The Lecture Notes in Physics Part 7 pot

116 V. Zeitlin ∂ t  u J  +  0 −J −3 −10  ∂ a  u J  =  v 0  . (4.32) The eigenvalues of the matrix in the l.h.s. of (4.32) are μ ± =±J − 3 2 and the corresponding left eigenvectors are  1 , ±J − 3 2  . Hence, Riemann invariants are w ± = u ± 2J − 1 2 and we have ∂ t w ± + μ ± ∂ a w ± = v. (4.33) Expressions of original variables in terms of w ± are u = 1 2 (w + + w − ), J = 16 (w + − w − ) 2 > 0 ,μ ± =±  w + − w − 4  3 . (4.34) In terms of the derivatives of the Riemann invariants r ± = ∂ a w ± , we get ∂ t r ± + μ ± ∂ a r ± + ∂μ ± ∂w + r + r ± + ∂μ ± ∂w − r − r ± = ∂ a v = Q(a) − J , (4.35) which may be rewritten using Lagrangian derivatives along the characteristics d dt ± = ∂ t + μ ± ∂ a as dr ± dt ± + ∂μ ± ∂w + r + r ± + ∂μ ± ∂w − r − r ± = Q(a) − J . (4.36) Wave breaking and shock formation correspond to r ± →±∞in finite time. In terms of new variables R ± = e λ r ± , with λ = 3 128 log | w + − w − | , (4.35) may be rewritten as dR ± dt ± =−e −λ ∂μ ± ∂w ± R 2 ± + e λ ( Q(a) − J ) , (4.37) where ∂μ ± ∂w ± = 3 64 (w + − w − ) 2 > 0. The qualitative analysis of these generalized Ricatti equations shows that if initial relative vorticity Q − J = ∂ a v is sufficiently negative (anti-cyclonic), rotation does not stop wave breaking, which is taking place for any initial conditions. However, if the relative vorticity is positive (cyclonic case), as well as the derivatives of the Riemann invariants at the initial moment, there is no breaking. An example of wave breaking due to the geostrophic adjustment of the unbalanced jet is presented in Fig. 4.2. 4 Lagrangian Dynamics of Fronts, Vortices and Waves 117 −2L −L 0 L 2L 0 Vmax Vjet Fig. 4.2 Wave breaking and shock formation (right panel) during adjustment of the unbalanced jet (left panel, top to bottom: consecutive profiles of the free surface with time measured in f −1 units). Length is measured in deformation radius units: L = R d = gH f 4.2.1.6 “Trapped Waves” in 1.5d RSW: Pulsating Density Fronts The above-established supra-inertiality of the spectrum of the small perturbations around a balanced 1.5d RSW front means the absence of trapped waves, and, hence, the attainability of the adjusted state by evacuating the excess of energy via inertia- gravity wave emission (eventually with shock formation). There exist, however, the RSW fronts, where the wave emission is impossible. These are the lens-type con- figurations with terminating profile of fluid height. Such RSW configurations are used to model oceanic double density fronts, either outcropping or incropping, e.g. Griffiths et al. [10]. In Lagrangian description (4.9) the evolution of a double RSW front corresponds to positive h I terminating at x = x ± . Adjustment of such fronts, therefore, should proceed without outward IGW emission. An example of adjusted front treated in literature is given in Fig. 4.3. A family of exact unbalanced pulsating solutions is known for such fronts (Frei [9]; Rubino et al. [22]). Let us make the following ansatz: X(x, t) = xχ(t), h I (x) = h 0 2  1 − x 2 L 2  ,v I (x) = x, (4.38) Fig. 4.3 An example of equilibrated double density front 118 V. Zeitlin where h 0 ,,L are constants. Plugging (4.38) into (4.9) and non-dimensionalizing with the timescale f −1 and the length-scale L gives the following ODE for χ: ¨χ + χ − γ χ 2 = μ, (4.39) where γ is the Burger number gh 0 f 2 L 2 and μ = 1 +  f . Integrating (4.39) once gives ˙χ 2 2 + P(χ) = E, P(χ) = χ 2 2 − μχ + γ χ , (4.40) where the integration constant E is expressed in terms of initial conditions χ(t = 0) = 1, ˙χ(t = 0) = U: E = U 2 2 + 1 2 − μ + γ. (4.41) Equation (4.40) may be integrated in elliptic functions. The “potential” P(χ) being convex, the solution for χ is finite amplitude and oscillating with supra-inertial frequency. The minimum of P corresponds to the front in geostrophic equilibrium and constant χ = 1. Thus, the adjustment (initial-value) problem for double density fronts will result, in general, in a pulsating solution, whereas relaxation to the steady state is possible only due to viscous effects (shocks). 4.2.2 Axisymmetric Case 4.2.2.1 Governing Equations and Lagrangian Invariants Axisymmetric RSW motion is described in cylindrical coordinates by fields depend- ing on radial variable only. As in the rectilinear case, it is possible to reduce the whole dynamics to a single PDE for a Lagrangian variable R(r, t), the distance to the center of a “particle” (or rather a particle ring) initially situated at r. We first rewrite the Eulerian RSW equations in cylindrical coordinates (r,θ) and assume exact axial symmetry: (∂ t + u r ∂ r )u r − u θ  f + u θ r  + ∂ r h = 0 , (∂ t + u r ∂ r )u θ + u r  f + u θ r  = 0 , (4.42) ∂ t h + 1 r ∂ r (ru r h) = 0 . Here u r , u θ are the radial and azimuthal components of velocity. Note that the adjusted stationary state changes character as compared to the rectilinear case: it 4 Lagrangian Dynamics of Fronts, Vortices and Waves 119 verifies conditions of the cyclo-geostrophic balance and not of the purely geostrophic one: u θ  f + u θ r  = ∂ r h, u r = 0. (4.43) Multiplying the second equation in (4.42) by r, we recover the conservation of angu- lar momentum: (∂ t + u r ∂ r )  ru θ + f r 2 2  = 0 , (4.44) which replaces the conservation of geostrophic momentum in the plane-parallel case. Equation (4.42) can be rewritten using the Lagrangian coordinate R(r, t). Integrating (4.44) gives R(r, t) u θ (r, t) + f R 2 (r, t) 2 = ru θ I (r) + f r 2 2 ≡ G(r), (4.45) where u θ I is the initial azimuthal velocity profile. Using the above expression we get u θ  f + u θ R  = 1 R  G − f R 2 2  f + G R 2 − f 2  = 1 R 3  G 2 − f 2 R 4 4  . (4.46) The mass conservation is expressed by the following relation: h(r, t) R(r, t) dR = h I (r) rdr. (4.47) With the help of (4.46), (4.47) and the definition ˙ R(r, t) = u r (r, t), the radial momentum equation becomes ¨ R + f 2 4 R − 1 R 3 G 2 + 1 ∂ r R ∂ r  rh I R ∂ r R  = 0 , (4.48) to be solved with initial conditions R(r, 0) = r, ˙ R(r, 0) = u r I . The stationary part of this equation defines the adjusted, slow states. The fast motions are axisymmetric IGW. Indeed, for small perturbations about the state of rest: R(r, t) = r + φ(r, t), (4.49) with |φ|r , h I (r) = 1 and u θ I (r) = 0, the following equation is obtained after some algebra: 120 V. Zeitlin ¨ φ + f 2 φ − ∂ r φ r − ∂ 2 rr φ + φ r 2 = 0 . (4.50) If solutions are sought in the form φ(r, t) = ˆ φ(r) e iωt , (4.50) yields, after a change of variables, the canonical equation for the Bessel functions. The familiar axisym- metric wave solutions involving Bessel functions J 1 then follow: φ(r, t) = CJ 1 (  ω 2 − f 2 r) e iωt + c.c., (4.51) where C is the wave amplitude. The whole program of the previous section may be carried on as well in cylindri- cal coordinates, with similar conclusions. We present below only the case of the axisymmetric density fronts (Sutyrin and Zeitlin [23]). 4.2.2.2 Axisymmetric Density Fronts and Radial “pulson” solutions We make the following ansatz in (4.48): h I (r) = h 0 2  1 − r 2 L 2  , R(r, t) = rφ(t), u θ I (r) = r,  = const. (4.52) Then by non-dimensionalizing the system in the same way as for the rectilinear fronts, introducing the Burger number γ , and denoting M = 1 2 +  f we get ¨ φ + φ 4 − M 2 φ 3 − γ φ 3 = 0, (4.53) to be solved with initial conditions φ(0) = 1, ˙ φ(0) = u r I . A drastic simplification of this equation is provided by the substitution φ 2 = χ which immediately gives the equation of the harmonic oscillator with shifted equilibrium position: ¨χ + χ −4E = 0, E = u 2 r I 2 + 1 8 + M 2 + γ 2 > 0. (4.54) The “radial pulson” solution (cf. Rubino et al. [21] for a derivation in Eulerian framework) satisfies the initial conditions χ(0) = 1, ˙χ(0) = 2u r I and is given by χ(t) = 4E + (1 − 4E) cost + (2u r I + 1 −4E) sin t. (4.55) The crucial difference between the radial and rectilinear pulson, thus, is that the for- mer always has inertial frequency and thus represents nonlinear inertial oscillations, while the latter is always supra-inertial. 4 Lagrangian Dynamics of Fronts, Vortices and Waves 121 4.3 Including Baroclinicity: 2-Layer 1.5d RSW 4.3.1 Plane-Parallel Case 4.3.1.1 Governing Equations and General Properties of the Model To introduce the baroclinic effects in the dynamics in the simplest way we consider the two-layer rotating shallow water model. We use the rigid lid upper boundary condition and again consider for simplicity a flat bottom. In this case the equations governing the motion of two superimposed rotating shallow-water layers of unper- turbed depths H 1,2 , H 1 +H 2 = H and densities ρ 1,2 in Cartesian coordinates under hypothesis of no dependence of y (straight two-layer fronts) are ∂ t u i + u i ∂ x u i − f v 1 + ρ −1 i ∂ x π i = 0 , (4.56a) ∂ t v i + u i ( f + ∂ x v i ) = 0 , (4.56b) ∂ t h i + ∂ x ((h i u i ) = 0 , i = 1, 2 (4.56c) π 1 + g  (ρ 1 h 1 + ρ 2 h 2 ) = π 2 , (4.56d) h 1 + h 2 = 1, (4.56e) where no sum over repeated index is understood, π i are the pressures in the layers, g  = ρ 2 −ρ 1 ρ 2 +ρ 1 g is the reduced gravity and h i are the variable layers depths. A sketch of the 2-layer 1.5d RSW is presented in Fig. 4.4. The Lagrangian invariants of equations (4.56a), (4.56b) and (4.56c) are potential vorticities and geostrophic momenta in each layer: Q i = f + ∂ x v i h i , M i = fx+∂ x v i , i = 1, 2. (4.57) For any solution of system (4.56a), (4.56b), (4.56c), (4.56d) and (4.56e), constraint (4.56e) imposes that x v2(x,t) u2(x,t) . g Ω ρ2 ρ1 h2(x,t) v1(x,t) (x,t) u1 h1(x,t) Fig. 4.4 Schematic representation of the 2-layer 1.5d RSW model 122 V. Zeitlin ∂ x (h 1 u 1 + h 2 u 2 ) = 0. (4.58) Hence, the barotropic across-front velocity is U = h 1 u 1 + h 2 u 2 H = U(t). (4.59) Choosing the boundary condition of absence of the mass flux across the front sets U = 0. The geostrophic equilibria are stationary solutions: u i = 0,v i = 1 f ρ i ∂ x π i , i = 1, 2 ,π 2 = π 1 + g(ρ 1 h 1 + ρ 2 h 2 ). (4.60) The fast motions in the linear approximation are internal inertia-gravity waves prop- agating along the interface between the layers. By linearizing about the rest state h 1 = H 1 , h 2 = H 2 , u 1,2 = 0,v 1,2 = 0, the dispersion relation for the waves with frequency ω and wavenumber k follows: ω 2 (ω 2 − f 2 − c 2 e k 2 ) = 0 . (4.61) Here c 2 e = g  H e is the phase speed of the waves, H e = (ρ 2 −ρ 1 )H 1 H 2 ρ 1 H 1 +ρ 2 H 2 is the equivalent height for the baroclinic modes of the model. As in the one-layer model, conditions for existence and uniqueness of the adjusted state can be obtained as conditions for existence and uniqueness of solutions to the PV equations (LeSommer et al. [14]). These equations can be combined to give two ordinary differential equations for the depths of the layers: g  f h  1 − (Q 2 +rQ 1 ) h 1 =− ( − f (1 −r) + HQ 2 ) , (4.62a) g  f h  2 − (Q 2 +rQ 1 ) h 2 =− ( f (1 −r) +rH Q 1 ) , (4.62b) where notation r = ρ 1 /ρ 2 for the density ratio of the layers has been introduced and the prime denotes the x  - differentiation. An essential difference of these equations from their one-layer counterpart is that the forcing terms at the r.h.s. are not constant. They, nevertheless, may be analysed by the same method as in 1dRSW. For an equation of the form h  − R(x) h =−S(x), the existence and uniqueness of solutions are guaranteed if R and S have constant asymptotics at ±∞. Further- more, the solution is positive if R and S are positive. Hence, for the initial states with localized PV anomalies such that Q 1 ≥ 0 and Q 2 ≥ (1 −r) f/H , (4.63) the above equations have unique solutions h 1 and h 2 that are everywhere positive. 4 Lagrangian Dynamics of Fronts, Vortices and Waves 123 A crucial simplification of the rigid-lid 2-layer equations follows from the fact the pressures π i may be eliminated from (4.56a), (4.56b) and (4.56c). Indeed by using (4.58) and (4.56e) and (4.56d) we get, again under the hypothesis of zero overall across-front mass flux: ∂π 1 ∂x =  h 1 ρ 1 + h 2 ρ 2  −1  f (h 1 v 1 + h 2 v 2 ) − ∂ ∂x  h 1 u 2 1 + h 2 u 2 2  − gh 2 ρ 2 ∂ ∂x ( ρ 1 h 1 + ρ 2 h 2 )  , (4.64) ∂π 2 ∂x =  h 1 ρ 1 + h 2 ρ 2  −1  f (h 1 v 1 + h 2 v 2 ) − ∂ ∂x  h 1 u 2 1 + h 2 u 2 2  + gh 1 ρ 1 ∂ ∂x ( ρ 1 h 1 + ρ 2 h 2 )  . (4.65) One can use (4.64), (4.65) in order to reduce the system to four equations for four independent variables u 2 , h 2 ,v 2 and v 1 , i.e. lower (heavier)-layer variables plus upper-layer jet velocity: ∂u 2 ∂t + u 2 ∂u 2 ∂x − f v 2 + ρ 1 ρ 2 h 1 + ρ 1 h 2  f (h 1 v 1 + h 2 v 2 ) − ∂ ∂x  h 1 u 2 1 + h 2 u 2 2  + g(ρ 2 − ρ 1 ) ρ 1 h 1 ∂h 2 ∂x  = 0, (4.66) ∂h 2 ∂t + u 2 ∂h 2 ∂x + h 2 ∂u 2 ∂x = 0, (4.67) ∂v 2 ∂t + u 2 ∂v 2 ∂x + fu 2 = 0, (4.68) ∂v 1 ∂t + u 2 ∂v 1 ∂x + (u 1 − u 2 ) ∂v 1 ∂x + fu 1 = 0, (4.69) where u 1 = h 2 u 2 h 2 − H , h 1 = H − h 2 . (4.70) 4.3.1.2 Lagrangian Approach to 2-Layer 1.5d RSW We start from the system (4.66), (4.67), (4.68), (4.69) and (4.70), taken for sim- plicity in the frequently used limit r → 1 and introduce the Lagrangian coordinate 124 V. Zeitlin X(x, t) corresponding to the positions of the fluid particles in the lower layer. In terms of displacements φ with respect to initial positions X(x, t) = x + φ(x, t). The corresponding Lagrangian derivative is d dt = ∂ ∂t +u 2 ∂ ∂x . The dependence of the height variable h 2 on the Lagrangian labels and transformation of its derivatives are obtained via the mass conservation in the lower layer: h 2 I dx = h 2 (X(x, t), t)dX. The subscript 2 will be omitted in what follows. As in the one-layer case, (4.68) expresses the conservation of the geostrophic momentum in the lower layer and allows to eliminate v 2 in terms of φ and its initial value: v 2 (x, t) + f φ(x, t) = v 2 I (x). (4.71) The 2-layer Lagrangian equations, thus, are ¨ X − f  1 − h H   v 2 I − v 1 − f (X − x)  − 1 X   h ˙ X 2 H − h   + g   1 − h H  1 X  h  = 0, (4.72) ˙v 1 − ˙ X 1 − h H  v  1 X  − f h H  = 0, (4.73) where h = h I (x) X  , prime and dot denote x- and t-differentiations, respectively, and g  = g ρ 2 −ρ 1 ρ 2 – the reduced gravity in the limit r → 1. 4.3.1.3 Symmetric Instability A qualitatively new phenomena appearing in the dynamics of fronts due to the baro- clinic effects is a specific symmetric instability, i.e. an instability developing without perturbations in the front-wise direction. This instability is frequently called inertial, the term “symmetric” being often reserved for its moist counterpart (e.g. Bennetts and Hoskins [2]). For simplicity, we will consider the particular case of the initial conditions in the form of a barotropic jet with h 2 I = H 2 = const., v 2 I = v 1 I = v I (x). By introducing the notation α 1 = H 1 H ,α 2 = H 2 H ,α 1 + α 2 = 1 we have in non-dimensional form: ¨ φ + φ α 1 + φ  1 + φ  +  α 1 + φ  1 + φ  (v 1 − v I ) − α 2 1 + φ   ˙ φ 2 α 1 + φ    − γ 1 (1 + φ  ) 4 φ  = 0 , (4.74) ˙v 1 − 1 α 1 + φ  ˙ φv  1 − α 2 α 1 + φ  1  ˙ φ = 0, (4.75) where  = Ro = V fL is the Rossby number based on the typical jet velocity V and typical jet width L, γ = Bu = g  Hα 1 α 2 f 2 L 2 is the Burger number. 4 Lagrangian Dynamics of Fronts, Vortices and Waves 125 Equations (4.74), (4.75) are to be solved with initial conditions φ(x, 0) = 0, ˙ φ(x, 0) = u 2 I ,v 1 (x, 0) = v I . The initial jet v 1 = v I ,φ= 0, if non-perturbed: u I = 0isa solution. System (4.74), (4.75) in the linear approximation gives ¨ φ + α 1 φ + α 1 ξ 1 − γφ  = 0 , (4.76) ˙ ξ 1 − v  I α 1 ˙ φv  I − α 2 α 1 1  ˙ φ = 0 , (4.77) where we introduced v 1 − v I = ξ . Hence, φ + ˙ φ(1 + v  I ) − γ ˙ φ  = 0 . (4.78) Using the variable ψ = ˙ φ, renormalizing x with √ γ and looking for the solution ψ ∝ e iωt , we get the quantum-mechanical Schrödinger equation: ∂ 2 xx ψ +(E − V(x))ψ = 0 (4.79) for a particle having the energy E = ω 2 and moving in the potential V(x) = 1+v  I . It is worth noting that Burger number plays the role of the Planck constant squared. It is known (e.g. Landau and Lifshits [13]) that in the case of quantum mechanical potential well there are both propagating solutions corresponding to the continuous spectrum ω 2 ≥ 1 and trapped in the well, localized solutions corresponding to the discrete spectrum Min(V (x)) < ω 2 < 1. As is easy to see, the potential well cor- responds to the region of anticyclonic shear. Hence, the trapped modes are localized there, oscillating at sub-inertial frequencies. If the potential is deep enough (strong enough anticyclonic shear), non-oscillatory unstable modes with ω 2 < 0 appear and therefore a specific instability arises. This is the symmetric instability which is thus intricately related to the presence of trapped modes inside the front. It should be noted that the known explicit solutions of the Schrödinger equations for some potentials, e.g. cosh −2 potential (e.g. Landau and Lifshits [13]) may be used for analytical studies of symmetric instability. The Lagrangian equations (4.74), (4.75) provide a convenient framework for studying the nonlinear stage of this instability. 4.3.1.4 Equatorial 2-Layer 1.5d RSW in Lagrangian Variables As in the one-layer case, the Lagrangian description may be also applied to the equatorial zonal flows. The equatorial counterparts of (4.72), (4.73), with obvious interchanges between the zonal (u) and meridional (v) components of velocity and respective Lagrangian coordinates, are [...]... to use as independent variables the positions of the particles in ¯ ¯ the adjusted state ( X , Z ), rather than the initial positions (x, z) When this change of variables is made in (4.128), two terms which express the thermal wind relation in the adjusted state cancel out Furthermore, it is convenient to express the gradients ¯ of v and θ in the adjusted state through the geopotential φ, making explicit... coincides with the classical scenario of Hoskins and Bretherton [12], with the only difference that in their example the parameters of the system were driven towards the singular case by an adiabatic change due to external deformation field 4.4.3 Trapped Modes and Symmetric Instability in Continuously Stratified Case The Lagrangian approach is also efficient for studying symmetric/inertial instability in. .. (4.94e) Here they are written in the atmospheric context using potential temperature θ and the so-called pseudo-height vertical coordinate (Hoskins and Bretherton [12]), θr is a normalization constant For oceanic applications potential temperature should be replaced by density and the sign in the hydrostatic relation (4.94c) should be changed, z then becomes the ordinary geometric coordinate Potential... collaborators [5 7] In (M, θ) coordinates, the thermal wind relation takes the form: f Hence a potential g ∂Z ∂X = ∂θ θr ∂ M (4.113) for the final positions of the fluid particles may be introduced: X= g ∂ , θr ∂ M Z= f ∂ ∂θ (4.114) The Jacobian of the transformation from (x, z) to (X, Z ) can be rewritten as ∂(X, Z ) ∂(M, θ) = 1, ∂(M, θ) ∂(x, z) (4.115) from which we can obtain, replacing X and Z by their expressions... (standard) justification of the one-layer reduced-gravity model and a possibility to calculate baroclinic corrections to the one-layer RSW solutions For h example, the pulsating front solution presented in Sect 4.2 is a zero-order in H solution of (4 .72 ), (4 .73 ), but corrections will appear in the next orders, in particular the non-zero velocity field v1 in the thick upper layer They may be calculated order... absence of symmetric instability, the adjusted state exists and is unique in the absence of boundaries It is to be emphasized that the criterion is the same as for fronts in 1- and 2-layer RSW An alternative Lagrangian formulation using the geostrophic and isentropic coordinates (M, θ) as independent variables in the Monge–Ampère equation was extensively used in the literature, in particular by Cullen... always the same, but its width depends on the vertical wavenumber n: the smaller the vertical scale of the waves, the wider the potential The Schrödinger equation (4.139) has a continuous and a discrete spectrum of eigen¯ values ω 2 The potential (1 + ) tends to one as X → ∞; hence, for a given n, we have • continuous spectrum of solutions with ω > 1 This part of the spectrum is doubly degenerate (two independent... resembles the homogeneous part of the Sawyer–Eliassen equation (e.g Holton [11], p 275 ) except for the term with the double-time derivative, which makes (4.134) prognostic In the Sawyer–Eliassen equation, this term is absent because the fast time has been filtered out by balanced scaling, making the equation diagnostic Like in the two-layer case above, we take the simplest example of a barotropic jet The. .. obey the thermal wind relation f A potential g ∂θ ∂M = ∂z θr ∂ x (4.99) may be introduced for balanced states, such that M = f −1 ∂ , ∂x θr ∂ g ∂z θ= (4.100) 2 In fact, is an “extended” geopotential given as = φ + f 2 x2 The fast motions are internal inertia gravity waves Their dispersion relation may 2 be easily obtained in the case of linear background stratification θ0 (z) = N θr z by g linearization... (4.126) The positions of isentropic surfaces and isotachs may be easily obtained from knowing explicit final positions of the fluid particles (4.123), (4.124) and using the Lagrangian conservation of θ and M It may be thus shown (Plougonven and Zeitlin [ 17] ) that singularity corresponds to intersecting isentropes, as shown in Fig 4.5 and in nite gradients of v We thus have a frontogenesis process, which in . ρ 2 h 2 ). (4.60) The fast motions in the linear approximation are internal inertia-gravity waves prop- agating along the interface between the layers. By linearizing about the rest state h 1 = H 1 ,. collaborators [5 7] . In (M,θ)coordinates, the thermal wind relation takes the form: f ∂ X ∂θ = g θ r ∂ Z ∂ M . (4.113) Hence a potential  for the final positions of the fluid particles may be introduced: X. and General Properties of the Model To introduce the baroclinic effects in the dynamics in the simplest way we consider the two-layer rotating shallow water model. We use the rigid lid upper boundary condition