Deep-Sea Research Part II xxx (xxxx) xxx–xxx Contents lists available at ScienceDirect Deep-Sea Research Part II journal homepage: www.elsevier.com/locate/dsr2 Large-scale oceanic currents as shallow-water asymptotic solutions of the Navier-Stokes equation in rotating spherical coordinates ⁎ A. Constantina, , R.S. Johnsonb a Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria b School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom ARTICLE INFO ABSTRACT Keywords: We show that a consistent shallow-water approximation of the incompressible Navier-Stokes equation written in Incompressible Navier-Stokes equation a spherical, rotating coordinate system produces, at leading order in a suitable limiting process, a general linear Ekman-type solution theory for wind-induced ocean currents which goes beyond the limitations of the classical Ekman spiral. In Rotating spherical coordinates particular, we obtain Ekman-type solutions which extend over large regions in both latitude and longitude; we present examples for constant and for variable eddy viscosities. We also show how an additional restriction on our solution recovers the classical Ekman solution (which is valid only locally). 1. Introduction this approach is still valid only in the neighbourhood of a point on the surface of the sphere, i.e. it is purely local. Of course, the formulation of The Ekman spiral (Ekman, 1905) is one of the foundations of physical oceanic flows in rotating, spherical coordinates is not new; see, for example, oceanography, being the basis for the discussion of ocean circulation in non- Marshall et al. (1997) and Veronis (1973). However, typical of this earlier equatorial regions; see Samelson (2011). The familiar approach in the work, the discussion of the Ekman spiral is purely local (as in Veronis, presentation of this theory is via the f-plane approximation for a homo- 1973), or the full spherical-coordinate system is developed for the purposes geneous flow, under the assumption of a depth-independent eddy viscosity; of numerical simulation (as in Marshall et al., 1997) – no detailed analysis of see Vallis (2006). This fundamental solution can be extended (following the governing equations is attempted. Sverdrup; see Sverdrup, 1947), by invoking the -plane approximation. The In this paper, we show how a new (but equally valid; cf. Constantin predicted (local) spiralling structure of the wind-induced horizontal com- and Johnson, 2018b, 2018c) asymptotic approximation leads to a more ponent of the ocean's velocity, combining speed decay with anticyclonic general linear theory which is not restricted to small regions on the rotation of direction, both for increasing depth, has been confirmed as surface of the sphere; indeed, the flow region can be as large aswe qualitatively accurate by observations and measurements; see McWilliams wish, enabling gyres, for example, to be represented, suitably driven by and Huckle (2016). However, a number of issues need to be addressed in a wind-stress pattern over large regions of the oceans. Furthermore, this order to accommodate ocean flows on the large scale. While density stra- formulation also enables us to be precise about the assumptions that tification appears to have limited impact on these flows – so weconsider lead to the classical Ekman solution, and why it is necessarily restricted constant-density flow here – the importance of depth-dependent eddy to the neighbourhood of a point. viscosities has been highlighted in Cronin and Kessler (2009) and Cronin We note that, while the small departure of the shape of the Earth and Tozuka (2016). In this paper, we include this property and also address from sphericity is altogether negligible for the study of large-scale two further issues: Is the Ekman spiral inherently a purely local phenom- ocean flows (see Wunsch, 2015), the local curvature effects are relevant enon? What are the effects of using the (correct) spherical geometry? An in some situations (see Paldor, 2015) and, for this reason, it is then attempt to study the effects of spherical geometry has been madein necessary to go beyond the flat-geometry setting of the f-plane ap- Constantin and Johnson (2018b); this approach produces results that go proximation. The standard approach that incorporates a (weak) con- well beyond the classical Ekman spiral, incorporating more detail than is tribution from the curvature is the -plane approximation; see Vallis possible in the classical tangent-plane approximation, and providing a wide (2006). However, in contrast to the f-plane approximation, the -plane range of Ekman-type solutions. These new results are easily seen to be a equations represent a consistent approximation of the governing consequence of using, consistently, a spherical coordinate system. However, equations for geophysical flows only near the Equator; see the ⁎ Corresponding author. E-mail address: [email protected] (A. Constantin). https://doi.org/10.1016/j.dsr2.2018.12.007 0967-0645/ © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/). Please cite this article as: Constantin, A., Deep-Sea Research Part II, https://doi.org/10.1016/j.dsr2.2018.12.007 A. Constantin, R.S. Johnson Deep-Sea Research Part II xxx (xxxx) xxx–xxx discussion in Dellar (2011). And there is an additional complication: the classical Ekman theory breaks down in equatorial regions; we refer to Boyd (2018) and to Constantin and Johnson (2015, 2016a) for dis- cussions of ocean flows near the Equator, where the vanishing ofthe meridional component of the Coriolis force favours zonal flow in the atmosphere and the ocean. Furthermore, nonlinear effects play an im- portant role – which is in marked contrast to the typical mid-latitude ocean flows. All these various considerations provide the motivation for an in-depth study of the linear theory of wind-generated ocean currents at mid-latitudes, and this within the framework of shallow-water flow in a spherical, rotating coordinate system. 2. Governing equations We introduce a set of (right-handed) spherical coordinates, (,,)r : r Fig. 1. The spherical coordinate system for mid-latitude oceanic flows, with [ /2, /2] [,] [0, 2 ) is the distance (radius) from the centre of the sphere, is the 2 2 the angle of latitude, the azimuthal angle (the angle of angle of latitude and [0, 2 ) is the angle of longitude. (We use primes, longitude) and r the distance from the origin. The unit vectors e and e are not throughout the formulation of the problem, to denote physical (dimen- defined at the two poles. sional) variables; these will be removed when we non-dimensionalise.) The unit vectors in this -system, for all points but the two poles, are p= Ps ( , ,t ) on r= R+ h ( , ,t ), (3) (,,)e e er , respectively, and the corresponding velocity components are (,,)u v w , while P is the pressure and is the (constant) density; e and e e points from West to East, from South to North and r upwards (see h u h v h Fig. 1). The -system is associated with a point fixed on the sphere w = + + on r= R+ h ( , ,t ), t r cos r (4) (other than the two poles) which is rotating about its polar axis with an 5 −1 angular speed 7.29× 10 rad s . respectively, where Ps is the pressure at the surface. For the flows of The Navier-Stokes equation and the equation of mass conservation interest here the horizontal velocity field should decay rapidly with are (see Gill, 1982) depth in the near-surface layer of the region R d (,) r R+ h (,,)t r= R d (,) u v , where is the + + + w (,,)u v w t r cos r r impermeable, solid, stationary bed with 1 + (u v tan+ u w , u 2 tan+ v w , u2 v 2 ) u= v= w = 0 on r= R d ( , ); (5) r + 2 (v sin+ w cos ,u sin ,u cos ) this bottom condition is, in this context, more naturally replaced by the + r 2 (0, sin cos , cos2 ) decay of the horizontal velocity field below the surface layer. (Of 1 1 p 1 p p 2 course, close to the solid, bottom boundary, we have a corresponding = , ,+ (0, 0,g )+ 1 (u , v , w ) r cos r r Ekman layer, but that is not relevant in the development that we pre- 1 (u , v , 2 w cos2 2v sin cos )+ 22 (0,w , v ) sent here.) For free surfaces that are fairly close to spherical, the wind- 2 2 2 r cos r stress at the surface is specified by + 2 2 (w 2tan, v u tan,u ) r 2 cos (,,)t = u , 1 1 r d u v 2 w + r 1 ,, on r= R+ h ( , ,t ), dr r r r r r r (,,),t = v 1 d 1 w w 2 1 r + 2 , , 0 , (6) r dr r cos (1) with the surface wind stress (,)1 2 related to the vertical eddy visc- and osity by 1 = |(1 , 2 )| on the surface, where is a (dimensional) U 1 u 1 1 constant in the form of a quadratic function of the wind velocity wind at + (v cos ) + (r2 w )= 0 , 2 10 m above the sea surface: (,)1 2 = cD air UUwind| wind | with air the r cos r cos r r (2) −3 density of air (about 1.2 kg m ) and with cD 0.0013 a (dimension- respectively, where p (,,,)r t is the pressure in the fluid, the less) drag coefficient; see the discussion in Wenegrat et al. (2014). For −1 density, and example, a wind speed of 10 m s , often recorded for the trade winds of the equatorial Pacific, corresponds to a stress of about−2 0.1Nm .