DISCRETE AND CONTINUOUS doi:10.3934/dcds.2019273 DYNAMICAL SYSTEMS Volume 39, Number 11, November 2019 pp. 6261{6276 THE VORTICITY EQUATION ON A ROTATING SPHERE AND THE SHALLOW FLUID APPROXIMATION Vikas S. Krishnamurthy∗ Faculty of Mathematics, University of Vienna Oskar-Morgenstern-Platz 1 1090 Vienna, Austria Abstract. The material conservation of vorticity in fluid flows confined to a thin layer on the surface of a large rotating sphere, is a central result of geophysical fluid dynamics. In this paper we revisit the conservation of vorticity in the context of global scale flows on a rotating sphere. Starting from the vorticity equation instead of the Euler equation, we examine the kinematical and dynamical assumptions that are necessary to arrive at this result. We argue that, in contrast to the planar case, a two-dimensional velocity field does not lead to a single component vorticity equation on the sphere. The shallow fluid approximation is then used to argue that only one component of the vorticity equation is significant for global scale flows. Spherical coordinates are employed throughout, and no planar approximation is used. 1. Introduction. The evolution of the vorticity field in a fluid flow is governed by the dynamical vorticity equation, and understanding the behavior of the vorticity often provides crucial information about the nature of the flow [1, 13]. The material conservation of vorticity [17] plays an important role in our understanding of many geophysical flows of interest [9, 21], see [10, 15,4,5] for some recent studies in this regard. The vorticity equation is obtained by eliminating the pressure (or geopotential) terms from the Euler equations, which are the governing equations of fluid flow. An alternative formulation is the streamfunction-vorticity formulation of the equations of motion [13], which also forms the basis for numerical approaches to geophysical problems [7]. The theoretical importance of vorticity may be contrasted with the fact that the vorticity is usually calculated from velocity measurements, rather than being measured directly [7]. In this paper, we focus on the vorticity equation itself, for a discussion of vortex solutions in geophysical flows, see [16], for a discussion of vortex motion on a sphere see [19, 11]. In the context of geophysical flows, the earth can be considered to be a sphere with a radius equal to the mean radius of the oblate spheroidal earth, R 6368 km [21,7, 20]. A spherical coordinate system is a natural choice to describe≈ flows on the surface of the earth. For an early study utilising spherical coordinates to study waves on a rotating sphere see [12], and for more recent studies with applications to geophysical flow features such as the Equatorial Undercurrent and the Antarctic 2010 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35. Key words and phrases. Vorticity equation, rotating sphere, two-dimensional flow, ideal fluid dynamics, curvature effects. The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin. ∗ Corresponding author. 6261 6262 VIKAS S. KRISHNAMURTHY Circumpolar Current, see [10,6]. We are mostly concerned with application of our discussion of the vorticity equation to large scale, global flows in the ocean, and for such flows the horizontal velocity scales are taken to be U = 0:1 m/s [21,7]. We can calculate the time scale T from the global length scale R and the velocity scale U to be T = R=U = 6:368 107 s, or equivalently T 2:02 years. This is in agreement with the time scale corresponding× to the large scale≈ motions (eg. gyres) in the ocean [20]. The earth rotates with an angular velocity Ω = 7:29 10−5 rad/s which corresponds to the duration of a sidereal day [7]. We evaluate× below a few numbers for later use in this paper. This first set of numbers is valid for large scale flows in the ocean: U 2 UΩ = 2:46 10−16 s−2 and = 1:14 10−12 s−2: (1) R2 × R × If we consider large scale flows in the atmosphere, then the velocity scale changes 5 to Ua = 10 m=s [21], and the time scale changes to Ta = 6:638 10 s 7:4 days. This time scale is in accordance with the time scale for large scale× motions≈ in the atmosphere (eg. cyclones) [7]. The numbers in (1) become U 2 U Ω a = 2:46 10−12 s−2 and a = 1:14 10−10 s−2: (2) R2 × R × The ratios of different length scales in geophysical flows are of central importance in theoretical analyses. Let H = 10 km be the typical vertical (i.e. radial) length scale for both oceanic and atmospheric flows. We can use the same length scale in both cases since, the average depth of the oceans is O(10 km) (and not o(10 km)), and the height of the atmospheric layer relevant to weather phenomena is also O(10 km) [7, 20]. The length scale H is much smaller than the radius of the earth R. We define the shallow fluid approximation as the case when in an expansion of the small parameter H " 1:57 10−3; (3) ≡ R ≈ × we only retain terms of the leading order. The large scale flow of a fluid on the sur- face of a stationary or rotating sphere can effectively be regarded as two-dimensional flow due to the smallness of ". The flow of water and air in the earth's oceans and atmosphere is a prototypical example of such flows. For our purposes, we consider the fluid flow to be incompressible since, the vari- ation of density in the ocean is very small, as is the variation in density in the first 10 km of the atmosphere above the surface of the earth [7, 20]. The effects of viscosity are neglected throughout this paper since the relevant Reynolds number is very large, O(1010), and the Ekman number is very small, O(10−15). This paper is organised as follows. We consider first fluid flow on the surface of a stationary sphere in 2. We discuss the general vorticity equation in an inertial ref- erence frame in 2.1 xand consider the two-dimensional planar vorticity equation in 2.2. In 2.3, wex consider two-dimensional flows on a stationary sphere and obtain thex requiredx form of the velocity field and the corresponding vorticity equation. We introduce scaling arguments to neglect the horizontal (i.e. zonal and meridional) components of the vorticity equation. In 3, we discuss fluid flow on the surface of a rotating sphere. The vorticity equationx for flow in a non-inertial reference frame is discussed in 3.1 and two-dimensional flow on a rotating sphere is considered in 3.2, once againx making use of scaling arguments. The streamfunction-vorticity formulationx of the governing equations is derived in 4. An appendix with vector x THE VORTICITY EQUATION ON A ROTATING SPHERE 6263 calculus identities and their application to vector calculus algebra in spherical co- ordinates has been provided at the end. These identities are used throughout the rest of this paper. 2. Flow on the surface of a stationary sphere. 2.1. Vorticity equation for flow in an inertial reference frame. Consider the three-dimensional flow of an inviscid fluid in an inertial frame of reference. We use vector notation and take the position vector in the inertial frame to be denoted by r, time by t, the density of the fluid by ρ = ρ(r; t), the velocity field by u = u(r; t), the pressure field by p = p(r; t) and the body forces acting on the fluid (per unit mass of the fluid) by F = F (r; t). The slight abuse of notation committed here by providing the same names to variables and their corresponding functions should not cause any confusion. Conservation of momentum for this fluid flow takes the form of the Euler equation @u rp + (u · r)u = + F ; (4) @t − ρ and conservation of mass takes the form of the `continuity equation' @ρ + r · (ρu) = 0: (5) @t Here r is the gradient operator in three dimensions. The reader may consult the standard textbooks of fluid dynamics for derivation and discussion of these formulas [1, 13]. We denote the material derivative operator i.e. the derivative following a fluid element, by D @ = + (u · r): (6) Dt @t The material derivative may act on any scalar or vector quantity of interest. We can rewrite (4) in terms of the material derivative as Du rp = + F ; (7) Dt − ρ with the material derivative acting on the velocity field in this case. We can also rewrite (5) using (6) and (A.1c) as Dρ + ρr · u = 0; (8) Dt with the material derivative acting on the density field in this case. The vorticity field ! = !(r; t) for a fluid flow is defined as ! = r u; (9) × and has the interpretation of being twice the local angular velocity in the fluid [13]. An equation of motion for the vorticity field may be derived by taking the curl of (4). Assuming that the derivatives are continuous allows us to apply the curl and the time derivative interchangeably, and we get @! rp + r (u · r)u = r + r F ; (10) @t × − × ρ × 6264 VIKAS S. KRISHNAMURTHY Using the standard vector calculus identities (A.1) we can rewrite some of the terms in this equation as 1 r (u · r)u = r r(u · u) + ! u (11a) × × 2 × = r (! u) (11b) × × = (u · r)! (! · r)u + !(r · u); (11c) − rp rρ rp r = × : (11d) − × ρ ρ2 Substituting into (10), we obtain the vorticity equation in an inertial frame of ref- erence: @! rρ rp + (u · r)! (! · r)u + !(r · u) = × + r F : (12) @t − ρ2 × The basic texts [1, 13, 21] may be consulted for discussion and interpretation of the various terms in the vorticity equation.