Baroclinic Sea Level

Sea level and its horizontal gradient are an expression of oceanic volume, heat content, and currents. Large‐scale currents have historically been viewed as mostly “baroclinic,” and tides as “barotropic,” respectively, in the loose sense of being strongly related to the oceanic density distribution or not. In particular, the evolution of the barotropic velocity is influenced by a horizontal pressure‐gradient force that depends on the gradient of a particular depth‐weighting of the density field as well as on the gradient of the sea‐surface elevation ζ; hence, even the tides must be viewed as a product of the coupled interaction of barotropic and baroclinic fields. The purpose of this note is to give dynamical precision to the distinction between barotropic and baroclinic contributions to ζ and the surface pressure‐gradient force −g∇ζ, and, in the particular case of the tides, demonstrate their combined barotropic‐baroclinic interactions with a realistically forced, high‐resolution simulation of the Pacific Ocean circulation. While the different tidal sea‐level contributions manifest a horizontal scale separation (e.g., more barotropic at larger scales; more baroclinic surface pressure‐gradient force at smaller scales), there are cross‐mode corrections in both at the level of tens of percent. The proposed barotropic‐baroclinic decomposition is generally relevant to the sea‐level expression of oceanic currents.


Introduction
Changing sea-level elevation ζ is perhaps the most readily measurable and measured oceanic property.Averaged over surface gravity waves, it expresses the tides and, at lower frequencies, the horizontal pressure gradient force for surface currents ⃗ u.At even lower frequencies it expresses climate changes in global temperature and ice volume.
Sea level is traditionally measured with coastal gauges.In recent decades satellite measurements of elevation and Earth's gravity field have greatly expanded our view, and we now have the prospect of even higher-resolution measurements of ζ with new missions.
What is the three-dimensional reality that underlies these surface expressions?To answer this question, measurements are limited and models must be employed, that is, geographically and dynamically realistic computational simulations of the oceanic current and density fields.
As is common in oceanic circulation models with a weakly compressible equation of state, we make the Boussinesq approximation to the pressure-gradient force, where ρ is density, p is pressure, the arrow denotes a 3D vector, and ϕ = p/ρ 0 is the geopotential function or normalized dynamic pressure with ρ 0 a constant.(Note: this precludes one definition, used in compressible fluid dynamics, of a baroclinic-barotropic decomposition based on whether or not there is vorticity generation by ⃗ ∇ρ × ⃗ ∇p ≠ 0, because the curl the pressure gradient force is zero in a Boussinesq fluid; see Pedlosky (1987)).
In the large-scale, low-frequency circulation, the sea level ζ is often also called the surface dynamic height (relative to mean sea level), and its density-normalized (by ρ 0 ), horizontal pressure-gradient force is approximately in balance with the local Coriolis force at the surface z = ζ (i.e., with a geostrophic current), ϕ s is the dynamic pressure there.The first equality in Equation 2 is based on hydrostatic balance and the assumption that the air-sea interface is a surface of constant total pressure neglecting atmospheric surface pressure variations; see Griffies and Greatbatch (2012) for a more fundamental discussion of sea level variation.The surface values are denoted by the subscript s; g is the gravitational constant; f is the Coriolis frequency; ̂⃗z is the unit vector in the upward vertical direction; u gs is the surface geostrophic current; the caret denotes a unit vector; and horizontal vectors are bold face, while 3D vectors have an arrow symbol.An accompanying vertical momentum relation for the 3D dynamically relevant pressure is hydrostatic balance in an integrated form, where the buoyancy field is b = g(1 ρ/ρ 0 ), ρ is density, and ρ 0 is a constant reference value.In a loose approximation, when only density measurements are available, it is sometimes assumed that the horizontal pressure-gradient force ∇ϕ and ∇b vanish at depth (i.e., below z = h nm , a "level of no motion"), so that the surface dynamic height is entirely due to the interior buoyancy variations b: Here the upper vertical integration limit at z = ζ has been further approximated by 0, the mean surface elevation (i.e., ζ ≪ h nm ).While Equation 4 is not fully accurate for basin and mesoscale currents due to its neglect of ∇ϕ at depth, together with Equation 2 it indicates how interior b can influence ζ, hence u.The dynamical influence of b is referred to as baroclinicity, so ζ nm in Equation 4 is a type of baroclinic sea level.However, in this situation of a surface-intensified geostrophic current and no motion at depth, both the barotropic and baroclinic currents defined in Equations 5 and 6 are non-zero; so ζ nm is a mixed barotropic-baroclinic sea level.(See the end of Section 2.2 for a further remark.) For the tides, again ∇ϕ s = g∇ζ at the relevant frequencies.Tides have an astronomical and self-interaction gravitational forcing that is essentially independent of depth within the ocean.The associated response in u to this forcing is called the "external" tide, which in this paper will be equated with the depth-averaged velocity, a.k.a. the barotropic current, u bt , where H(x) is the resting depth of the ocean.In many papers the external tide is also referred to as the "surface" tide, although that is ambiguous with respect to the barotropic-baroclinic decomposition.The remainder is the "internal" tide, associated with the baroclinic horizontal current, where both u and u bc are functions of depth.The existence of an internal tide is evident in interior time series of b, which often show strong oscillations at or near the tidal frequencies.As will be explained in Section 2, ζ reflects both barotropic and baroclinic current dynamics.
A simple approximate model for the external tide is to associate the dynamical response to the gravitational forcing with u(z) = u bt .With an approximation of an incompressible mass balance (i.e., ⃗ because ρ variations are small compared to ρ 0 ) and the kinematic boundary conditions at the solid bottom and top free surface, the column-integrated continuity relation is Integrated over the area of the domain, this relation implies that the ocean has a constant volume (ignoring rivers and other surface freshwater volume fluxes, all of which have small effects on tidal time scales).Notice that Equation 7 has no explicit dependency on b, while the pressure-gradient force, ∇ϕ, from Equation 3 does so, in particular in the evolution equation for the u bt .This is the sense in which the evolution of (ζ, u bt ) is not dynamically decoupled from (b, u bc ), as further explained in Section 2.2.
With a further approximation that b = 0, Equation 3 implies that ∇ϕ(z) = ∇ϕ s = g∇ζ, and a horizontal momentum balance can be formulated entirely in terms of u bt , which together with Equation 7 is called the Shallow-Water Equations.It can also be called a barotropic tidal model with ζ = ζ bt .One of the first satellite tidal products was G. Egbert et al. (1994) that fits a Shallow-Water model to altimetric measurements.This model is dynamically inconsistent with Equation 3 as it neglects any baroclinic effect from b ≠ 0, and it is quite different from Equation 4; nevertheless, it has been widely considered useful as an estimate of ζ at large spatial scales comparable to the width of oceanic basins, associating it with the barotropic tide.Furthermore, a model of the barotropic tide by itself is dynamically incomplete, and any such model, with whatever mixture of dynamics and measurements, has to confront the important matter of energy conversion to the baroclinic tide that occurs in stratified waters over variable topography (G.Egbert & Ray, 2000) but is absent in a Shallow-Water or any other only barotropic model (Stammer et al., 2014).
From the perspective of this paper, a sufficiently accurate model for tides and other currents is the Boussinesq, hydrostatic, incompressible (i.e., non-divergent ⃗u) equations that also contain a realistic seawater equation of state and necessary forcing and damping effects.In this system, Equations 3 and 5-7 are correct relations, while Equations 2 and 4 are only approximations for the indicated circumstances.The specific question posed here is how the sea level should be partitioned into barotropic and baroclinic components and, more specifically, how the astronomically forced external and internal tides should be partitioned dynamically.The theoretical answer is provided in Section 2.2 and is illustrated with Pacific-basin simulations in Section 3.2 using the Regional Oceanic Modeling System (ROMS) (Shchepetkin & McWilliams, 2005) that embodies the dynamical assumptions listed in this paragraph.ROMS is a model that explicitly makes the barotropic-baroclinic decomposition as proposed here, and separates these components in its time-stepping algorithm.
For the general circulation, mesoscale eddies, and other currents in hydrostatic balance, our proposed partition is mostly consistent with common practice, and it indicates how low-frequency changes in oceanic mass and buoyancy distributions are expressed in sea level.

Background
Multiple approaches have been taken for determining the barotropic-baroclinic decomposition of the tides, and this is a commonly discussed topic.The direct approach of evaluating Equations 5 and 6 is rarely available from measurements of the full water column, and it would not directly show the decomposition of sea level.As with dynamic height in the context of large-scale circulation Equation 2, tidal sea level is relevant dynamically to the velocity field through its relation to the horizontal pressure-gradient force at the surface.
One approach to the decomposition is by Colosi and Munk (2006), who analyzed very long time series of ζ(t) at two tide gauges in Hawaii.They devised a statistical model for the shape of the frequency spectrum that assumed that the barotropic component is entirely a "coherent" spectrum line (i.e., in phase with the astronomical forcing, a.k.a. the equilibrium tide), while the baroclinic component has both a coherent line and a smoothly varying "incoherent" shape in nearby frequencies in association with refraction caused by spatial variations in subtidal b; the latter split is measured by the variance in the phase at the tidal frequency.They concluded that for the M 2 frequency ζ bt is much larger than ζ bc , while the latter has comparable magnitudes in its coherent and incoherent parts.This is an adynamical analysis in that it does not connect to the governing fluid equations.In particular, it has no information about g∇ζ, which is the horizontal pressure-gradient force.Savage et al. (2017) analyze a realistic global simulation model (HYCOM) and define a "steric" sea-level anomaly by They then associate ζ st with ζ bc and define a "non-steric" (barotropic) residual, ζ bt = ζ ζ bc .Their conclusions regarding the tidal sea-level decomposition are qualitatively consistent with those of Colosi and Munk (2006) and have the advantage of global coverage.Notice the functional similarity between Equation 9and the lowfrequency approximation ζ nm in Equation 4, apart from a difference of integration range.In our view this definition, while motivated by a conception of seawater compressibility, is not dynamically defensible, as further explained in Section 2.2.As pointed out in Zaron and Ray (2023), another way to express the incompleteness of this barotropic-baroclinic partition of sea-level is through the relation of surface dynamic pressure, ϕ s = gζ, the steric height ζ st in Equation 9, and the bottom pressure ϕ( H) through its dependency on buoyancy, viz., Kelly ( 2016) opens with "The de facto standard is to define surface tides as depth-averaged pressure and horizontal velocity and internal tides as the residuals," which we almost agree with as a definition of the barotropic tide.He then proceeds, as his main topic, to define vertical modes, associating mode 0 with the barotropic mode and modes 1, 2, …with the baroclinic modes.Part of his paper is to include a correct free-surface boundary condition in the modal calculation, even though that introduces a modest discrepancy with the principle quoted here.(With a rigid-lid boundary condition, it does conform; see Appendix A).Thus, the linear, conservative, freesurface, gravest (a.k.a."barotropic") eigenmode u o for the hydrostatic Primitive or Boussinesq equations does not exactly coincide with the depth-averaged u bt defined here in Equation 5, and the depth-average of the " baroclinic" eigenmodes, u n , n ≥ 1, are not exactly zero, again in contrast with u bc here, although these eigenmodes nearly have these depth-averaged attributes and their differences with the rigid-lid eigenmodes, which have these exact attributes, are slight.However, this difference does allow the u 0 eigenmode to formally escape the discrepancy of non-zero ζ 0 while still satisfying the free-surface continuity Equation 7; however, in practice using a diagnostic relation of ζ n = ϕ n (0)/g works well even with rigid-lid modes.With this diagnostic extension, the rigid-lid barotropic and baroclinic eigenmodes are consistent with our proposed general decomposition principle (Section 2.2).
In general we find vertical modes somewhat problematic as a representation for realistic situations because they presume as background the local value for Coriolis frequency f, resting depth H, and sub-tidal stratification profile N 2 (z) = db/dz, all of which are geographically variable (temporally, too, for N 2 ).There is an implicit assumption that these environmental variables are slowly varying in space and time, which sometimes they are not.These modes comprise a formally complete basis set for arbitrary vertical profiles at each location, and thus any general dynamical model (e.g., the Primitive Equations) could be projected onto them at the cost of inter-mode coupling terms for all processes not consistent with their defining eigenmode problem.Our primary criticism is that this approach effectively presumes that the ocean behaves similarly to the eigenmodes without significant modification by nonlinearity, forcing, or dissipation.
Nevertheless, several papers carry out modal analyses of baroclinic sea level and report useful skill in fitting to altimetric measurements filtered at tidal frequencies.The High Resolution Empirical Tide (HRET) (Ray & Zaron, 2016;Zaron, 2019;Zaron et al., 2022) fits horizontal plane waves to the sea level and extends this to other fields using the conservative eigenmodes.The Coupled-mode Shallow Water (CSW) model (Kelley et al., 2021;Kelly et al., 2016;Savage et al., 2020) projects a linear Primitive Equation model onto the conservative eigenmodes with topographic inter-mode coupling and parameterized linear damping, using climatological stratification field and the observed surface tide as a topographic source term for the baroclinic modes.
In practice the most common approach for decomposing tidal ζ, both for satellite measurements and models, is on the basis of horizontal scale content (Carrere et al., 2021;Ubelmann et al., 2022).The conservative, linear eigenmodes for a flat, resting ocean at the tidal frequencies-whose vertical structure is consistent with Kelly (2016)-have a very large horizontal wavelength of O(10 4 ) km for the barotropic mode, in contrast with O (10 2 ) km for the baroclinic modes, which is near the baroclinic deformation radius ∼Nh pyc /f, where h pyc is the depth of the main pycnocline.Of course, real tidal dynamics are forced and damped, if not also nonlinear, but this criterion does provide a heuristically plausible framework for the decomposition.
Thus, we conclude that none of the existing approaches for making a barotropic-baroclinic decomposition is sufficiently fundamental by the standard of a realistic oceanic circulation model (Section 1), even though many of them have come to sensible and mutually consistent conclusions about the physical characteristics of the tides.
Our proposed decomposition is in the next section.

Dynamical Decomposition
The fundamental basis for a barotropic-baroclinic decomposition in a model like ROMS is in terms of the horizontal velocity u, that is, Equations 5 and 6.While Equation 7suggests that sea level is associated with the barotropic velocity, the evolution equation for u bt cannot be closed entirely in terms of the sea level as its pressuregradient force.Rather, the barotropic horizontal momentum equation has the form of with a depth-averaged pressure-gradient force P bt ; the dots indicate the non-pressure forces elided here (Shchepetkin & McWilliams, 2005).Using Equation 3, we can evaluate this barotropic force to be where the residual from the sea-level gradient term is defined as the baroclinic contribution to the total barotropic force.Extending this definition below the surface again using Equation 3, the vertical profile of the baroclinic pressure-gradient force is with a surface value of Thus, the total surface pressure-gradient force is as expected, from a rearrangement of Equation 12.The relations Equations 11-15 make it clear that the evolution of the barotropic current is influenced by the buoyancy field as well as the sea level; that is, the barotropic and baroclinic currents have a coupled dynamics in a stratified ocean, as in Equations 8 and 15, and ζ and P cannot be uniquely associated with either component alone.
To make the coupling explicit, ζ changes due to a divergence in the barotropic transport, u bt changes due to the depth-averaged pressure-gradient force involving both ζ and b, and this then feeds back onto further changes in ζ; Journal of Advances in Modeling Earth Systems 10.1029/2023MS003977 meanwhile b changes due to advection by both u bt and u bc .Alternatively expressed, b enters into both P bt and P b , and we can expect its variations to influence the evolution of both u bt and u bc .
The governing momentum equations contain a pressure-gradient force, not the pressure per se.With the Boussinesq approximation where |b| ≪ g and in deep water where |ζ| ≪ H, the two expressions Equations 13 and 14 for P bc can be simplified by setting ζ ≈ 0. However, for variable H(x), P bc cannot be expressed in the form of a baroclinic sea-level gradient, because ∇ does not commute with H; hence, we cannot identify a ζ bc such that P bc (ζ) = g∇ζ bc .
If we manipulate Equation 14, we can write with a baroclinic pseudo sea level, and a residual contribution to the baroclinic pressure-gradient force, that is associated with resting-depth gradients, ∇H ≠ 0. (In a modal representation, R bc is related to the topographically induced coupling between modes.)Again, one can simplify these expressions with the approximation In fact, ζbc is equivalent to minus the buoyancy contribution to the depth-averaged (i.e., barotropic) pressure from Equation 3; that is, which itself is equal to g times the barotropic pseudo sea level ζbt = ζ ζbc .The existence of R bc ≠ 0 in Equation 16shows that the depth-averaged pressure gradient differs from the gradient of the depth-averaged pressure.
Notice that R bc vanishes for a flat bottom, whence with this ζ simplification, when ∇H = 0.This partition in Equation 16is intended only to demonstrate the ζbc component.In particular, note that the unpartitioned P bc (ζ) in Equation 14does not have any direct dependency on ∇H; rather, that arises only in the partitioned expressions.The baroclinic pseudo sea level ζbc has a partial similarity with the steric ζ st in Equation 9 through its dependency only on b, not ζ, but, as previously remarked, it differs by the extra vertical integral.
Thus, we propose a dynamical decomposition of the horizontal pressure-gradient force Equations 12-15 based on the barotropic-baroclinic decomposition of u, rather than a direct decomposition of sea level itself except where H is flat.The baroclinic contribution (e.g., ζbc in Equation 21) can be compared with the baroclinic expressions for ζ nm in Equation 4 and for ζ st in Equation 9; it has similar ingredients but it adds another vertical integral.With most b(z) profiles the different ζ values will be quantitatively different but similar in magnitude.
This decomposition is valid for all frequencies.The application to tides is perhaps the most timely one with the prospect of new altimetric satellites with higher spatial resolution.To make this the focus, the expressions in this section should be temporally filtered to isolate the tidal frequencies.With the simplification |ζ| ≪ H, these expressions are linear in ζ and b, which makes the filtering task easier.Furthermore, the decomposition does not depend on calculating vertical modes (Kelly, 2016), although that is a further analysis option.And, it makes no assumption apriori about the spatial scale content of the barotropic and baroclinic components.For the special case of conservative linear, rigid-lid, tidal eigenmodes, Appendix A shows that the relations in this section yield the familiar modal results.
Finally The choice here for the dynamically relevant decomposition of the surface pressure-gradient force has some similarity with the long-standing discussion about the role of bathymetry in large-scale circulation.Several alternative interpretive frameworks have been adopted, all correct and variously helpful for physical understanding: the vertical curl of the depth-averaged horizontal momentum balance (i.e., the barotropic balance), featuring the Joint Effect of Baroclinicity and Relief (JEBAR) (Mellor, 1999;Sarkisyan & Ivanov, 1971); the curl of the depth-integrated momentum balance (i.e., the transport balance), featuring the Bottom Pressure Torque (BPT) (Molemaker et al., 2015;Song & Wright, 1998); and the depth-integral of the curl of the momentum balance (i.e., the vorticity balance), featuring the Bottom Stress Divergence Torque (BSDT) (Capó et al., 2023;Jagannathan et al., 2021).These alternatives arise from the non-commutativity of the vertical integral or average and the horizontal gradient of H, as in Equations 16-18.For the tides the vorticity or circulation tendency is less relevant than the force, hence the focus there is on the pressure-gradient force, using the depth-averaged decomposition in Equations 11-15.

Pacific Simulation
The UCLA version of ROMS (the Regional Oceanic Modeling System; Shchepetkin and McWilliams (2003) and Shchepetkin and McWilliams (2005)) is a terrain-following oceanic circulation model.It uses third-order upwind advection algorithms for the horizontal advection of tracers and momenta.These advection schemes have a dissipative discretization error that is hyper-diffusive or -viscous in nature and automatically scales with resolution, negating the need for an explicitly prescribed horizontal smoothing or regularization term.Vertical advection is computed with a fourth-order spline advection scheme.Unresolved mixing processes are parameterized with a the K-profile parameterization in the surface and bottom boundary layers, combined with a Richardson number based parameterization in the interior (Large et al., 1994).
The simulation that is the foundation of the investigation in this paper is a basin-scale simulation of the full Pacific Ocean with a nominal horizontal grid resolution of dx = 6 km.It is a high-resolution descendant of the wellvalidated simulation in Lemarié et al. (2012) with additional tidal forcing.Strictly speaking, this is still a regional simulation that needs to be forced at its lateral boundaries, which are most extensive in the south.The information for these open boundaries is derived from the GLORYS reanalysis data set (Verezemskaya et al., 2021) that is provided at a resolution of 0.083°and a time interval of 1 day.The GLORYS data are interpolated in space to the computational grid and interpolated in time at each time-step while the model is running.We refer to this type of computations as an "online" computation.This is in contrast with pre-and postprocessing of data before or after the model run, which we refer to as an "offline" computation.The GLORYS

Journal of Advances in Modeling Earth Systems
10.1029/2023MS003977 MCWILLIAMS ET AL.
data does not contain tidal information, and the basin-scale simulation is tidally forced at the open boundaries with sea-surface elevations and tidal barotropic currents from the TPXO9 analysis (G.D. Egbert & Erofeeva, 2002).In addition to this tidal forcing at the lateral boundaries, the model is forced with a surface geopotential forcing.The surface geopotential is a combination of the astronomical tide and the self-attraction and loading effect (Arbic et al., 2018).The self-attraction and loading are the result of geopotential anomalies that arise from the evolving sea surface elevation itself as well as the deformation of Earth's crust under the influence of the tidal motions (Arbic et al., 2018).Atmospheric forcing is obtained from the ERA5 global reanalysis (Hersbach et al., 2020).This data set is available at a nominal 0.25°spatial resolution and hourly intervals.The COARE formulation (Fairall et al., 2011) is used to compute momentum and tracer fluxes from atmospheric variables using a bulk approach.The use of sufficiently high-frequency atmospheric forcing permits realistic levels of near inertial internal waves below the mixed layer, which are essential to a correct representation of the kinetic energy budget in the ocean (Barkan et al., 2021;Shcherbina et al., 2013).A more complete description of the tidal simulation and its analyses will be reported in separate papers.

Tidal Pressure-Gradient Force
We now focus on the tidal components of the Pacific simulation, and even more particularly on the lunar semi-diurnal (M2) component that on average has the largest amplitude among the components.They are extracted by time filtering the model output at the M2 frequency of 2.237 × 10 5 cycles per second: the M2 signal is defined as the complex Fourier amplitude of a single frequency in this time series whose length is an integer multiple of its period.
Eight months of model output data are analyzed here, which is sufficient to accurately extract the M2 signal with its 466 cycles.
The purpose of this paper is to decompose the surface pressure-gradient force, g∇ζ, into its barotropic and baroclinic components.Furthermore, using the approximation Equation 17, we can even decompose the sea level ζ itself in Equation 19.We find a postiori that the influence of R bc in Equation 18 is rather small on larger scales, even compared to ζbc itself.This comparison is made in terms of a surface potential field Z that satisfies the Poisson equation, with zero Neumann boundary conditions; more is said about R bc near the end of this section.
The sea-level decomposition is in Figure 1.In these plots only a single phase in the M2 cycle is shown, but it is representative of the scales and patterns of the tide throughout its cycle.As expected, the ζ field appears smooth on the basin scale, and it is visually similar to the barotropic pseudo sea level ζbt ; however, their difference, ζbc , is not particularly small (i.e., about 20% in amplitude), and this difference represents the dynamical inconsistency in modeling the barotropic tide without including the buoyancy variations that represent the dynamical influence of modal coupling.The basin-scale pattern of ζbc is quite different from that of ζbt ; thus, there is little evidence of "compensation" between these components (cf., ζ nm ).Furthermore, in ζbc the smaller scale structure is visually evident as "ripples" at approximately the mesoscale baroclinic deformation radius length of O( 100) km (and smaller), especially in the western Pacific, where the baroclinic tidal amplitude is very strong, but also around other islands and ridges in the central and equatorial Pacific.The analogous surface pressure-gradient forces are in Figure 2 with averaging over the M2 tidal cycle.Now the interior patterns are dominated by mesoscale structures that are quite inhomogeneously related to island and topographic generation sites, again as expected from baroclinic tidal generation by energy conversion from the astronomically forced tides at those sites.Many of the edge patterns are associated with shallow shelves and coasts where the barotropic tide is both amplified and dissipated.Most of the interior mesoscale patterns are mostly associated with P bc , that is, the baroclinic tide, but there are locations where P bt is not small, for example, especially near undersea ridges.Its interior magnitude can be nearly half of that of P bc , and it exhibits both mesoscale and broader signals.Some of the former are at topographic generation sites for the internal tide, but they can also extend farther away along what may be baroclinic "ray" paths, indicating some barotropicbaroclinic coupling in the interior.The common practice of interpreting mesoscale tidal signals in ∇ζ as entirely baroclinic, mostly based on the linear eigenmode decomposition (Section 2.1 and Appendix A), is a fairly good approximation, but not a perfect one because P bt is not uniformly smaller than P bc .
Together, Figures 1 and 2 provide the most direct indication of barotropicbaroclinic dynamical coupling in that the scale content of their separate components do not strictly correspond to the eigenmode expectation of barotropic large spatial scales and baroclinic mesoscales, although they do so somewhat.While some of this cross-scale content might be attributed to cross-mode topographic coupling (as in the CSW model; Section 2.1), the spatial patterns do not wholly resemble the topography.
A further decomposition of the barotropic pressure gradient force P bt is in Figure 3.With Equation 20, This shows that the gradient of the depth-averaged pressure is almost everywhere larger than the part due to the non-commutativity between the depth-average and the gradient in forming P bt associated with topographic variations, that is, R bc in Equation 18.The exceptions are near the island and ridge lines where ∇H is large.Alternatively expressed, ζbc is usually larger than Z from Equation 22(not shown).Thus, for many purposes, ζbc can be viewed as the dynamically relevant baroclinic sea level field without attending to the influence of ∇H ≠ 0.
These figures demonstrate that there is important dynamical coupling between the barotropic and baroclinic tides throughout most of the domain, beyond the particular topographic locations where baroclinic generation occurs.We intend to present a fuller and more phenomenological interpretation of the heterogeneous tidal signals, especially for the complex spatial patterns in P bc and g∇ζ (Figure 2) in a later paper.

Summary and Conclusions
A dynamically consistent barotropic-baroclinic decomposition of the pressure-gradient force is based on the definition of the barotropic horizontal velocity u bt as the depth-averaged current and the baroclinic velocity u bc (z) is the residual from the total u.This prescription uniquely relates the sea-level ζ to the vertical structure of the interior currents that are in hydrostatic balance.It implies there is a significant buoyancy influence on the (depthaveraged) pressure-gradient force for the barotropic current P bt , as well as on the baroclinic P bc (z), which is the residual of the total force ∇ϕ(z).The barotropic force P bt is the familiar -g∇ζ force as well as a contribution from the double depth integral of the density field in Equations 12-14-but not simply a steric-height contribution, g∇ζ st , with only a single vertical integral in Equation 9.
At the surface the pressure gradient force cannot be decomposed into sealevel gradients because of variations of oceanic depth (i.e., due to the noncommutativity of a horizontal gradient and a depth average).However, in a high-resolution Pacific simulation, an approximate partition into pseudo sea level components, ζ = ζbt + ζbc , is fairly accurate.
At basin scales the tidal sea level ζ is mostly barotropic, and at mesoscales the surface pressure gradient g∇ζ is mostly due to-P bc .While this approximate scale partition can be anticipated from the linear eigenmodes at tidal frequencies, it is by no means exact due to the dynamical coupling between barotropic and baroclinic tidal components.The widespread, albeit inhomogeneous, spatial extent of large gradients in P bt indicates that the modal dynamical coupling is not limited only to the relatively narrow regions of barotropic-baroclinic energy conversion with strong u bt and ∇H magnitudes and sufficient abyssal stratification.This paper's purpose is to make a precise barotropic-baroclinic decomposition for sea level and the surface pressure gradient force in a way that can be matched to general circulation model solutions, thus going beyond various semi-empirical models (Section 2.1) that have been fitted to altimetric measurements.This is illustrated here with a Pacific simulation in Section 3, but there remain future tasks to assess its accuracy vis a vis measurements and more approximated models.
Historically, the Shallow-Water Equations have been considered as a useful approximate model for the barotropic tides (as they are for tsunamis and storm surges).In this paper we show that this view has serious limitations in its accuracy, both because the Shallow-Water approximation lacks an expression for baroclinic energy conversion and because of the sometimes strong dynamical coupling through the pressure gradient force and buoyancy field.Similarly its bottom-drag dissipation rate in deep water cannot be well represented.Going forward, more care needs to be taken in interpreting a tidal decomposition.While this is difficult in measurements because of the requirement for depth-averaging, it is feasible in 3D models such as the one used here.The best future tidal products will be made by data assimilation within such models, placing the burden of accuracy heavily on the model skill.

Appendix A: Vertical Modes
As an illustration of the implications of the formulas in Section 2.2, consider the simple situation of linear, conservative eigenmodes over a flat bottom.We will follow the notation of Kelly (2016) (i.e., K16) and use rigidlid modes with their usual diagnostic interpretation that the dynamic pressure at z = 0 is equal to gζ; that is, for mode n, the sea level is for n = 0, 1, 2, …. φ n (z) is the separable vertical eigenfunction for pressure and horizontal velocity (cf., K16, Equation 2a; n.b., the notation there is ϕ n instead of φ n ).The K16 convention on the units for the modal amplitudes is UNIT[p n ] = m 2 s 2 and UNIT[φ n ] = 1 (i.e., non-dimensional).Here n = 0 is the barotropic mode, and n ≥ 1 are the baroclinic modes.(Compared to the more general free-surface modes in K16, Section 2, the differences are immaterial here).

ζ
≈ 0. Notice that ζbc in Equation 17 differs from ζ st in Equation 9 by an extra vertical integral associated with the vertical averaging in the barotropic momentum equation.Analogous to Equation 8, we define a companion pseudo barotropic sea level by ζbt = ζ ζbc .(19)

Figure 1 .
Figure 1.(top) Sea level ζ [m], (middle) barotropic pseudo sea level ζbt = ζ ζbc [m], and (bottom) baroclinic pseudo sea level ζbc [m] for a single phase of the M2 tide in the Pacific Ocean.Note the reduced colorbar range for ζbc and the more evident small-scale fluctuations.

Figure 2 .
Figure 2. Cycle-averaged amplitude of (top) surface pressure-gradient g|∇ζ| [m s 2 ], (middle) barotropic surface pressure-gradient magnitude |P bt | [m s 2 ], and (bottom) baroclinic surface pressure-gradient magnitude |P bc | [10 5 m s 2 ] for the M2 tide in the Pacific Ocean.Note the reduced colorbar range for |P bt |, which is the depth-averaged force for the barotropic mode.

Figure 3 .
Figure 3. Decomposition of the cycle-averaged barotropic surface pressuregradient amplitude |P bt | [m s 2 ] in Equation 12 (shown as middle panel of Figure 2) into its two parts associated with the depth-averaged pressure, |∇ϕ bt |, and the non-commutative interaction of the buoyancy field with the topographic gradient |R bc | in Equation 18 (bottom) for the M2 tide in the Pacific Ocean.Note the reduced colorbar range for R bc .
, in the low-frequency context of many upper-ocean currents where ζ nm is relevant for surface-intensified geostrophic bu(z) (Section 1), ζ = ζ nm , and ζ bt = ζ nm ζbc , where ζbc has the same sign as but is smaller in magnitude than ζ nm .Thus, ζ bt is reduced (i.e., partly "compensated") compared to ζ, and the vertical isopycnal displacements in the interior, η ≈ b/N 2 , have the opposite sign as the sea level ζ. u bt and u bc have the same sign in the upper ocean and approximately cancel at depth.(This is not the tidal situation).