12.1 Governing Equations and Approximations 647 Evidently, the Boussinesq approximation of (12.13) is ∂ω  ∂t + ∇×(ω  × u  )=2Ω ×u  + ∇σ ×g + ν∇ 2 ω  . (12.22) Further approximations require scale analysis to identify the leading-order mechanisms. To this end we nondimensionalize (12.22). Let L and U be the characteristic length and velocity scales of the relative fluid motion under consideration so that the time scale is L/U, we obtain (dimensionless relative quantities are denoted by the asterisk) Ro  ∂ω ∗ ∂t ∗ + ∇ ∗ × (ω ∗ × u ∗ )  =2k ·∇ ∗ u ∗ − 1 Fr ∇ ∗ σ ×e z + Ek∇ ∗2 ω ∗ , (12.23a) where, with the Reynolds number defined by Re = UL/ν as usual, Ro = U ΩL ,Ek= ν ΩL 2 = Ro Re ,Fr= ΩU g (12.23b) are the Rossby number, Ekman number,andFroude number, respectively. The Rossby number measures the relative importance of the relative and planetary vorticities. Then, since Ek ∼ Ro/Re, except within the terrestrial boundary layer where the viscous effect is significant, large-scale flows with Ro = O(1) or smaller all have Ek  1 (for ocean there is Ek =10 −14 ). Thus we ignore the viscosity in the rest of this chapter. The orders of magnitude of the Rossby numbers for several typical flows on the earth, along with their characteristic length and velocity scales, are listed in Table 12.1. The most intense atmospheric vortices are hurricanes (typhoons in the North-Western Pacific) and tornados at small and large Ro, respectively, which are associated with extreme and hazardous weather events. Table 12.1. The orders of magnitude of the Rossby numbers for typical geophysical fluid flows flow phenomena length scales velocity scales orders of Ro bath-stub vortex 1 cm 0.1 m s −1 10 5 dust devil 3 m 10 m s −1 3 ×10 4 tornado 50 m 150 m s −1 3 ×10 4 hurricane 500 km 50 m s −1 1 low-pressure system 1,000 km 1.5 m s −1 10 −1 oceanic circulation 3,000 km 1.5 m s −1 3 ×10 −3 From Lugt (1983) 648 12 Vorticity and Vortices in Geophysical Flows 12.1.3 The Taylor–Proudman Theorem If the inviscid fluid motion is barotropic, the Rossby number will be the only dimensionless parameter. It is then evident from (12.23a) that a small-Ro flow has yet another remarkable feature: lim Ro→0 2k ·∇u ∗ = 0. (12.24) Namely, any slow steady motion in a rapidly rotating system tends to be in- dependent of the axial position. This result was first pointed out by Hough (1897) according to Gill (1982a,b), then by Proudman (1916), and then ex- perimentally confirmed by Taylor (1923). Taylor’s flow visualization photos with a rotating dish are shown as Plate 23 of Batchelor (1967). A drop of colored fluid is quickly drawn out into a thin cylindrical sheet parallel to the rotating axis. More astonishingly, a short obstacle moving at the bottom of the dish can carry an otherwise stagnant column of the fluid with it (the Tay- lor column), see the sketch of Fig. 12.4. Although not in mathematical rigor, this result is now known as (cf. Batchelor 1967). The Taylor–Proudman Theorem . Steady motions at small Rossby num- ber must be a superposition of a two-dimensional motion in the lateral plane and an axial motion which is independent of the axial position. The considerable significance of the Taylor–Proudman theorem in geophys- ical flows was later realized, and since then many experiments and numerical simulations have confirmed the tendency of the flow to be two-dimensionalized as Ro → 0; e.g., Carnevale et al. (1997) and references therein. Recent studies have explored into the dynamic process toward two-dimensionalization and into rotating turbulence, e.g., Wang et al. (2004), Chen et al. (2005), and references therein. We remark that the Taylor–Proudman theorem can be understood in a dif- ferent way. If an inviscid and barotropic relative flow is steady with nonlinear W Fig. 12.4. The Taylor column in a rapidly rotating fluid 12.1 Governing Equations and Approximations 649 advection being identically zero, then the motion must be exactly described by the theorem even if Ro is not small. 2 12.1.4 Shallow-Water Approximation We return to the general inviscid equations and from now on drop the prime for relative quantities. While for planetary-scale motion the spherical coordinates (φ, θ, r) of Fig. 12.1 are appropriate (cf. Batchelor 1967), our main concern will be the motion with characteristic horizontal length L ∼ R∆θ of order of 100 km or larger (∆θ is the latitude variation around a reference value θ 0 ), but still much smaller compared to the earth radius R. Namely, if D is the average depth of the atmosphere and oceans, we have L R  1, |∆θ|1, D L =   1, (12.25a,b,c) by which some further simplifications can be made. First, inequality (12.25a) implies that the effect of the earth curvature can be neglected, thus we may replace the spherical coordinates (φ, θ, r)in Fig. 12.1 by local Cartesian coordinates (x, y, z) on the earth surface, with velocity components (u, v, w). This simplifies the inviscid version of (12.12) to Du Dt +2Ω π w − 2Ω ⊥ v = − 1 ρ ∂ ˜p ∂x , (12.26a) Dv Dt +2Ω ⊥ u = − 1 ρ ∂ ˜p ∂y , (12.26b) Dw Dt − 2Ω π u = − 1 ρ ∂ ˜p ∂z − g, (12.26c) whereΩ ⊥ and Ω π are given by (12.10), and ˜p = p + ρφ c is the modified pressure. Secondly, let U be the characteristic horizontal velocity. Then (12.25c) implies w = O(U)andω x ,ω y = O(ω z ). Retaining the O(1) terms only in (12.26) then leads to the shallow-water approximation for large-scale geophysi- cal flows, where the fluid motion is basically horizontal (horizontal components are denoted by subscript π). The bottom boundary of the flow is allowed to have slowly-varying topography z = h B (x, y), and the upper free boundary z = η(x, y, t) may similarly have slow tidal motions at the scale of L,see Fig. 12.5. We now combine the shallow-water model and Bousinnesq approximation. Let ˜p = p 0 (z)+p  as before and denote p  = ρ 0 (z)π, such that only the 2 For example, Carnevale (2005, private communication) noticed that this would be the case if the relative motion is a steady axisymmetric and inviscid pure vortex, strictly governed by (6.18). 650 12 Vorticity and Vortices in Geophysical Flows W sinq z,w g L p = const. y,v h(x,y,t ) h(x,y,t ) h B (x,y) D x,u Fig. 12.5. Shallow-water approximation for large-scale geophysical fluid motion vertical gradient of p 0 balances the gravitational force and that of π balances the vertical acceleration Dw/Dt. Then integrating (12.18) from z to η yields ˜p(x, y, z, t)=ρ 0 g[η(x, y, t) − z]+ρ 0 π(x, y, z, t). (12.27) But, there is ∇ π π ∼ D L ∂π ∂z ∼ D L D π w Dt ∼  2 U 2 L , which can be dropped. On the other hand, in (12.26a) we have |Ω π w||Ω ⊥ v|, so the Ω π -term can be dropped. Another Ω π -term in (12.26c) should then be dropped simultaneously, for otherwise the energy conservation would be vio- lated (e.g., Salmon 1998). Consequently, the Coriolis force is solely controlled by 2Ω ⊥ =2e z Ω sin θ ≡ f = e z f, (12.28) where f is called the Coriolis parameter. In contrast, Ω π only contributes to ∇φ c as a modification of the “vertical” direction and the “acceleration g due to gravity” (see the context following (12.9)). Therefore, by using (12.27) with ∇ π π dropped, and denoting u = v + we z with horizontal velocity v =(u, v) independent of z, the momentum equation is simplified to D π v Dt + fe z × v = −g∇ π η = − 1 ρ 0 ∇ π p, (12.29) where and below D π /Dt ≡ ∂/∂t + v ·∇. Thus, the horizontal fluid motion is somewhat like a Taylor column. 3 3 The quasi two-dimensional feature exists in shallow-water approximation even without system rotation. This feature is then enhanced by the rotation if the Rossby number is small. 12.1 Governing Equations and Approximations 651 Moreover, although w is neglected in (12.28), as in the boundary layer theory the continuity equation ∇·u = 0 has to be exactly satisfied: ∇ π · v ≡ δ = − ∂w ∂z . (12.30a) But since δ equals the rate of change of the cross area A of a vertical fluid column, which is in turn related to that of the column height h(x, y, t)= η(x, y, t) −h B (x, y), we have ∂w ∂z = − 1 A D π A Dt or δ = − 1 h D π h Dt . (12.30b) Combining this and (12.30a) yields ∂h ∂t + ∇ π · (vh)=0. (12.31) Equations (12.29) and (12.31) are the primitive equations in shallow-water approximation. Note that w depends on z only linearly. Then, (12.25b) implies that, in considering the variation of the Coriolis parameter at latitude, it suffices to retain the first two terms of the Taylor expansion: f(θ)  2Ω sin θ 0 +2Ω(θ − θ 0 )cosθ 0 =2Ω sin θ 0 +2Ω cos θ 0 y R ≡ f 0 + β 0 y, (12.32a) where β 0 ≡ 2 R Ω cos θ 0 ,y= R(θ − θ 0 ). (12.32b) Then (12.29) is reduced to D π v Dt +(f 0 + β 0 y)e z × v = −g∇ π η = − 1 ρ 0 ∇ π p, (12.33) which is called β-plane model although (x, y) vary along the sphere, and is accurate near x = 0. The motion caused by the variation of f with θ is called the β-effect. Simpler than this, if L/R is negligible, we may simply take f = f 0 and obtain an approximation called the f -plane model. It is of interest to look at the dimensionless form of (12.29) made by the horizontal characteristic length L and velocity U, but η is set to be η =Dη ∗ . Then Ro ∂v ∗ ∂t ∗ + e z × v ∗ = −  Fr ∇ ∗ π η ∗ ,= D L , (12.34) where Fr is the same as in (12.23b) and Ro = U/fL is the local Rossby number. Sometimes the Froude number is alternatively defined as F  ≡ U √ gD or F  ≡ U ND , (12.35a) 652 12 Vorticity and Vortices in Geophysical Flows where N is the buoyancy frequency and, as will be seen in Sect. 12.3.3, √ gD is the phase velocity of the gravity wave, which for D = 2 km is about 440 m s −1 . Then (12.34) can be rewritten Ro ∂v ∗ ∂t ∗ + e z × v ∗ = − Ro F 2 ∇ ∗ π η ∗ . (12.35b) Thus, for a steady flow with Ro  F 2 , the Taylor–Proudman theorem holds. Finally, it is often convenient to replace (12.29) by the equivalent equations for vertical vorticity ζ and horizintal divergence δ. Denote ζ a = e z (ζ + f)as the absolute vertical vorticity and recast (12.29) to ∂v ∂t + ζ a × v = −g∇ π  η + 1 2g |v| 2  , so that its curl yields ∂ζ a ∂t + v ·∇ π ζ a + ζ a δ =0. (12.36) Thus, along with the horizontal divergence of (12.29) and substituting h(x, y, t)=η(x, y, t) −h B (x, y) into (12.31), see Fig. 12.5, we obtain the prim- itive shallow-water equations for three scalar variables (ζ,δ,η): ∂ζ ∂t + fδ + βv = −∇ π · (vζ), (12.37a) ∂δ ∂t − fζ + βu + g∇ 2 π η = −∇ π · (v ·∇ π v), (12.37b) ∂η ∂t −∇ π · (vh B )=−∇ π · (vη), (12.37c) where the nonlinear advection terms are put on the right-hand side for clarity. Of course we may also introduce scalar potential χ(x, y, t) and stream function ψ(x, y, t)by v = ∇ π χ + e z ×∇ψ, (12.38) such that δ = ∇ 2 π χ and ζ = ∇ 2 π ψ, and use (χ, ψ, η) as dependent variables. 12.2 Potential Vorticity In geophysical flows the potential vorticity defined by (3.106) plays a key role. In a rotating system its general definition is P ≡ 1 ρ (ω +2Ω) ·∇φ (12.39) for any scalar φ satisfying Dφ/Dt = 0. When the absolute flow is circulation- preserving or ∇φ is perpendicular to the vorticity diffusion vector ∇×a,the 12.2 Potential Vorticity 653 Ertel potential-vorticity theorem stated in Sect. 3.6.1 holds true and has been proved extremely useful: DP Dt =0. (12.40) Rossby (1936, 1940) was the first to introduce the concept of potential vorticity to the fluid motion in oceans. Quite soon afterwards Ertel (1942) established independently the theorem now associated with his name, but with φ being a meteorological quantity. For a review of this early history and later developments see Hoskins et al. (1985). In this section we first introduce the barotropic (Rossby) potential vor- ticity and illustrate its rich consequences in both atmospheric and oceanic motions. We then turn to baroclinic (Ertel) potential vorticity and go beyond the conservation condition, to see in what situation and how the combined distributions of entropy and Rossby–Ertel potential vorticity can characterize and uniquely determine the entire atmospheric motion. 12.2.1 Barotropic (Rossby) Potential Vorticity Consider the shallow-water approximation and observe that (12.36) has the same form as (3.98c) for two-dimensional compressible and circulation- preserving flow. Moreover, a comparison of (12.30b) and the general continuity equation (2.40) clearly indicates that the role of variable density in the latter is now played by the variable fluid-layer depth. Therefore, by inspecting the two-dimensional and circulation-preserving version of the Beltrami equation (3.99), namely (3.137), we see at once that, as the Beltramian form of the vorticity equation for barotropic flow under the shallow-water approximation, there is (Rossby 1936, 1940) D π Dt  f + ζ h  =0. (12.41) The quantity in the brackets is referred to as the barotropic potential vortic- ity or Rossby potential vorticity P , of which the corresponding Lagrangian invariant scalar φ, see (12.39), can be found by inspecting (12.30b) where A and h are independent of z.Thus,w depends on z linearly: w =(h B − z)ϑ π + w B , where w B is the vertical velocity at the bottom z = h B (x, y) determined by the no-through condition w B = D π h B Dt = v ·∇h B . Hence, by (12.30b), it follows that w − v ·∇h B = D π Dt (z − h B )= z − h B h D π h Dt , 654 12 Vorticity and Vortices in Geophysical Flows from which we find φ = z − h B h , D π φ Dt =0. (12.42) The simple conservative equation (12.41) has very clear physical meaning and significant consequence. First, for constant h, if a vertical fluid column (since the flow is treated as inviscid, the vorticity can terminate at upper and lower boundaries) on the northern hemisphere moves northward into a region of larger f, its relative vorticity will decrease, and vice versa if it moves southward. Then, for constant f>0 (say, a column in a rotating tank of fluid or carried by a west wind), the absolute vertical vorticity ζ a in the tube must be proportional to h. Larger h must be associated with the stretching of the tube, and vice versa. Thus, assuming |ζ| < |f|, if initially ζ = 0, as the column moves to a deeper h a ζ>0 must be produced, just as if it moves southward with constant h; while at shallower h a ζ<0 will be created just as if it moves northward. These relative rotating flows are called cyclone and anticyclone in meteorology. In short, a vorticity column stretching produces cyclonic vor- ticity, and its shrinking produces anticyclonic vorticity. Therefore, a proper choice of the bottom topography in a rotating-tank experiment may lead to the same dyanamic effect as the Coriolis parameter varies (e.g., Carnevale et al. 1991a,b; Hopfinger and van Heijst 1993), which provides a convenient method to experimentally study the motion of barotropic vortices. Moreover, assume for simplicity the flow is along a latitude line θ 0 with v = 0. Then if at some x-location the air raises up due to a local high tempera- ture and form a low pressure region at low altitude so that the surrounding air merges toward this location, then by (12.41) at the low-pressure center a cy- clone can be formed. But when the up-raising air stream reaches high altitude to form a local high-pressure region, the air stream will move away to sur- rounding regions, so by (12.41) an anticyclone can be formed. Therefore, the cyclone in low-pressure region and anticyclone in high-pressure region appear in pair, see the sketch of Fig. 12.6. 12.2.2 Geostrophic and Quasigeostrophic Flows In terms of the streamfunction introduced by (12.38), (12.41) can be cast to ∂P ∂t +[ψ,P]=0, (12.43) where [ψ,·] is the horizontal Jacobian operator similar to that used in Sect. 6.5.1 for two-dimensional flow. However, this equation alone cannot be solved for ζ and h which are still coupled with δ in (12.37). In studying large-scale geophysical flows approximations derived from but simpler than (12.37) are often used. The relative importance of the inertial force and Coriolis force depends on the local Rossby number Ro = U/fL. For the earthwehaveΩ =7.29 × 10 −5 s −1 and f =1.03 × 10 −4 s −1 at θ =45 ◦ .If 12.2 Potential Vorticity 655 (a) (b) (c) Fig. 12.6. The formation of a cyclone (a) and anticyclone (b) due to the potential- vorticity conservation, and their connection (c) by vertical airstream. From Lugt (1983) L ∼ 1, 000 km, the inertial force could be comparable with the Coriolis force only if U ∼ 100ms −1 , which seldom occurs. Therefore, as a crude approxi- mation at Ro  1, we neglect the relative acceleration so that the pressure gradient or free-surface elevation is solely balanced by the Coriolis force. The flow under this balance is called the geostrophic flow,whichmustbesteady: f 0 e z × v = −g∇ π η or f 0 v = ge z ×∇η. (12.44) By (12.38), we see χ =0sothev-field is incompressible. The streamlines must be perpendicular to the pressure gradient, or along a streamline the pressure is constant. Moreover, ψ must be proportional to η up to an additive constant; so we write ψ = λ 2 ∆η ∗ ,λ 2 ≡ gD f 0 , ∆η ∗ ≡ η − D D . (12.45) The scalar λ has the dimension of length and is known as the Rossby de- formation radius, which characterizes the behavior of rotating flow subject to gravitational restoring force. The large-scale flows on the earth, such as the westerly belt at middle latitudes, the atmospheric low-pressure sys- tem, hurricane or typhoon, and oceanic circulation, all have small Rossby numbers and to the leading order can be modeled as geostrophic flow. Note that a geostrophic flow satisfies the condition of the Taylor–Proudman theorem. Despite its simplicity, the steady solution (12.44) involves no evolution and is useless in weather and ocean prediction. To find the the next-order approximation called the quasigeotrosphic flow, denote h ∗ B = h B /D such that h = D(1 + ∆η ∗ − h ∗ B ), (12.46) 656 12 Vorticity and Vortices in Geophysical Flows and assume Ro = U f 0 L  1,β ∗ ≡ βL f 0  1, |∆η ∗ |, |h ∗ B |1. (12.47) Then we can substitute the geostrophic ψ–η relation (12.45) into the potential vorticity equation (12.41), retain the leading-order terms, to remove η in an iterative way. Denote y ∗ = y/L and replace P by  P = DP in (12.43), we find (Salmon 1998)  P = D h (f + ζ)  f 0 (1 + β ∗ y ∗ + ∇ 2 π ψ/f 0 )(1 − ∆η ∗ + h ∗ B )  (∇ 2 π − λ −2 )ψ + f + f 0 h ∗ B = f 0 + P  , where f 0 (1 + h ∗ B ) is a background potential vorticity and P  =(∇ 2 π − λ −2 )ψ + f 0 h ∗ B + βy (12.48) is the potential vorticity due to the relative motion. Recall that λ −2 ψ =∆η ∗ in geostrophic model represents the stretching effect of vertical vorticity due to the depth variation caused by the upper-boundary elevation. Then the governing equation for quasigeotrosphic flow with a single unknown ψ follows from substituting (12.48) into (12.43): (∇ 2 π − λ −2 ) ∂ψ ∂t +[ψ,∇ 2 π ψ + f 0 h ∗ B ]+β ∂ψ ∂x =0, (12.49) which has been widely utilized in studies of large-scale barotropic geophysical vortices and will serve as the basic equation for most of our analyses in the rest of this chapter. We stress that solving (12.49) for specified initial and bound- ary conditions is a typical geophysical vorticity-vortex dynamics problem. An important example will be given in Sect. 12.3.4. 12.2.3 Rossby Wave In the atmosphere and ocean there are many types of waves, such as internal- gravity wave, tropographic wave, Kelvin wave, etc. which are modified by the system rotation. Weak wave solutions are obtained from the linearized governing equations. For example, in the shallow-water approximation, by neglecting the nonlinear advections in (12.37), and using the (χ, ψ) expression of v given by (12.38), one can obtain coupled wave equations for (χ, ψ, η), which permitting free traveling waves. If we assume a constant f = f 0 and β = 0 so that it suffices to consider only the x-wise traveling wave of the form exp[i(kx −ωt)], then for h B = 0 one finds a dispersive relation (e.g., Pedlosky 1987; Gill 1982a,b; Salmon 1998) ω[ω 2 − (f 2 0 + gDk 2 )] = 0. (12.50) [...]... the potential vorticity inversion holds up to O(F 4 lnF ) In the worked out examples of potential -vorticity inversion by McIntyre and Norton (2000), Fmax has reached about 0.5–0.7 12.3 Quasigeostrophic Evolution of Vorticity and Vortices Sections 12.1 and 12.2 have set a basic framework for large-scale geophysical vorticity dynamics, of which the key is the concept and conservation of Rossby–Ertel potential... would vanish and an initially axisymmetric vortex would remain so Thus, it is natural to decompose ω into a vortex part and a coupled β-caused residual anomaly, denoted by ψ = Φ + φ, ω = Q + q Then, Sutyrin and Flierl (1994) simplify the formulation by replacing the smooth ω-field by nested vortex patches of piecewise constant potential vorticity, so that the vortex patch theory and contour dynamics of... Quasigeostrophic Evolution of Vorticity and Vortices (a) (b) (c) (d) (e) 673 (f) Fig 12 .14 Vorticity evolution of a monopolar vortex with a perturbation of type (12.46) with n = 2, µ = 0.1, and σ = 0.25 Nondimensional time are (a) t = 0, (b) t = 25, (c) t = 50, (d) t = 75, (e) t = 100, and (f ) t = 200 From Kloosterziel and Carnevale (1999) where n = 2 is the azimuthal wave number, and µ and σ are adjustable... long been recognized, and made routine use of, in another field, that of classic aerodynamics” (e.g Prandtl and Tietjens 1934; Goldstein 1938; Lighthill 1963; Batchelor 1967; Saffman 1981) 12.2 Potential Vorticity 661 Hoskins et al (1985) stress that, as in deducing the flow field from a given vorticity distribution, given an IPV map that by (12.56) is the product of absolute vorticity and static stability,... Outside the vortex core the relative vorticity 678 12 Vorticity and Vortices in Geophysical Flows ω vanishes and the potential vorticity is P = f /H In the lower layer, there is no vortex initially Then 1 ∇2 ψ1 + λ−2 (ψ2 − ψ1 ) = P1 , rR 2 1 ∇2 ψ2 + λ−2 (ψ1 − ψ2 ) = 0, 2 (12.89b) R λ (12.89a) (12.89c) √ where λ = Λ/ 2, P1 = ω0 = HP0 − f is the relative vorticity From... same strength and core radius Two baroclinic vortices with similar potential vorticity but different strength and core radii will interact asymmetrically with larger critical distance, and the weaker vortex was always drawn out to surround the stronger one Overman and Zabusky (1982) described the asymmetric interaction as “entrainment” of the region of greater vorticity density (the stronger vortex core)... when α > 1.85 Kloosterziel and Carnevale (1999) have examined a monopolar vortex with α = 3.0 and initial state shown in Fig 12.14a They added a disturbance of the form ω = µ cos(nθ) exp − (αrα − 2)2 2σ 2 (a) (12.82) (b) (c) , (d) Fig 12.13 Multipole structures (a) vortex monopole; (b) vortex dipole; (c) symmetrical vortex quadrupole; (d) vortex tripole Reproduced from Voropayev and Afanasyev (1994) 12.3... criterion The simplest stable vortex structure is monopolar vortex consisting of circular streamlines about a common center, with positive or negative vorticity Its simplest model is circular vortex patch A more commonly seen but more complicated monopolar vortex consists of a vortex core of certain rotating sense surrounded by the vorticity of opposite sign (similar to the Taylor vortex) These large-scale... process as addressed in Sect 12.3.1 and shown in Fig 12.10 A class of relatively simple isolated vortical structures has often been used to study the instability of a monopolar vortex (Carton and McWilliams 1989; Carnevale and Kloosterziel 1994), which has smooth vorticity and velocity distributions Its dimensionless form is α 1 ω = ω0 1 − αrα e−r , 2 (12.80) 672 12 Vorticity and Vortices in Geophysical Flows... to turn around each other and merge to a larger vortex due to viscosity Figure 12.12 confirms the key role of Ds/Dt in the formation of sheet-like and patch-like structures Shown in the figure are the sign of Ds/Dt and vorticity contours In the positive regions the isovorticity lines are highly clustered, implying large vorticity gradient, while in the negative regions the isovorticity lines are quite . (Ertel) potential vorticity and go beyond the conservation condition, to see in what situation and how the combined distributions of entropy and Rossby–Ertel potential vorticity can characterize and. or ∇φ is perpendicular to the vorticity diffusion vector ∇×a,the 12.2 Potential Vorticity 653 Ertel potential -vorticity theorem stated in Sect. 3.6.1 holds true and has been proved extremely useful: DP Dt =0 wave in the westerly belt. Solid and dashed marks represent the ambient potential vorticity f and newly produced relative vorticity ζ,respec- tively. The larger and smaller sizes of the solid marks