Selected Topics in Fluid Dynamics & Advanced Hydrodynamics Flows in rotating systems (Dated: May 26, 2020) Recommended reading: Lautrup (Chapter 20) & Acheson (Chapter 8.5) The motivation for studying rotating flows is threefold. Firstly, it is important to assess the effect of rotation on flows in an industrial context. Secondly, and more importantly, we often witness flows in rotating frames of reference in our everyday experience. The effects of rotation are particularly pronounced in geophysical flows. As observers of these, we are necessary in a non-inertial frame of reference Even if the corrections to acceleration for such an observer are small compared to gravity, for large scale, slow flows, such as in the ocean or atmosphere, they can shape the flows we see in a considerable way. Thirdly, astrophysical flows are known to be deeply impacted by the effects of (often fast) rotation of their components. The main new players are two ficticious forces (they are ficticious from the point of view of an intertial observer, but embark on a carousel to convince yourself that they can be painfully real!): the centrifugal force and the Coriolis force. Thus, when in a non-inertial, rotating frame, we need to correct the Newton’s second law by incorporating these two forces. Here, we will only consider steady rotation, so Ω = const. For unsteady rotation, revisiting any Classical Mechanics lecture will be helpful to include dΩ=dt terms. VELOCITY AND ACCELERATION IN STEADY ROTATION In Exercises, we will discuss the derivation of the equations of motion in a steadily rotating frame of reference. Considering two frames, one inertial (x0; y0; z0) and one rotating (x; y; z). By examining the acceleration in the rotating frame, we find that a = a0 − 2Ω × u − Ω × (Ω × x) (1) Question: Estimate the magnitude of these terms (as compared to gravity as the natural scale for acceleration) for a carousel with R = 5 m and rotating at one turn per 6 seconds. Are they significant? We will now asses their significance for Earth. At typical values the centrifugal force can be estimated as Ω2R, with R being now the Earth’s radius, which compares to gravity as Ω2R=g ≈ 1=291, so it is negligibly small. It leads to a slight flattening of the rotating planet 2 perpendicularly to its rotation axis, but in the geophysical context it does not play a major role and, as we will show, it can be incorporated into effective pressure. The Coriolis force is different. Let us consider a point on the surface of the Earth at a latitude θ and introduce a local flat-Earth coordinate system with the x-axis pointing towards the East, y pointing North, and z being vertical. The rotational velocity of the Earth then is Ω = Ω(0; sin θ; cos θ). The Coriolis acceleration gC in terms of the local velocity u = dx=dt reads 0 1 uy cos θ − uz sin θ B C C B C g = 2Ω B −ux cos θ C : (2) @ A ux sin θ At the poles the Coriolis force is always horizontal, and at the equator it is always vertical for purely horizontal motion. The vertical component of the Coriolis force is very small compared to gravity. For example, for a jet aircraft flying westwards at middle latitudes at a speed close to the velocity of sound, it amounts to about 0.3% of gravity, similarly to the centrifugal force. Thus it can be neglected for most Earthly flows of interest. For typically small vertical flow velocities, we will also neglest the Coriolis force due to uz. So we will adopt an assumption that 0 1 uy B C C B C g = 2Ω? B−uxC ; (3) @ A 0 with Ω? = Ω cos θ. Concluding, in a flat-Earth coordinate system the Coriolis force indeed looks as if the Earth was rotating about the local vertical with the angular velocity Ω?. At middle ◦ − −1 latitudes θ ≈ 45 , we can take Ω? ≈ 10 4 s , rendering the typical period of rotation of a Foucault’s pedulum [1] being 2π=Ω? ≈ 34 h. FLOW IN ROTATING FRAMES Applying the modified acceleration to the Navier-Stokes equation (exercise), we can show that the resulting governing equations for the flow become @u 1 + u · ru + 2Ω × u − Ω × (Ω × x) = − rp + νr2u; (4) @t ρ r · u = 0: (5) Moreover, upon defining the effective pressure P (another exercise), the flow equations may be recast as @u 1 + u · ru = −2Ω × u − rP + νr2u; (6) @t ρ r · u = 0; (7) 3 where for the calculation of the Coriolis term we can only use the local, vertical angular velocity Ω?. Dimensionless numbers: Reynolds, Rossby, and Ekman It is again interesting to study the limits of this equation. Taking the flow velocity scale U and the length scale L, we can define, as usually, the Reynolds number UL Re = : (8) ν In the limit of large Reynolds number, Re 1, the advective term dominates over the viscous term, and we can generally neglect the effects of viscosity and the last term in Eq. (6). Remember though, that this approximation will generally only work far away from boundaries, where boundary layers will be formed! From the remaining terms, it is tempting to compare the relative importance of the advective term and the Coriolis acceleration, which leads to (exercise) the Rossby number ju · ruj U Ro = ≈ ; (9) 2jΩ × uj 2ΩL named after a Swedish-born US meteorologist Carl-Gustav Arild Rossby (1898-1957). In this idealised case, the Coriolis force will only be significant for low Ro, so for U . 2ΩL, when the translational flow velocity is comparable or smaller to the local rotation-induced velocity. Example Consider ocean currents and weather cyclones, which are relatively steady rotating flows. With U ∼ 1 m/s (ocean) or U ∼ 10 m/s (wind), and the typical local angular velocity of 2Ω ≈ 10−4 Hz, we get Rossby numbers of Ro ≈ 0:01 (ocean) or Ro ≈ 0:1 (cyclones). We conclude that both are thus dominated by the Coriolis force, which plays the key role in their evolution. For a typical swimming human, with L ∼ 1 m and U ∼ 1 m/s, we find Ro ≈ 104, which justifies the complete neglect of the Coriolis force in this case. Interestingly, in systems where gravity is simulated by rotation (such as a space station), the Coriolis force will cause surprises when playing ball games (see Lautrup). For lower Reynolds numbers, it becomes important to compare the relative strength of viscous and Coriolis effects. The relevant dimensionless combination (exercise) is the Ekman number jνr2uj ν Ro Ek = = = ; (10) 2jΩ × uj 2ΩL2 Re named after a Swedish oceanographer Vagn Walfrid Ekman (1874– 1954). When the Ekman number is small, the viscous force is small compared to the Coriolis force. This is automatically the case for moderate Ro and large Re. In the sea and the atmosphere, the huge Reynolds number 4 makes the Ekman number negligible but, warned by our experience with boundary layers, we may expect the Ekman number to be larger close to boundaries. Indeed, it normally is of order unity close to bounding surfaces where viscosity dominates over advection, and competes with the Coriolis force. This gives rise to a particular boundary layer, called the Ekman layer, which we will discuss further. GEOSTROPHIC FLOW In most natural flow context, we may neglect both the Rossby ansd Ekman numbers, since Ro 1 and Ek 1. This means we are disregarding both viscous and advective terms in the Navier-Stokes equations. The flow is completely dominated by the Coriolis force and called geostrophic. For a steady flow, we end up with a particularly simple equation 1 − 2Ω × u − rP = 0; (11) ρ which has a nice hydrostatic analogy – the effective pressure gradient must always balance the Coriolis force. However, the Coriolis force has a rather special form, and this has far-reaching consequences. The most spectacular one is that geostrophic flow is essentially two-dimensional. To examine this, notice that u · rP = 0; (12) which implies that the effective pressure is constant on streamlines. Moreover, if the motion is horizontal, the gravitational force plays no role in the pressure, and effective pressure is the same as the hydrostatic pressure. This means that streamlines and isobars coincide in geostrophic flow. Thus, in a typical weather map, the direction of the wind can be readily read off from the isobars. Now, to account for the sign of the wind direction, we may use our knowledge of the Coriolis force direction, which makes winds in the Northern hemisphere turn anti-clockwise around the low pressure areas (cyclones) and clockwise around the high pressure areas (anti-cyclones). The wind velocity may be determined from the geostrophic equation (11) using Ω · u = 0 to find Ω × rP u = : (13) 2Ω2ρ In the absence of the Coriolis force, the air flow would happen along the pressure gradients only, from the high to low pressure regions. The 2D character of geostrophic flow may be seen as follows. Taking a scalar product of Eq. 11 with Ω, we find @P (Ω · r)P = Ω = 0; (14) @z 5 so the effective pressure is constant along the axis of rotation. In a constant gravitational field, using the definition of P and again disregarding the centrifugal contribution, we find the hydrostatic pressure as p = P (x; y) − ρgz: (15) The fact that the effective pressure is invariant under translations along the rotation axis extends to the whole flow. Taking the curl of Eq. (11) and using incompressibility, we find @u 0 = r × (Ω × u) = Ω(r · u) − (Ω · r)u = −Ω : (16) @z Thus the flow field u is a function of x and y only and is two-dimensional! This result is called the Taylor-Proudman theorem.