Horizontal gravity in ocean Ekman transport

Three-dimensional gravity vector field g (= i g λ + j g φ + k g z ) in geodesy has been greatly simplified to a uniform vertical vector (-g 0 k ) in oceanography with ( λ , φ , z ) the (longitude, latitude, height), ( i , j , k ) the corresponding unit vectors, and g 0 = 9.81 m/s 2 . Recent studies by the author show such simplification incorrect. The horizontal gravity is important in ocean dynamics. Along the same path, the horizontal gravity is included into the classical Ekman layer dynamics with constant eddy viscosity and depth-dependent-only density ρ ( z ) represented by an e-folding near-inertial buoyancy frequency. A new Ekman spiral and in turn a new formula for the Ekman transport are obtained. With the horizontal gravity data from the global static gravity model EIGEN-6C4 and the surface wind stress data from the Comprehensive Ocean-Atmosphere Data Set (COADS), the Ekman transport due to the horizontal gravity is crucial and cannot be neglected.


Introduction
The seminal paper by Ekman [1] laid the foundation for the modern oceanography through modeling the turbulent mixing in upper ocean as a diffusion process similar to molecular diffusion, with an eddy viscosity K, which was taken as a constant with many orders of magnitude larger than the molecular viscosity. The turbulent mixing generates ageostrophic component of the upper ocean currents (called the Ekman spiral), decaying by an e-folding over a depth as the current vector rotate to the right (left) in the northern (southern) hemisphere through one radian. Along with the Ekman spiral, the Ekman transport was identified.
The Ekman theory was established using the uniform gravity (-g0k, g0 = 9.81 m s -2 ) with k the unit vector in vertical (positive upward). However, the real gravity is well-known in geodesy as a three-dimensional vector field [g = gh + gzk] with gh the horizontal gravity vector. Recent research [2][3][4][5][6] show that the horizontal gravity gh is comparable to the horizontal pressure gradient force, Coriolis force, and surface wind stress curl. Thus, the feasibility of using the uniform gravity (-g0k) in the ocean Ekman layer dynamics needs to be investigated. Oceanographers are referred to the appendices for information about the horizontal gravity.

Dynamic equation with real gravity
Steady-state large-scale ocean dynamic equation with the Boussinesq approximation (replacing density ρ by a constant ρ0 except ρ being multiplied by the gravity) and the real gravity is given by [3][4][5][6]   where R = 6.3781364×10 6 m, is the Earth radius; N is the geoid height relative to the normal where UE = (uE, vE) is the Ekman velocity. After substitution of (2)-(4) into (1), the horizontal component of (1) is represented by Baroclinicity (i.e., non-zero horizontal density gradient) and spatially varying eddy viscosity K affect the Ekman layer dynamics [7]. To limit the study on the effect of the horizontal gravity, the eddy viscosity K is assumed constant and the density varies in vertical only, i.e., there is no vertical shear of the geostrophic current /0 Furthermore, a special density ρ(z) is selected for this study as the e-folding near-inertial buoyancy frequency [8], 2 3 -1 0 00 0 ( ) exp , ( ) , 5.24 10 s , The second formula in (7a) becomes Substitution of (6) and (7b) into (5) leads to the Ekman dynamic equation with including the horizontal gravity The horizontal gravity, Ekman velocity, vertical shear of the Ekman velocity, and the Ekman volume transport are defined in complex variables where (gλ, gφ) are the longitudinal and latitudinal components of the horizontal gravity. The Ekman equation (8) is converted into the complex form Differentiation of (10) with respect to z and use of (7b) and (9) lead to the equation for the vertical shear of the Ekman velocity

Boundary conditions
The turbulent momentum flux should be continuous at the ocean surface (z = 0), where (τλ, τφ) are the surface wind stress components. The Ekman velocity U and vertical shear W need to be satisfied by the lower boundary condition,

Ekman solutions in complex form
Eq.(11) with the boundary conditions (12) and (13a) has an exact solution Here, DE is the Ekman layer depth; and  is proportion to the ratio between the Ekman layer depth (DE) and the e-folding depth (d) of the buoyancy frequency  . Integration of (14) with respect to z from z = - to z and use of the lower boundary condition (13b) lead to the Ekman spiral, 7 Integration of (16) with respect to z from - to 0 leads to the complex form of the Ekman Simplification due to small parameter χ The eddy viscosity K inferred from Ekman theory and the time-averaged stress was directly estimated from 0.054 m 2 s -1 [9]

Data sources
Two datasets were used to identify importance of the horizontal gravity on the Ekman layer

Global Ekman transport
The calculated global Ekman transport has different patterns due to wind stress MW (Fig.   1a) and due to horizontal gravity MG (Fig. 2a). The intensities of the Ekman transport components |MW| (Fig.1b) and |MG| (Fig. 2b) have different horizontal distributions and strengths. The histogram of |MW| (Fig. 1c) shows near Gamma distribution with the shape parameter of 1 and scale parameter of 2, and with the mean and standard deviation (946.8, 11 993.9) kg m -1 s -1 . However, the histogram of |MG| (Fig. 2c) shows near Weibul distribution with the shape parameter of 1.5 and scale parameter of 10, and with the mean and standard deviation (4,098, 5,836) kg m -1 s -1 . With the density ρ(z) given by (6) (i.e., Garrett-type efolding near-inertial buoyancy frequency), the global mean Ekman transport is around four times larger due to the horizontal gravity |MG| than due to the surface wind stress |MW|.

Global non-dimensional E number
Larger Ekman transport due to the horizontal gravity |MG| than due to the surface wind stress |MW| is also shown in the world ocean distribution of E values (Figure 3a)  and horizontal gravity (MG) are identified using the two independent datasets: COADS for wind stress (τ), and EIGEN-6C4 geoid height (N) for the horizontal gravity. Note that the larger Ekman transport due to the horizontal gravity than due to the surface wind stress is only for the specially selected density filed represented by the e-folding near-inertial buoyancy frequency, not for the density in the real ocean. However, it shows that the horizontal gravity is an important forcing term in the ocean Ekman layer dynamics.