNAL NAME
Fig. 8. The dependence of the Rossby wave phase speed on thevertical structure 𝜑 𝑛 , as discussed in section 3b. The first internal modeis plotted. Two columns with 𝛾 / 𝐻 = ,
10 and two rows 𝛾 / 𝐻 = , 𝛾 = 𝛾 = 𝛾 =
10 and 𝛾 =
0. The non-dimensional phase speed ˜ 𝑐 is given by ˜ 𝑐 = 𝑐 /( 𝛽 𝐿 𝑑 ) and the vertical wavelength is given by 𝜈 = 𝜋 /√ 𝜆 . The horizontalwavevector k points in the 𝜃 = ◦ direction and has a magnitudegiven by 𝑘 𝐿 𝑑 = .
1. Other parameters are as in figure 4. which shows that the phase speed is inversely proportionalto the square vertical wavenumber, 𝜆 . As shown by Rhines(1970), bottom topography increases the speed of Rossbywaves by modifying the bottom boundary condition andhence decreasing the square vertical wavenumber, 𝜆 . Fig-ure 8 shows that boundary buoyancy gradients at the uppersurface can also modify the Rossby phase speed. Notably,when buoyancy gradients are present at the upper surface,bottom topography now decreases the phase speed. For ex-ample, the phase speed decreases as we increase 𝛾 from 𝛾 = 𝛾 =
10 in figure 8(d). Of course,surface buoyancy gradients in the ocean arise in the pres-ence of background mean flows and so equation (58) for 𝑐 𝑥 no longer holds. This analysis suggests that it would beinteresting to investigate how the Rossby wave phase speeddepends on surface buoyancy gradients.In the above analysis, we have assumed 𝛾 𝑗 ≠ 𝑗 = ,
2. Suppose now that 𝛾 = 𝑗 =
2; the resulting modes form a basis for 𝐿 ⊕ C (i.e., there is now one “missing” degree of freedom).Condition (i) above holds if 𝛾 > 𝛾 < 𝛾 = 𝛾 ≠ 𝛾 and 𝛾 vanishthen we recover the results of section 3a.3) The general time-dependent solutionAt some wavevector k , the observed vertical structurenow has the form 𝜓 k ( 𝑧, 𝑡 = ) = Ψ ( 𝑧 ) (59)where Ψ is a twice continuously differentiable function.For such functions we can write Ψ = ∞ ∑︁ 𝑛 = (cid:104) Ψ , 𝜑 𝑛 (cid:105)(cid:104) 𝜑 𝑛 , 𝜑 𝑛 (cid:105) 𝜑 𝑛 . (60)so that the time-evolution is 𝜓 k ( 𝑧, 𝑡 ) = ∞ ∑︁ 𝑛 = (cid:104) Ψ , 𝜑 𝑛 (cid:105)(cid:104) 𝜑 𝑛 , 𝜑 𝑛 (cid:105) 𝜑 𝑛 ( 𝑧 ) e − i 𝜔 𝑛 𝑡 . (61)Again, it is the above expression, which is valid only in lin-ear theory with a quiescent background state, that gives thegeneralized Rhines modes 𝜑 𝑛 physical meaning. Outsidethe linear theory of this section, the generalized Rhinesmodes do not have any physical interpretation and insteadmerely serve as a mathematical basis for 𝐿 ⊕ C . c. The vertical velocity eigenvalue problem Let ˆ 𝑤 ( 𝑧 ) = ˆ 𝑤 𝜒 ( 𝑧 ) where 𝜒 ( 𝑧 ) is a non-dimensionalfunction. For the Rossby waves with isentropic bound-aries of section 3a (the traditional baroclinic modes), thecorresponding vertical velocity modes satisfy − d 𝜒 d 𝑧 = 𝜆 (cid:32) 𝑁 𝑓 (cid:33) 𝜒 (62)with vanishing vertical velocity boundary conditions 𝜒 ( 𝑧 𝑗 ) = { 𝜒 𝑛 } ∞ 𝑛 = form an orthonormal basis of 𝐿 with orthonor-mality given by 𝛿 𝑚𝑛 = 𝐻 ∫ 𝑧 𝑧 𝜒 𝑚 𝜒 𝑛 (cid:32) 𝑁 𝑓 (cid:33) d 𝑧. (64)One can obtain the eigenfunctions, 𝜒 𝑛 , by solving theeigenvalue problem (62)–(63) or by differentiating thestreamfunction modes 𝜙 𝑛 according to equation (A12).The quasigeostrophic vertical velocity eigenvalue problem(62)–(63) is often used as an alternative to the streamfunc-tion eigenvalue problem (40) (e.g., Chelton et al. 1998;Ferrari et al. 2010).31) Relating to the gravity wave problemThe quasigeostrophic vertical velocity eigenvalue prob-lem (62)–(63) is isomorphic to the vertical velocity eigen-value problem for internal gravity waves with rigid bound-aries (e.g. Gill 1982, section 6.11). In the Rossby waveproblem, the eigenvalue 𝜆 is given by the dispersion rela-tion (41), while in the internal gravity wave problem, theeigenvalue 𝜆 is given by 𝑐 = 𝜆𝑓 (65)where 𝑐 gravity is the speed of internal gravity waves.However, the isomorphism is lost once boundary dynam-ics are activated. In the case of gravity waves, boundarydynamics are activated in the presence of a free-surface[see Kelly (2016) and Yassin (in prep.)]. In contrast, onecan never apply a free-surface boundary condition to the continuously stratified quasigeostrophic problem as it isasymptotically inconsistent (Pedlosky 1982, section 6.9,p. 345). Instead, there is a genuine rigid-lid at the levelof the continuously stratified quasigeostrophic approxima-tion .2) Quasigeostrophic boundary dynamicsAs seen earlier, boundary buoyancy gradients activateboundary dynamics in the quasigeostrophic problem. Inthis case, boundary conditions for the quasigeostrophicvertical velocity problem (62) become −(− ) 𝑗 𝛾 𝑗 𝑘 d 𝜒 d 𝑧 (cid:12)(cid:12)(cid:12) 𝑧 𝑗 = 𝜆 (cid:20) 𝜒 | 𝑧 𝑗 + (− ) 𝑗 𝛾 𝑗 d 𝜒 d 𝑧 (cid:12)(cid:12)(cid:12) 𝑧 𝑗 (cid:21) (66)(see the appendix). The resulting modes { 𝜒 𝑛 } ∞ 𝑛 = forman orthonormal basis of 𝐿 ⊕ C and satisfy a peculiarorthogonality relation given by equation (A18).In the presence of boundary buoyancy gradients, ˆ 𝜓 andˆ 𝑤 become proportional at the boundaries [see equation(A15)]. Indeed, figure 9 shows that, while for the 𝑛 > 𝜓 𝑛 and ˆ 𝑤 𝑛 are related by differentiation,for the two boundary-trapped modes ( 𝑛 = , 𝜓 and ˆ 𝑤 are nearly identical. It follows that, inthe limit 𝛾 𝑗 → − , we obtain a stationary vertical velocitystep-mode 𝜒 step 𝑗 = 𝜙 step 𝑗 . (67)Similarly, in the limit of | 𝛾 𝑗 | → ∞ , we obtain an infinitelyfast boundary-trapped vertical velocity mode with a verti-cal e-folding of 𝑘 − .
4. Eigenfunction expansions
Motivated by the Rossby waves of the previous sec-tion, we now investigate various sets of normal modes for In contrast, one can include a free surface in stacked shallow-waterquasigeostrophic models. See chapter 5 in Vallis (2017) for more details. quasigeostrophic theory. Let { 𝜙 𝑛 } ∞ 𝑛 = be a collection of 𝐿 normal modes, and assume 𝜓 k ( 𝑧, 𝑡 ) is twice continuouslydifferentiable in 𝑧 . Completeness in 𝐿 implies that theeigenfunction expansion 𝜓 exp k , defined by 𝜓 exp k ( 𝑧, 𝑡 ) = ∞ ∑︁ 𝑛 = 𝜓 k 𝑛 ( 𝑡 ) 𝜙 𝑛 ( 𝑧 ) (68)where 𝜓 k 𝑛 = (cid:104) 𝜓 k , 𝜙 𝑛 (cid:105) (69)satisfies ∫ 𝑧 𝑧 | 𝜓 k ( 𝑧 ) − 𝜓 exp k ( 𝑧 )| d 𝑧 = . (70)Significantly, the vanishing of the integral (70) does notimply 𝜓 k = 𝜓 exp k since the two functions may differ at somepoints 𝑧 ∈ [ 𝑧 , 𝑧 ] .We now list physically desirable properties for a col-lection of normal modes. In general 𝜓 k ≠ 𝜓 exp k , but thislack of equality is not desirable physically. Hence, our firstcondition is that the expansion 𝜓 exp k be pointwise equal to 𝜓 k on the closed interval [ 𝑧 , 𝑧 ] , 𝜓 k ( 𝑧 ) = 𝜓 exp k ( 𝑧 ) for 𝑧 ∈ [ 𝑧 , 𝑧 ] . (71)If such pointwise equality is our only requirement for thenormal modes, then it is not necessary to restrict ourselvesto quasigeostrophic normal modes. Indeed, any basis of 𝐿 or 𝐿 ⊕ C satisfying equation (71) will suffice. However,we also require that the resulting series must be differen-tiable (we will see this requirement excludes 𝐿 bases) andthat the normal modes partition the energy, volume poten-tial vorticity, and surface potential vorticity amongst thevarious modes. That is, we insist that the normal modesdiagonalize the energy and potential enstrophy integrals asin Section 2e. This diagonalization requirement restrictsus to quasigeostrophic normal modes, and we will showthat only the 𝐿 ⊕ C modes satisfy all our requirements. a. The four possible 𝐿 modes There are only four 𝐿 bases in quasigeostrophic theorythat diagonalize the energy. All four sets of correspondingnormal modes satisfy the differential equation − dd 𝑧 (cid:32) 𝑓 𝑁 d 𝜙 d 𝑧 (cid:33) = 𝜆 𝜙 𝑧 ∈ ( 𝑧 , 𝑧 ) , (72)but differ in boundary conditions according to the follow-ing. • Baroclinic modes : Vanishing vertical velocity at bothboundaries (Neumann),d 𝜙 ( 𝑧 ) d 𝑧 = , d 𝜙 ( 𝑧 ) d 𝑧 = . (73) The reader should consult Yassin (in prep.) for appropriate refer-ences regarding the mathematical facts used in this section. AMS JOURNAL NAME
Fig. 9. The first six vertical velocity normal modes 𝜒 𝑛 (thin grey lines) and streamfunction normal modes 𝜑 𝑛 (black lines) (see section 3c).The propagation direction is 𝜃 = ◦ with a wavenumber of 𝑘 𝐿 𝑑 =
2. The remaining parameter are as in figure 4. Note that 𝜒 𝑛 and 𝜑 𝑛 arenearly indistinguishable for the boundary-trapped modes 𝑛 = , 𝑛 >
1. Theeigenvalue in the figure is non-dimensionalized by the deformation radius 𝐿 𝑑 . • Anti-baroclinic modes : Vanishing pressure at bothboundaries (Dirichlet), 𝜙 ( 𝑧 ) = , 𝜙 ( 𝑧 ) = . (74) • Surface modes : (mixed Neumann/Dirichlet) 𝜙 ( 𝑧 ) = , d 𝜙 ( 𝑧 ) d 𝑧 = . (75) • Anti-surface modes : (mixed Neumann/Dirichlet)d 𝜙 ( 𝑧 ) d 𝑧 = , 𝜙 ( 𝑧 ) = . (76)All four sets of modes are missing two modes. Each bound-ary condition of the formd 𝜙 ( 𝑧 𝑗 ) d 𝑧 = , (77)implies a missing step-mode while a boundary conditionof the form 𝜙 ( 𝑧 𝑗 ) = , (78)implies a missing boundary-trapped exponential mode. b. Expansions with 𝐿 modes We here examine the utility of eigenfunction expansionsin terms of 𝐿 modes, considering both pointwise equalityand differentiability. Recall that the geostrophic streamfunction 𝜓 is proportional to pres-sure (e.g., Vallis 2017, section 5.4).
1) Pointwise equality on [ 𝑧 , 𝑧 ] We now consider the convergence behaviour of the seriesexpansion 𝜓 exp k of 𝜓 k in terms of 𝐿 modes. For all foursets of 𝐿 modes, if 𝜓 k is twice continuously differentiablein 𝑧 , we obtain equality in the interior 𝜓 k ( 𝑧 ) = 𝜓 exp k ( 𝑧 ) for 𝑧 ∈ ( 𝑧 , 𝑧 𝑧 ) . (79)The behaviour at the boundaries depends on the boundaryconditions the modes 𝜙 𝑛 satisfy. If the 𝜙 𝑛 satisfy thevanishing pressure boundary condition at the 𝑗 th boundary 𝜙 𝑛 ( 𝑧 𝑗 ) = 𝜓 exp k ( 𝑧 𝑗 ) = 𝜓 k ( 𝑧 𝑗 ) . Hence, 𝜓 exp k generallyhas a jump discontinuity at the 𝑗 th boundary and is notequal to 𝜓 k on the closed interval [ 𝑧 , 𝑧 ] . In contrast, ifthe 𝜙 𝑛 satisfy a zero vertical velocity boundary conditionat the 𝑗 th boundary d 𝜙 𝑛 ( 𝑧 𝑗 ) d 𝑧 = 𝜓 k ( 𝑧 𝑗 ) = 𝜓 exp k ( 𝑧 𝑗 ) . (83)Consequently, of the four sets of 𝐿 modes, only withthe baroclinic modes do we obtain the pointwise equality 𝜓 k ( 𝑧 ) = 𝜓 exp k ( 𝑧 ) on the closed interval [ 𝑧 , 𝑧 ] . Indeed,with the baroclinic modes, the convergence of the series Ψ exp to Ψ on [ 𝑧 , 𝑧 ] is uniform and so no Gibbs phe-nomenon is present.However, even though we can represent 𝜓 k in terms ofthe baroclinic modes, we are unable to represent the cor-responding velocity 𝑤 k in terms of the vertical velocity5baroclinic modes since the modes vanish at both bound-aries. Analogous considerations show that only the anti-baroclinic vertical velocity modes (see the appendix) canrepresent arbitrary vertical velocities.2) Differentiability of the series expansionAlthough we obtain pointwise equality on the wholeinterval [ 𝑧 , 𝑧 ] , we have lost two degrees of freedom inthe expansion process. Recall that the degrees of freedomin the quasigeostrophic phase space are determined by thepotential vorticity. The volume potential vorticity, 𝑞 k , isassociated with the 𝐿 degrees of freedom while the surfacepotential vorticities, 𝑟 k and 𝑟 k , are associated with the C degrees of freedom.The series expansion 𝜓 exp k of 𝜓 k in terms of the baro-clinic modes is differentiable in the interior ( 𝑧 , 𝑧 ) . Conse-quently, we can differentiate the 𝜓 exp k series for 𝑧 ∈ ( 𝑧 , 𝑧 ) to recover 𝑞 k , that is, 𝑞 k = ∞ ∑︁ 𝑛 = 𝑞 k 𝑛 𝜙 𝑛 (84)where 𝑞 k 𝑛 = −( 𝑘 + 𝜆 𝑛 ) 𝜓 k 𝑛 . (85)However, 𝜓 exp k is not differentiable at the boundaries, 𝑧 = 𝑧 , 𝑧 , so we are unable to recover the surface potentialvorticities, 𝑟 k and 𝑟 k . Two degrees of freedom are lostby projecting onto the baroclinic modes. The energy at wavevector k is indeed partitioned be-tween the modes 𝐸 k = ∞ ∑︁ 𝑛 = ( 𝑘 + 𝜆 𝑛 ) 𝜓 k 𝑛 (86)and similarly for the potential enstrophy 𝑍 k = ∞ ∑︁ 𝑛 = ( 𝑘 + 𝜆 𝑛 ) 𝜓 k 𝑛 . (87)However, as we have lost 𝑟 k and 𝑟 k in the projection pro-cess, the surface potential enstrophies 𝑌 k and 𝑌 k , definedin equation (34), are not partitioned. To see that 𝜓 exp k is non-differentiable at 𝑧 = 𝑧 , 𝑧 , suppose that theseries 𝜓 exp k is differentiable and that d 𝜓 k ( 𝑧 𝑗 )/ d 𝑧 ≠ 𝑗 = ,
2. Butthen 0 ≠ d 𝜓 k ( 𝑧 𝑗 ) d 𝑧 = ∞ ∑︁ 𝑛 = 𝜓 k 𝑛 d 𝜙 𝑛 ( 𝑧 𝑗 ) d 𝑧 = , which is a contradiction. c. Quasigeostrophic 𝐿 ⊕ C modes Consider the eigenvalue problem − dd 𝑧 (cid:32) 𝑓 𝑁 d 𝜑 d 𝑧 (cid:33) = 𝜆 𝜑 for 𝑧 ∈ ( 𝑧 , 𝑧 ) (88a) − 𝑘 𝜑 + (− ) 𝑗 𝐷 − 𝑗 (cid:32) 𝑓 𝑁 d 𝜑 d 𝑧 (cid:33) = 𝜆 𝜑 for 𝑧 = 𝑧 𝑗 (88b)where 𝐷 and 𝐷 are non-zero real constants. This eigen-value problem differs from the generalized Rhines eigen-value problem (50) in that 𝐷 𝑗 are generally not equal tothe 𝛾 𝑗 defined in equation (51). The inner product (cid:104)· , ·(cid:105) in-duced by the eigenvalue problem (88) is given by equation(53) with the 𝛾 𝑗 replaced by the 𝐷 𝑗 .Smith and Vanneste (2012) investigate an equivalenteigenvalue problem to (88) and conclude that, when 𝐷 and 𝐷 are positive, the resulting eigenfunctions form abasis of 𝐿 ⊕ C . However, such a completeness result isinsufficient for the Rossby wave problem of section 3b, inwhich case 𝐷 𝑗 = 𝛾 𝑗 and 𝛾 𝑗 can be negative. Subsequently,Yassin (in prep.) examines the general theory of eigen-value problems with 𝜆 -dependent boundary conditions. Aconsequence of the theory is that the eigenfunctions of (88)form a complete basis of 𝐿 ⊕ C when 𝐷 , 𝐷 ≠ 𝐷 𝑗 = 𝛾 𝑗 where 𝛾 𝑗 is givenin equation (51). The surface-aware basis of Smith andVanneste (2012) correspond to the case 𝐷 , 𝐷 >
0. Allfour 𝐿 bases of section 4a, such as the traditional baro-clinic modes and the surface modes, are singular limitswhen | 𝐷 𝑗 | → | 𝐷 𝑗 | → ∞ . d. Expansion with 𝐿 ⊕ C modes When 𝐷 , 𝐷 in the eigenvalue problem (88) are finiteand non-zero, the resulting eigenmodes { 𝜑 𝑛 } ∞ 𝑛 = form abasis for the vertical structure phase space 𝐿 ⊕ C . Thus,the projection 𝜓 exp k ( 𝑧 ) = ∞ ∑︁ 𝑛 = 𝜓 k 𝑛 𝜑 𝑛 ( 𝑧 ) (89)where 𝜓 k 𝑛 = (cid:104) 𝜓 k , 𝜑 𝑛 (cid:105)(cid:104) 𝜑 𝑛 , 𝜑 𝑛 (cid:105) (90)is an equivalent representation of 𝜓 k . Not only do we havepointwise equality 𝜓 k ( 𝑧 ) = 𝜓 exp k ( 𝑧 ) for 𝑧 ∈ [ 𝑧 , 𝑧 ] , (91)but the series 𝜓 exp k is also differentiable on the closed inter-val [ 𝑧 , 𝑧 ] . Thus given 𝜓 exp k , we can differentiate to obtain6 AMS JOURNAL NAME both 𝑞 k and 𝑟 𝑗 k and thereby recover all quasigeostrophicdegrees of freedom. Indeed, we have 𝑞 k ( 𝑧, 𝑡 ) = ∞ ∑︁ 𝑛 = 𝑞 k 𝑛 ( 𝑡 ) 𝜑 𝑛 ( 𝑧 ) (92) 𝑟 𝑗 k ( 𝑧, 𝑡 ) = ∞ ∑︁ 𝑛 = 𝑟 𝑗 k 𝑛 ( 𝑡 ) 𝜑 𝑛 ( 𝑧 𝑗 ) (93)where 𝑞 k 𝑛 = −( 𝑘 + 𝜆 𝑛 ) (cid:104) Ψ , 𝜑 𝑛 (cid:105)(cid:104) 𝜑 𝑛 , 𝜑 𝑛 (cid:105) (94) 𝑟 𝑗 k 𝑛 = 𝐷 𝑗 𝑞 k 𝑛 (95)for 𝑗 = ,
2. In fact, we have just obtained an expansionof the relative potential vorticity distribution of Bretherton(1966) given by equation (24), 𝔮 k = ∞ ∑︁ 𝑛 = 𝑞 k 𝑛 + ∑︁ 𝑗 = 𝐷 𝑗 𝛿 ( 𝑧 − 𝑧 𝑗 ) 𝜑 𝑛 . (96)In addition, the energy, 𝐸 k , volume potential enstrophy, 𝑍 k , and surface potential enstrophies, 𝑌 k and 𝑌 k , arepartitioned (diagonalized) between the modes 𝐸 k = ∞ ∑︁ 𝑛 = ( 𝑘 + 𝜆 𝑛 ) 𝜓 k 𝑛 , (97) 𝑍 k + 𝐻 ∑︁ 𝑗 = 𝐷 𝑗 𝑌 𝑗 k = ∞ ∑︁ 𝑛 = ( 𝑘 + 𝜆 𝑛 ) 𝜓 k 𝑛 . (98)
5. Discussion a. Projecting model/observational data
If one seeks modes to project model or observationaldata, which modes should be chosen? If our only concernis reproducing the shape of the vertical structure, then wecan use any 𝐿 basis (not just those listed in this article)where the basis functions do not vanish at the boundary.Alternatively, any 𝐿 ⊕ C basis can be used and thesemodes will ensure the differentiability of the series expan-sion as well.However, it is typically desirable that the projectionalso diagonally partitions the energy and potential vorticityamongst the eigenmodes. Of the four quasigeostrophic 𝐿 modes discussed in section 4a, only the baroclinic modesare capable of exactly representing a function and parti-tioning the energy. The baroclinic modes, however, canonly partition the volume potential vorticity, 𝑞 k , amongstthe modes whereas the surface potential vorticities, 𝑟 𝑗 k ,are lost in the projection process. In addition a series ex-pansion in terms of the baroclinic modes will generally notbe differentiable at the boundaries. For the 𝐿 ⊕ C modes, any set of modes obtained fromthe eigenvalue problem (88) with 𝐷 , 𝐷 ≠ 𝑞 k , and the surface potential vorticities, 𝑟 𝑗 k , amongstthe modes. Therefore, in a non-linear ocean with non-isentropic boundaries, there are infinitely many ways (pa-rameterized by 𝐷 and 𝐷 ) of decomposing the verticalstructure into normal modes. We have no physical argu-ment prefering one decomposition over another.Indeed, we emphasize that modal projection is a math-ematical procedure that merely provides an alternate rep-resentation of the vertical structure as a sum of normalmodes. For instance, the presence of the barotropic modein some decompositions does not imply that there is any co-herent barotropic motion. Rather, it is only in the appropri-ate linear setting with quiescent background, as discussedin section 3, that such an interpretation holds.We also note that the projection process in terms ofthe baroclinic modes can be highly misleading. For in-stance, consider a realistic ocean with strong surface quasi-geostrophic dynamics at the upper surface and weak inte-rior dynamics. Most of the energy in this ocean is as-sociated with motion having large upper surface potentialvorticity, 𝑟 k , and small volume potential vorticity, 𝑞 k .However, a projection onto baroclinic modes will partitionthe energy amongst the traditional baroclinic modes andmay give the incorrect impression that the motion is alldue to the interior potential vorticity 𝑞 k . b. What vertical mode does the altimeter reflect? It is the misleading nature of the baroclinic modes thatis presently the basis of the debate on what mode a sur-face altimeter signal reflects. Returning to the controversyfrom the introduction between Wunsch (1997) and Lapeyre(2009), we only state that the question of what "mode" thesurface signal represents is itself ill-posed. Outside ofthe linear theory of section 3, the concept of a "mode" issomewhat arbitrary and certainly does not correspond toany coherent motion.An alternate and more physical question is the follow-ing: Is the altimeter signal due to interior (induced by 𝑄 ) orsurface (induced by 𝑅 ) dynamics? However, it is not pos-sible to answer this question using modal decompositionsalone. c. Galerkin approximations with 𝐿 modes Both the 𝐿 baroclinic modes and the 𝐿 ⊕ C modeshave infinitely many degrees of freedom. In contrast, nu-merical simulations only contain a finite number of de-grees of freedom. Consequently, it should be possible touse baroclinic modes to produce a Galerkin approximationto quasigeostrophic theory with non-trivial boundary dy-namics. Such an approach has already been proposed byRocha et al. (2015).7Projecting 𝜓 k onto the baroclinic modes produces a se-ries expansion, 𝜓 exp k , that is differentiable in the interiorbut not at the boundaries. By differentiating the series inthe interior we obtain equation (85) for 𝑞 k 𝑛 . If instead weintegrate by parts twice and avoid differentiating 𝜓 exp k , weobtain 𝑞 k 𝑛 = −( 𝑘 + 𝜆 𝑛 ) 𝜓 k 𝑛 − 𝐻 ∑︁ 𝑗 = 𝑟 𝑗 k 𝜙 𝑛 ( 𝑧 𝑗 ) . (99)The two expressions (85) and (99) are only equivalent when 𝑟 k = 𝑟 k =
0. For non-zero 𝑟 k and 𝑟 k , the singular natureof the expansion means we have a choice between equations(85) and (99).By choosing equation (99) and avoiding the differen-tiation of 𝜓 exp k , Rocha et al. (2015) produced a least-squares approximation to quasigeostrophic dynamics thatconserves the surface potential enstrophy integrals (33).This is a conservation property underlying the approxima-tion’s success.
6. Conclusion
In this article, we have studied all possible non-continuum collections of streamfunction normal modesthat diagonalize the energy and potential enstrophy. Thereare four possible 𝐿 modes: the baroclinic modes, theanti-baroclinic modes, the surface modes, and the anti-surface modes. Additionally, we developed a family of 𝐿 ⊕ C bases parameterized by the parameters 𝐷 , 𝐷 . If 𝐷 𝑗 = 𝛾 𝑗 , where 𝛾 𝑗 is given by equation (51) for 𝑗 = ,
2, theresulting modes are the vertical structure of Rossby wavesin a quiescent ocean with prescribed boundary buoyancygradients. We have also examined the associated 𝐿 and 𝐿 ⊕ C vertical velocity modes.For the streamfunction 𝐿 modes, only the baroclinicmodes are capable of exactly representing any quasi-geostrophic state on the interval [ 𝑧 , 𝑧 ] , whereas for thevertical velocity 𝐿 modes, only the anti-baroclinic modesare capable. However, in both cases, the resulting eigen-function expansion is not differentiable at the boundaries, 𝑧 = 𝑧 , 𝑧 . Consequently, while we can recover the vol-ume potential vorticity density, 𝑞 k , we cannot recover thesurface potential vorticity densities, 𝑟 k and 𝑟 k . Thus,we lose two degrees of freedom when projecting onto thebaroclinic modes. In contrast, 𝐿 ⊕ C modes providean equivalent representation of the function in question.Namely, the eigenfunction expansion is differentiable onthe closed interval [ 𝑧 , 𝑧 ] so that we can recover 𝑞 k , 𝑟 k , 𝑟 k from the series expansion.A natural application of the proposed 𝐿 ⊕ C normalmodes is to the study of weakly non-linear wave-interactiontheories of geostrophic turbulence found in Fu and Flierl(1980) and Smith and Vallis (2001), extending their workto include prescribed surface buoyancy gradients. Another application is to the extension of equilibrium statistical me-chanical calculations (e.g., Venaille et al. 2012) to includesurface quasigeostrophic degrees of freedom. Both appli-cations will be reported in subsequent contributions. Acknowledgments.
We offer sincere thanks to StephenGarner, Robert Hallberg, Isaac Held, and Sonya Legg forcomments and suggestions that greatly helped our presen-tation. This report was prepared by Houssam Yassin underaward NA18OAR4320123 from the National Oceanic andAtmospheric Administration, U.S. Department of Com-merce. The statements, findings, conclusions, and recom-mendations are those of the authors and do not necessarilyreflect the views of the National Oceanic and AtmosphericAdministration, or the U.S. Department of Commerce.
Data availability statement.
APPENDIX
Polarization relations and the vertical velocityeigenvalue problem a. Polarization relations
The linear quasigeostrophic vorticity and buoyancyequations, computed about a resting background state, are 𝜕𝜁𝜕𝑡 + 𝛽 𝜕𝜓𝜕𝑥 = 𝑓 𝜕𝑤𝜕𝑧 , (A1) 𝜕𝑏𝜕𝑡 = − 𝑁 𝑤 (A2)in the interior 𝑧 ∈ ( 𝑧 , 𝑧 ) . The vorticity, 𝜁 , and buoyancy, 𝑏 , are given in terms of the geostrophic streamfunction via 𝜁 = ∇ 𝜓 (A3) 𝑏 = 𝑓 𝜕𝜓𝜕𝑧 , (A4)The no-normal flow at the lower and upper boundariesimplies 𝑓 𝑤 = u · ∇ 𝑔 𝑗 (A5)for 𝑗 = ,
2. Substituting equation (A5) into the linearbuoyancy equation (A2), yields the boundary conditions 𝜕 𝑡 𝑏 + u · ∇ (cid:18) 𝑁 𝑓 𝑔 𝑗 (cid:19) = 𝑧 = 𝑧 𝑗 . (A6)We now assume solutions of the form 𝜓 = ˆ 𝜓 ( 𝑧 ) 𝑒 k ( x ) e − i 𝜔𝑡 (A7)and similarly for the variables 𝜁, 𝑏, 𝑤, 𝑢, 𝑣 . Substitutingsuch solutions into equations (A1)–(A2) and using u = ˆ z × ∇ 𝜓 gives ˆ 𝜁 = − 𝑘 ˆ 𝜓 (A8)ˆ 𝑢 = − i 𝑘 𝑦 ˆ 𝜓 (A9)ˆ 𝑣 = i 𝑘 𝑥 ˆ 𝜓 (A10)8 AMS JOURNAL NAME for 𝑧 ∈ [ 𝑧 , 𝑧 ] , andˆ 𝑏 = − i 𝑁 𝜔 ˆ 𝑤 (A11)d ˆ 𝜓 d 𝑧 = − i 𝑁 𝑓 𝜔 ˆ 𝑤 (A12)d ˆ 𝑤 d 𝑧 = i 𝜔𝑓 (cid:20) 𝑘 + 𝛽 𝑘 𝑥 𝜔 (cid:21) ˆ 𝜓. (A13)for 𝑧 ∈ ( 𝑧 , 𝑧 ) . At the boundaries 𝑧 = 𝑧 , 𝑧 , we use equa-tions (A5) and (A6) to obtainˆ 𝑏 = − 𝑁 𝑓 𝜔 ˆ u · ∇ 𝑔 𝑗 (A14)ˆ 𝑤 = i 1 𝑓 ˆ u · ∇ 𝑔 𝑗 . (A15) b. The vertical velocity eigenvalue problem Taking the vertical derivative of (A13) and using (A12)yields − d 𝜒 d 𝑧 = 𝜆 (cid:32) 𝑁 𝑓 (cid:33) 𝜒 (A16)where ˆ 𝑤 = 𝑤 𝜒 ( 𝑧 ) and 𝜒 is non-dimensional. The bound-ary conditions at 𝑧 = 𝑧 𝑗 are −(− ) 𝑗 𝛾 𝑗 𝑘 d 𝜒 d 𝑧 = 𝜆 (cid:20) 𝜒 + (− ) 𝑗 𝛾 𝑗 d 𝜒 d 𝑧 (cid:21) (A17)as obtained by using equations (A13) and (A12) in bound-ary conditions (50b).As shown in Yassin (in prep.), the eigenfunctions { 𝜒 𝑛 } ∞ 𝑛 = of the above problem form an orthonormal ba-sis of 𝐿 ⊕ C when 𝛾 𝑗 ≠ 𝑗 = ,
2. In this case, theorthonormality condition is ± 𝛿 𝑚𝑛 = 𝐻 (cid:34)∫ 𝑧 𝑧 𝜒 𝑚 𝜒 𝑛 (cid:32) 𝑁 𝑓 (cid:33) dz − 𝑘 ∑︁ 𝑗 = 𝛾 𝑗 (cid:0) B 𝑗 𝜒 𝑚 (cid:1) (cid:0) B 𝑗 𝜒 𝑛 (cid:1) (A18)where B 𝑗 𝜒 = 𝜒 ( 𝑧 𝑗 ) + (− ) 𝑗 𝛾 𝑗 d 𝜒 ( 𝑧 𝑗 ) d 𝑧 . (A19) c. The vertical velocity 𝐿 modes Analogously with the streamfunction 𝐿 modes, we havethe following sets of vertical velocity 𝐿 modes. • Baroclinic modes : Vanishing vertical velocity at bothboundaries, 𝜒 ( 𝑧 ) = , 𝜒 ( 𝑧 ) = . (A20) • Anti-baroclinic modes : Vanishing pressure at bothboundaries, d 𝜒 ( 𝑧 ) d 𝑧 = , d 𝜒 ( 𝑧 ) d 𝑧 = . (A21) • Surface modes :d 𝜒 ( 𝑧 ) d 𝑧 = , 𝜒 ( 𝑧 ) = . (A22) • Anti-surface modes : 𝜒 ( 𝑧 ) = , d 𝜒 ( 𝑧 ) d 𝑧 = . (A23) References
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