7 Balanced Motion. 7.1 Return of the...scale analysis for hydrostatic balance! CSU ATS601 Fall 2015
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1 7 Balanced Motion We previously discussed the concept of balance earlier, in the context of hydrostatic balance. Recall that the balanced condition means no accelerations (balance of forces). That is, Dv/Dt = 0. Hydrostatic balance is a balance in the vertical direction, that is, no accelerations in the vertical momentum equation. What about the horizontal? In this section we will discuss the fundamental balances that emerge from the horizontal momentum equations with rotation. 7.1 Return of the...scale analysis for hydrostatic balance! In section 2.7, we saw that for a static background state hydrostatic balance represents a very good approximation in the atmosphere and ocean! However, you may wonder, how well does it hold for largescale atmospheric and oceanic flows? We begin by assuming a background state that is in hydrostatic balance that is a function of z only. Beginning exactly as we did for our derivation of the Anelastic system, we will decompose pressure and density into a background and perturbation component: p(x, t) =p 0 (z)+p 0 (x, t), (x, t) = 0 (z)+ 0 (x, t), where If we assume that 0 << 0 we can write the vertical momentum equation 0 =-g 0 (7.1) Dw Dt -g 0 0 zp 0 0 (7.2) following exactly the steps in the Anelastic derivation that lead to (5.32). We now see that the scaling presented in section 2.7 was too naive: whether hydrostatic balance represents a good approximation from a dynamical perspective (i.e. things that vary in time) depends on the magnitude of the term on the far right (the vertical derivative of pressure perturbations). z p 0 is typically much smaller z p, we need to perform new scaling to determine its size. Introducing the typical scales as before, we have for the horizontal momentum equations: D(u, v) Dt U T For the vertical momentum equation we have:, Coriolis terms fu, pressure gradient term p0 0 L Dw Dt W T, pressure gradient term p0 0 H In order to determine whether Dw/Dt can be neglected, we need to determine an approximate scale for p 0. We can get this from our scaling of the horizontal momentum equations. To do this, we need to distinguish between two cases: (7.3) (7.4) E. A. Barnes 55 updated 13:57 on Tuesday 13 th October, 2015
2 Case 1: f 6 1/T (neglect Coriolis terms) Case 2: f > 1/T (neglect advection terms) Case 1: neglect Coriolis terms From the horizontal momentum equations we obtain: U T p0 0 L ) p 0 0 LU T (7.5) Thus, for our scaling of Dw/Dt: p 0 0 H U T L H ) Dw zp 0 0 if W T U T L H that is if H 2 L 2 1 (7.6) where we have assumed that U/W L/H. Case 2: neglect advection terms From the horizontal momentum equations we obtain: fu p0 0 L ) p 0 fu 0 L (7.7) Thus, for our scaling of Dw/Dt: p 0 0 H ful H ) Dw zp 0 0 if W T ful H that is if 1 H 2 ft L 2 1 (7.8) where we have again assumed that U/W L/H. Note that in this case we are already assuming that f > 1/T and therefore ft > 1. In both Case 1 and Case 2, we see that hydrostatic balance (i.e. neglecting Dw/Dt in the vertical momentum equation) represents a good approximation for flows with small aspect ratios. Hence, the hydrostatic approximation is a small aspect ratio approximation. Note how in Case 2 the presence of rotation strengthens the hydrostatic approximation. That is, larger f makes the hydrostatic approximation more likely to hold. 7.2 Geostrophic balance Recall that the fundamental balance condition can be written as Dv/Dt = 0. We have already discussed the balance that results in the vertical momentum equations (i.e. hydrostatic balance), so now we will focus on the horizontal momentum equations. E. A. Barnes 56 updated 13:57 on Tuesday 13 th October, 2015
3 Setting D(u, v)/dt = 0 in the horizontal momentum equations gives: -fv and (7.9) This is a statement of an exact balance between the Coriolis acceleration and the pressure gradient acceleration. We define the resulting horizontal flow as geostrophically balanced flow or geostrophic flow. The velocity components associated with this flow are given special names u =(u g, v g ) (7.10) defined by f u - 1 r Hp, or u g - where f (0, 0, 2 sin ) and r H is the horizontal gradient operator. Non-divergence of geostrophic flow and v g For a fluid of horizontally uniform density, the geostrophic flow on an f-plane is non-divergent, that is: = 0 for = 0(z) and f = f 0 = const (7.12) Using mass continuity (and still assuming balanced flow), we can also obtain an equation for w in this case. That 0 w) = 0 (7.13) A situation with the boundary condition of no mass flux across the surface or top of the domain then results in w = 0 everywhere and the flow is purely horizontal. Note that under the above conditions, if the flow is non-divergent, we can define a streamfunction p ) u g f 0 and v (7.14) Flow is a perfect balance between the Coriolis and pressure accelerations - and is parallel to lines of constant pressure! E. A. Barnes 57 updated 13:57 on Tuesday 13 th October, 2015
4 Fig. 2.5 Schematic of geostrophic flow with a positive value of the Coriolis parameter f. Flow is parallel to the lines of constant pressure (isobars). Cyclonic flow is anticlockwise around a low pressure region and anticyclonic flow is clockwise around a high. If f were negative, as in the Southern hemisphere, (anti-)cyclonic flow would be (anti-)clockwise. Figure: Taken from Figure 2.6 of Vallis Geostrophy in pressure coordinates Since geostrophic balance is such a dominant player in large-scale dynamics, it is important to know the different ways it may show up in the equations of motion. In our description above, we used a vertical z-coordinate (height coordinate). Alternatively, in pressure coordinates geostrophic balance is written as f u g -r H,p or u g =- g and v g = g (7.15) where = gz (the geopotential). Note that density does not appear explicitly on the right-hand-side of the horizontal momentum equations, and thus, is in a simpler form than in height coordinates. On an f-plane, the geostrophic flow in pressure coordinates is non-divergent, that is r H,p u g = 0 (7.16) Recall that in height coordinates this was only true for uniform density in the horizontal - but we don t have to make that assumption in pressure coordinates! In pressure coordinates, the continuity equation: r v = r H,p = 0 (7.17) then = 0. (7.18) E. A. Barnes 58 updated 13:57 on Tuesday 13 th October, 2015
5 7.3 Rossby number History of Carl-Gustav Rossby Our discussion of geostrophic flow may lead you to wonder, under what conditions is D(u, v)/dt 0a good approximation? That is, under what conditions is the magnitude of the material acceleration small compared to the Coriolis acceleration in the horizontal momentum equations? Once again, to find out we will perform a scale analysis! Consider the u-equation (we will assume the same scaling for +(u r H)u v (7.19) The right-hand-side scales as: U T U 2 L UW H fu (7.20) If we assume (as we typically do for large-scale motions) that time scales advectively (T L/U) and W/U H/L, then the material derivative operator (the first three terms) scale as U 2 /L. This gives us U 2 L fu (7.21) The Rossby number (Ro), named after Carl-Gustav Rossby (a key figure in the development of modern geophysical fluid dynamics and whom the Rossby wave is named after), is then defined as the ratio of scales of the advective term (U 2 /L) and Coriolis terms (fu): Ro U2 /L fu = U fl 1 ft (7.22) using the fact that T L/U from having an advective time-scale. Small Rossby number (Ro << 1): the Coriolis acceleration dominates the material acceleration yielding geostrophic balance (i.e. u u g and v v g ). Large Rossby number (Ro >> 1): the Coriolis acceleration may be neglected (e.g. flows of small time scales or large spatial scales compared to their velocity). Intermediate Rossby number (Ro 1): all terms play an important role (gradient wind balance results) E. A. Barnes 59 updated 13:57 on Tuesday 13 th October, 2015
6 7.4 Taylor-Proudman effect, Taylor columns A hydrostatic fluid of uniform (constant) density flow turns out to be strongly constrained, as you will see below. Mixing image of tank experiments We begin by considering the vertical dependence of geostrophic flow for a hydrostatic fluid of constant density. Recall that geostrophic balance can be written as: f u - 1 r Hp, or u g - and v g (7.23) Define p/ 0 and we have f u g =-r H, or u g - v g (7.24) In terms of, hydrostatic balance turns =-g (7.25) Taking the vertical derivative of the equations for u g and v g and using hydrostatic balance leads g @y = = 0, g = 0 (7.26) Since we know that the geostrophic horizontal velocities are non-divergent we can once again use continuity to show that for an z w = 0. Thus, we g = g = 0, = 0 (7.27) While at first perhaps these equations look rather dull, they are actually rather surprising. They tell us that the flow has no vertical dependence! That, geostrophic balance that is due to rotation makes the fluid stiff in the vertical. This effect is known as the Taylor-Proudman effect named after Geoffrey Ingram (G.I.) Taylor and Joseph Proudman. According to the Taylor-Proudman effect, geostrophic flow of a barotropic hydrostatic fluid is twodimensional and does not vary in the vertical (more precisely, in the direction of the rotation vector). That is, fluid columns act is if they were rigid columns and move along contours of constant fluid depth (i.e. Taylor columns). Thus, the fluid will move around, but not over, a submerged obstacle. E. A. Barnes 60 updated 13:57 on Tuesday 13 th October, 2015
7 7.5 Thermal wind Thermal wind balance arises due to a combination of geostrophic and hydrostatic approximations. It is most easily derived in pressure coordinates, and so we will start there. Note that in this derivation, we allow the density to vary. Figure: (left) Taken from Figure 2.6 of Vallis. (right) Schematic of thermal wind, where the pink arrows show the zonal wind increasing with height and the colored shading denotes the surface temperature (blue is cold and orange is warm) Thermal wind in pressure coordinates In pressure coordinates, geostrophic balance is written as f u g =-r H,p (7.28) where is the geopotential ( = gz) and r p is the gradient operator along surfaces of constant pressure. If we invoke an f-plane approximation, than the geostrophic wind is non-divergent on pressure surfaces (as we showed in an earlier section), that is r H,p u g = 0 (7.29) Taking the vertical derivative (i.e. derivative w.r.t. p) of both sides of geostrophic balance in pressure coordinates leads to @p r H,p = r =- (7.30) where the final step comes from the hydrostatic balance equation in pressure coordinates with = 1/. E. A. Barnes 61 updated 13:57 on Tuesday 13 th October, 2015
8 Using the ideal gas law and the fact that the gradient is taking along surfaces of constant pressure, we obtain the thermal wind relationship in pressure-coordinates: In component form this is: = R p r H,pT (7.31) = and - = (7.32) Thermal wind balance: a horizontal temperature gradient is accompanied by a vertical shear of the horizontal wind in order for geostrophic balance to be fullfilled. The term thermal wind is actually a bit misleading, as it is a relationship between the temperature gradient and the wind shear, not the wind itself. Also, note that while it is often framed in terms of a horizontal gradient in temperature, it can also be thought of in terms of a horizontal gradient in density Thermal wind in height coordinates In height coordinates, thermal wind balance becomes g =-r Hb, or g and - g (7.33) where b is the buoyancy, b -g@ / 0 as in the anelastic equations Thermal wind in log-pressure height coordinates One defines log-pressure height coordinates using Z H ln(p 0 /p) where H = RT ref /g (the scale height). In this case, thermal wind becomes g Z =- R H r H,ZT (7.34) In component form, this can be written as =- g Z =- Z and = T Z = Z (7.35) E. A. Barnes 62 updated 13:57 on Tuesday 13 th October, 2015
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