Gravity is vertical in Geophysical Fluid Dynamics

Earth is a rotating, gravitating planet, with space-time variations in these properties that are small compared to their average values. In analyzing Earth’s atmospheric and oceanic dynamics, it is often assumed that the gravitational acceleration g and the rotation rate are constants and that the shape of the planet is a sphere, or at least a spheroid of rotation about the polar axis that bulges outward at the Equator. None of these simplifications is fully accurate. Nevertheless, a common theoretical practice is to act as if they are true, with the proviso that the fluid dynamical equations are fundamentally expressed in geopotential coordinates, with the vertical direction ẑ defined as parallel to the spatial gradient of the combined gravitationalrotational potential, −∇ , and with orthogonal horizontal coordinates, e.g., in the directions of longitude and latitude1. The advantages of this practice are to hide the complexity of the true gravity and rotation in a simple coordinate frame and, in particular, to be able to express the widespread approximate hydrostatic force balance between pressure gradient and density in only the vertical direction. Chu2 asserts that the scientific community has hitherto neglected a potentially important, “horizontal” component of Earth’s gravitational acceleration. This is a misrepresentation of the community-standard formulation of the equations of Geophysical Fluid Dynamics (GFD), discussed in the preceding paragraph. The author claims instead that the widespread practice is to describe spatial variations on Earth in an absolute spherical coordinate system. In such a coordinate system the elevations of geopotential surfaces vary as functions of latitude and longitude, and thus there is a horizontal component of the gravitational acceleration. The author asserts that this component of gravity has simply been neglected in previous studies of the ocean and atmosphere. In fact, the implicit use of geopotential coordinates in the standard formulation of the equations of GFD ensures that there is no horizontal component of gravity. Below we provide a brief overview of the history of geopotential coordinates in GFD, to demonstrate the extent to which Chu2 misrepresents this standard practice. We then explicitly consider the formulation of the equations describing ocean circulation in an absolute spherical coordinate system, as suggested by Chu2. We show that even in such a coordinate system Chu2 vastly overestimates the importance of the “horizontal gravity” terms. Finally, we show that subsequently published findings by Chu3,4, built upon the “horizontal gravity” formulation of Chu2, erroneously attribute modifications of the flow in oceanic and atmospheric Ekman layers to the effects of “horizontal gravity”.


"Horizontal gravity" does not modify oceanic Ekman transport
The analysis of Chu 1 Here 0 is the reference density, is the spatially-varying density, is the Coriolis parameter, is the horizontal velocity vector, is the pressure, ∇ 3 denotes the three-dimensional gradient operator, and denotes additional sources and sinks of momentum. Note that the gravity vector has both horizontal and vertical components, = ℎ +ˆ. Following Chu 1 , we write these as where is the elevation of the marine geoid, 0 is the reference gravitational acceleration, and ∇ ℎ denotes the horizontal gradient operator. The central issue with the analysis of Chu 1 lies in the identification of a "geostrophic" component that balances the horizontal pressure gradient via the Coriolis force: The remaining component of the flow balances the horizontal component of the gravity vector and the external forcing via the Coriolis force, and is referred to as the "ageostrophic" or "Ekman" flow: Here the superscript C21 denotes definitions made by Chu 1 . We argue that this is an inappropriate decomposition, and that in fact the horizontal component of gravity should be included in the "geostrophic" flow, rather than the "ageostrophic"/"Ekman" flow. The basis for this argument is that: (i) the horizontal component of the gravity vector is almost entirely compensated by the horizontal component of the hydrostatic pressure, so this separation produces components of the flow that largely cancel one another; (ii) the residual between the horizontal pressure gradient force and horizontal gravity produces horizontal flows throughout the water column, rather than just in the surface boundary layer; (iii) the formulation of Chu 1 implies that even in the absence of wind forcing = 0, there is still an ageostrophic Ekman transport driven by the "horizontal gravity", which contradicts established theory and measurements of the wind-driven ocean surface boundary layer 2 . Thus, both the horizontal pressure gradient and the "horizontal gravity" jointly determine the geostrophic flow external to the Ekman layer.
In support of this argument, first write the density as a perturbation relative to the constant reference density, Next, note that the gravity term in (1) corresponding to the reference density 0 can be written as an exact gradient, where the reference gravitational potential is given by We can therefore define a reference hydrostatic pressure 0 that exactly balances the component of gravity associated with the reference density, i.e.
= 0 +˜, This form of the horizontal momentum equation largely removes the issue of compensation between the horizontal pressure gradient force and the horizontal gravity force, which would otherwise greatly exaggerate the dynamical influence of the "horizontal gravity term". We now consider the horizontal flows produced by the action of each of the terms on the right-hand side of (9). Following Chu 1 , we consider an external forcing resulting from vertical eddy momentum fluxes, where is the eddy viscosity. The surface boundary condition is where is the surface wind stress, and we assume that the eddy viscosity becomes negligibly small outside the surface boundary layer, where bl denotes the thickness of the surface boundary layer. By integrating (9) from the surface to an arbitrary elevation < − bl , we can then write an expression for the transport above that elevation: Here = −˜/ 0 is the buoyancy. The last term in (13) is the familiar expression for the Ekman transport 3 ; this component of the transport is confined entirely to the surface boundary layer. In contrast, both the pressure gradient and "horizontal gravity" components of the transport in (13) vary, in general, throughout the water column. Thus the pressure gradient and "horizontal gravity" set the far-field flow, below the Ekman layer, and the external forcing term in (9) drives the ageostrophic, Ekman layer flow. Furthermore, arbitrary changes in the "horizontal gravity" term may be imposed simply by changing the reference density 0 . To be precise, consider a change in the reference density from 0 to ★ 0 = 0 + 0 , where 0 ≪ 0 and ( − 0 ) ≪ 0 . The buoyancy becomes where we have approximated ★ 0 ≈ 0 in the denominator. Thus (13) becomes, after some manipulations, Here˜★ is the perturbation pressure, defined relative to a reference pressure ★ 0 that hydrostatically balances the reference gravitational potential Φ ★ 0 = ★ 0 0 ( − ) (c.f. Eq. (8)). Eq. (15) implies that an arbitrary change in the reference density leads to a vertically-uniform addition to the "horizontal gravity"-driven component of the flow, and thus a vertically-integrated transport that increases linearly with depth. This implies that the "horizontal gravity"-driven component of the flow is ill-defined, and thus that analyzing this flow in isolation, or as part of the 'Ekman' transport (as done by Chu 1 ) is misleading.

2/4
To reconcile the apparent arbitrariness in the definition of the "horizontal gravity"-driven component of the flow in Eq. (15), we note that the perturbation pressure gradient may be written, using (8), as Substituting Eq. (16) into Eq. (15) we obtain, after some manipulations, Thus the shift in the reference density produces equal and opposite perturbations to the pressure gradient-driven and "horizontal gravity"-driven components of the flow. Simplifying Eq. (17) by cancelling these terms recovers Eq. (13). This highlights the close dynamical link between the pressure gradient and "horizontal gravity" terms, and indicates that they should be considered to set the geostrophic component of the flow in tandem. Based on these considerations, we argue that in an absolute spherical coordinate system, the appropriate decomposition of the flow would be to relate the geostrophic flow to the residual between the horizontal pressure gradient and "horizontal gravity", It follows that the "ageostrophic" or "Ekman" flow is Thus the "Ekman" flow is unchanged by the shift to absolute spherical coordinates.
Here 0 is the reference density, is the spatially-varying density, is the Coriolis parameter, is the horizontal velocity vector, is the pressure, ∇ 3 denotes the three-dimensional gradient operator, and denotes additional sources and sinks of momentum. Note that the gravity vector has both horizontal and vertical components, = ℎ +ˆ. The analysis of Chu 4 proceeds by identifying a "geostrophic" component of the flow that balances the horizontal pressure gradient via the Coriolis force: The remaining component of the flow balances the horizontal component of the gravity vector and the external forcing via the Coriolis force, and is referred to as the "ageostrophic" or "Ekman" flow: Here the superscript C21 denotes definitions made by 1 . The author then solves (22) for the vertical structure of the "ageostrophic" flow, − C21 , throughout the atmospheric boundary layer, under the canonical assumption that the external forcing takes the form of a vertical Laplacian operator with a constant eddy viscosity 3 . This decomposition, from which all of the analysis of Chu 4 is derived, is critically flawed. The "horizontal gravity" force, which appears in (20) only because an absolute spherical coordinate system is used, should contribute to the definition of the geostrophic flow, rather than the ageostrophic/Ekman flow. Far above the atmospheric Ekman layer, i.e. for ≫ , the eddy vertical viscosity may be neglected and thus = 0. Thus the far-field horizontal velocity is given by

3/4
The canoncial formulation of the atmospheric Ekman layer 3 describes the behavior of a viscous boundary layer in response to an externally-imposed, geostrophic flow. In the absolute spherical coordinate system used by Chu 4 the externally-imposed flow is given by (23). Therefore, in this coordinate system the appropriate decomposition would be to define the geostrophic flow as and the "ageostrophic" or "Ekman" flow as Thus the "Ekman" flow and pumping are unchanged by the shift to absolute spherical coordinates.