Barotropic geophysical flows and two-dimensional fluid flows: Conserved Quantities

Transcription

1 Barotropic geophysical flows and two-dimensional fluid flows: Conserved Quantities Di Qi, and Andrew J. Majda Courant Institute of Mathematical Sciences Fall 2016 Advanced Topics in Applied Math Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

2 Lecture 2: Conserved Quantities 1 Review: Some special exact solutions 2 Conserved quantities Conservation of energy and enstrophy Barotropic geophysical flows in a channel domain 3 Variational derivatives and an optimization principle 4 More equations for geophysical flows Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

3 Outline 1 Review: Some special exact solutions 2 Conserved quantities Conservation of energy and enstrophy Barotropic geophysical flows in a channel domain 3 Variational derivatives and an optimization principle 4 More equations for geophysical flows Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

4 The simplest set of equations that meaningfully describes the motion of two-dimensional geophysical flows under these circumstances is given by the: Barotropic Quasi-Geostrophic Equations: where Dq Dt = q + v q = D ( ) ψ + F (x, t) t q = ω + βy + h (x, y), ω = ψ, v = ( y ψ, x ψ) T. q: the potential vorticity; ψ: the stream function; v = ψ: the horizontal velocity field; ω = ψ: the relative vorticity; βy: the beta-plane effect from the Coriolis force; h (x): the bottom floor topography, given by ocean floor or mountain range; D ( ): various possible dissipation mechanisms F (x, t): additional external forcing. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

5 Reduced linear system for the stream function ψ Here we concentrate on finding special exact solutions with both forcing and dissipation with the stronger ansatz: we assume that q and ψ are linearly dependent q = µψ = ψ + βy + h The the solution of the barotropic quasi-geostrophic equations under the linear dependence assumption small and large scale of stream functions ψ = V 0 y + ψ (x, y, t) µ ψ = D ( ) ψ + F (x, t), t µψ = ψ + h (x), V 0 = β/µ. ψ = V 0 y + ψ = β µ y + Â k e ix k. k =0 Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

6 Outline 1 Review: Some special exact solutions 2 Conserved quantities Conservation of energy and enstrophy Barotropic geophysical flows in a channel domain 3 Variational derivatives and an optimization principle 4 More equations for geophysical flows Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

7 Conserved quantities Conserved quantities play a decisive role both in physics and in mathematics. They are especially important in studying general properties of solutions, and they are of great importance in the non-linear stability theory and statistical theory. Candidates for Physically Conserved Quantities: Kinetic Energy: E = 1 ˆ ψ A RV 2 (t); }{{}}{{} small scale mean Enstrophy and Potential Enstrophy: E = 1 ˆ ω 2, E = 1 ˆ q 2 ; 2 2 Large-scale Enstrophy: Q = (ω + h) 2 + βv (t) ; Generalized Enstrophy: ˆ Q = G (q). Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

8 Conservation of Kinetic Energy A nature candidate of Kinetic Energy: E = 1 2 ˆ v 2 = 1 2 ˆ ψ 2. Recall the dynamical equation q t + ψ q = D ( ) ψ + F (x, t). The change rate of kinetic energy in time ˆ ˆ de dt = ψ ψ t = ψ ˆ t ψ = ψ q t {ˆ } { ˆ } = ψ ψ q + ψd ( ) ψ + { ˆ } ψf (x, t) = {1} + {2} + {3}. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

9 Claim The net contribution from advection is zero, i.e. 1 = 0. Because the potential vorticity q q = + y+h, is not a periodic function in the presence of y, the actual calculation becomes subtle under periodic geometry. Since we get 1 q = + h + x = h x q = h x = 1 + h n + ( 2 2 x=2 = 0 x=0 ) 2 where n denotes the unit outer-normal to the periodic domain The boundary terms drop out since, and h are periodic. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

10 Indeed 2 = = Notice for j = 2k even, we have 1 j j = 2k = Likewise, if j = 2k + 1 is odd, we have 1 j j = 2k+1 = l d j 1 j j j=0 k k = = j 1 /2 j 1 /2 k k+1 = where we utilized the classical integration by parts formula F G = F G = F G for periodic functions F and G. Hence we have l 2 = d j j/2 2 j=0 j even l j=1 j odd d j j/2 j/2 k k j 1 /2 2 Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

11 Claim 3 We will assume that = 0. This means that we can rewrite in the alternative form of = curl F F = or =. More specifically, if takes the Fourier expansion = k 0 ˆ k ei k x then is given by = k 0 ˆ k k 2 ei k x and we have the desired alternative forms. It then follows that 3 = = curl F = F = v F or 3 = = = = Combining our estimates on the three terms and putting them back into the equation (1.74), we obtain the results on conservation of energy in the absence of the mean velocity field. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

12 36 Barotropic geophysical flows and two-dimensional fluid flows Theorem 1.1 (Kinetic energy identity) The solutions of the barotropic quasigeostrophic equations (1.1) with periodic boundary conditions satisfy the kinetic energy identity d 1 v 2 = dt 2 E + v F (1.74) where E = curl F = l j=0 j even d j j/2 2 + l j=1 j odd d j j 1 /2 2 Corollary 1.1 In the absence of dissipation and forcing, the kinetic energy of periodic flow is conserved. Remarks on the dissipation Case 1: Newtonian (eddy) viscosity (j = 2 only, d 2 = ). Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

13 Large-scale and Small-scale flow interaction Stream Function with time dependent mean flow: ψ = V (t) y + }{{} ψ }{{}. mean periodic The small-scale part of kinetic energy (assume D = 0, F = 0) d dt E small scale = d ˆ ˆ ˆ 1 ψ 2 = ψ ψ q + dt 2 ˆ ψ ˆ h = V (t) x ψ + V (t) x ψ. V (t) q x ψ Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

14 Large-scale and Small-scale flow interaction Stream Function with time dependent mean flow: ψ = V (t) y + }{{} ψ }{{}. mean periodic The small-scale part of kinetic energy (assume D = 0, F = 0) d dt E small scale = d ˆ ˆ ˆ 1 ψ 2 = ψ ψ q + dt 2 ˆ ψ ˆ h = V (t) x ψ + V (t) x ψ. V (t) q x ψ Notice that ˆ ψ ˆ x ψ = ˆ x ( ψ ) ψ = ( ) 1 x 2 ψ 2 = 0. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

15 Large-scale and Small-scale flow interaction Stream Function with time dependent mean flow: ψ = V (t) y + }{{} ψ }{{}. mean periodic The small-scale part of kinetic energy (assume D = 0, F = 0) d dt E small scale = d ˆ ˆ ˆ 1 ψ 2 = ψ ψ q + dt 2 ˆ ψ ˆ h = V (t) x ψ + V (t) x ψ. V (t) q x ψ Therefore we deduce ˆ d h dt E small scale = V (t) x ψ topographic stress Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

16 Since there is no dissipation and forcing, it is natural to postulate the assumption that the total energy is conserved This Implies in the case V 0 0 d dt E total = d dt A 1 R 2 V 2 (t) + d dt E small scale = A R V (t) d dt V (t) + V (t) ˆ h x ψ. d dt V (t) = h x ψ with ffl Ω f = 1 A R f. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

17 Since there is no dissipation and forcing, it is natural to postulate the assumption that the total energy is conserved This Implies in the case V 0 0 d dt E total = d dt A 1 R 2 V 2 (t) + d dt E small scale = A R V (t) d dt V (t) + V (t) ˆ h x ψ. d dt V (t) = h x ψ with ffl Ω f = 1 A R f. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

18 Extension for large-scale and small-scale flow interaction via topographic stress A) q = ψ + h + βy B) ψ = V (t) y + ψ C) D) q + J (ψ, q) = 0 t d dt V (t) = h x ψ Steady state solution µψ = ψ + h remains true ˆ ˆ ˆ ˆ h x ψ = h ψ x = ψ ψ x µ ψ ψ x = 1 ˆ ( ψ 2 µ (ψ ) 2) = 0. 2 x Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

19 In Chapter 5, we will consider a special type of solutions to (1.82) (1.85). This special type of solutions consists of a large-scale Mean Flow component and Large-Scale a Rossby Wave Enstrophy: component. The two components interact through topographic stress. We will find very rich examples of integrable and chaotic dynamics. The equations of large-scale and small-scale Q = βv (t) + 1 interaction via topographic stress enjoy another conserved quantity, the large-scale (ω + enstrophy h) 2. Q = V t h 2, for general periodic flows. Indeed d 1 + h 2 = + h dt 2 t = + h + h + y V t + h + h + y x = 1 + h 2 + h 2 x V t + h 2 2 x = x h x = h 2 x 2 + x dv t = A R dt Conservation of Large-Scale Enstrophy since h are periodic. Thus Q is also conserved. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

20 q + v q = 0 (1.88) t Conservation of Generalized Enstrophy Since q is conserved along the particle trajectory, a good guess is that the integral of any function of q, ˆi.e G q, is conserved. In particular, q2 is called potential enstrophy, whereas G (q). 2 is called the enstrophy. The quantities G q are called the generalized enstrophies. We will see the guess is true if = 0. But not necessarily true if 0. [0,2π] [0,2π] Indeed d G q = G q q dt t = = = = = = y=2 2 0 v q G q v G q div vg q (1.89) v n G q x G + h + y + y=0 x G + h x x 0 t G x 0 t + h x G x 0 t + h x 0 dx Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

21 Conservation of Generalized Enstrophy ˆ d G (q) = dt [0,2π] [0,2π] ˆ ˆ 2π 0 [0,2π] [0,2π] G (q). ψ x (x, 0, t) [G ( ψ (x, 0, t) + h (x, 0) + 2πβ) G ( ψ (x, 0, t) + h (x, 0)) dx]. The generalized enstrophy is conserved if and only if one of the following is true β = 0, i.e. there is no beta-effect; G is periodic with period 2πβ; ψ x (x, 0, t) 0, i.e. there exists special symmetry in the flow. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

22 Case II: Subtle issues in periodic geometry if 0. Summary In this case, of far fewer conserved quantities: exist. periodic geometry β 0 Sub-case A: 0 h 0 V t = 0. (i) Energy: E (ii) Enstrophy: (iii) Non-robust conserved quantities: 2 2 G q dx dy, if G q is periodic with period Sub-case B: 0 h 0 V t 0. We neglect the large mean equation in this case, since we have an independent dynamics for the small scale. (i) Single robust conserved quantity energy: E (ii) Neither enstrophy nor potential enstrophy 1 2 q 2 are conserved. (iii) Non-robust conserved quantities: 2 2 G q dx dy, provided G q is periodic with 0 0 period 2. Sub-case C: 0 h 0 V t 0. (i) Total mean energy: E 1 V (ii) Large-scale enstrophy: V t h 2. 2 (iii) Non-robust conserved quantities: 2 2 G q dx dy, provided G q is periodic with 0 0 period 2. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

23 Channel domain geometry Domain is given by the regime: 0 < x < 2π, 0 < y < π. Boundary conditions: v (x + 2π, y, t) = v (x, y, t), v 2 (x, π, t) = v 2 (x, 0, t) = 0. The solid wall boundary condition implies that the stream function satisfies: ψ (x, π, t) = A (t), ψ (x, 0, t) = B (t) = 0. The most general stream function satisfying the boundary condition has the form ψ (x, y) = Vy + ψ (x, y), ψ y=0,ψ = 0, where V = A/π. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

24 Channel domain geometry Domain is given by the regime: 0 < x < 2π, 0 < y < π. Boundary conditions: v (x + 2π, y, t) = v (x, y, t), v 2 (x, π, t) = v 2 (x, 0, t) = 0. The solid wall boundary condition implies that the stream function satisfies: ψ (x, π, t) = A (t), ψ (x, 0, t) = B (t) = 0. The most general stream function satisfying the boundary condition has the form ψ (x, y) = Vy + ψ (x, y), ψ y=0,ψ = 0, where V = A/π. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

25 Connection with the periodic geometry We extend ψ oddly in the y direction to the box [0, 2π] [ π, π], i.e. { ψ ψ (x, y), for y > 0, (x, y) = ψ (x, y), for y 0. Both ψ and the small-scale velocity ψ are periodic functions in x and y thanks to the odd extension. The Fourier representation of ψ cannot contain even terms in y ψ (x, y) = (a jk cos (jx) + b jk sin (jx)) sin (ky). k 1 j Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

26 q = + h + y = V t y + Barotropic quasi-geostrophic equations in a channel with topographic stress q d q = + J + q h + = y 0 t dt V t = V t y = + h (1.105) x q d + J q = 0 t dt V t = h (1.105) We note here that the area for the domain is 2 2 and hence x 2 2 is now the one utilized in the normalized integral in the dynamic equation for the large mean V. We Fromnote ourhere previous that the discussions area for the indomain Subsections is and henceon 2 conserved 2 is now the quantities, it in follows the normalized that the system integral (1.105) in thehas dynamic the following: equation for the large mean V. one utilized From our previous discussions in Subsections on conserved quantities, Conserved it follows quantities that the system (1.105) has the following: (i) Total mean energy Conserved quantities E = 1 (i) Total mean energy 2 V 2 t (1.106) 2 (ii) Large-scale enstrophy E = 1 2 V 2 t (1.106) 2 Q = V t h 2 (1.107) (ii) Large-scale enstrophy 2 (iii) Generalized enstrophies Q = V t h 2 (1.107) 2 G q (1.108) (iii) Generalized enstrophies for an arbitrary function G q, where q = + h + y. G q (1.108) Di The Qi, and proofs Andrew J. are Majda pretty (CIMS) much the same Conserved as in Quantities the periodic case and the conservation Sept. 22, / 40

27 1.4 Barotropic geophysical flows in a channel domain 47 Conservation of extrema of the potential vorticity potential vorticity. This is physically reasonable: since the potential vorticity is advected along the stream lines, it is periodic in the zonal x direction and the vertical boundaries are impermeable. Mathematically we let m be the maximum of the initial potential vorticity in the channel m = max q 0 (1.109) and we define G q = 0 { q m 2 if q m if q<m (1.110) Then G is differentiable and hence = G q is conserved. Notice that 0 = 0 since m is the maximum of the initial potential vorticity. Thus t = 0 for all t, which further implies q t m (1.111) for all t. On the other hand, for any > 0 we may define { q m + 2 if q m G q = 0 if q<m (1.112) Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

28 For any ɛ > 0 for all t. On the other hand, for any > 0 we may define G q = 0 { q m + 2 if q m if q<m (1.112) Then G is differentiable and hence = G q is conserved. Notice that 0 > 0 since m is the maximum of the initial potential vorticity and > 0. Thus t > 0 for all t, which further implies for all t and all > 0. Hence which leads to which leads to max q t m (1.113) max q t m (1.114) max q t = m = max q 0 (1.115) The conservation of the minimum can be verified the same way. max q (, t) = m = max q (, 0). Note that this argument also shows that we have conservation of the extrema for general periodic flows with = 0. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

29 The impulse and conserved quantities The impulse, I, defined by 48 Barotropic geophysical flows and two-dimensional fluid flows Note that I has the units of I = velocity. yω, Physically, where ω= ψ the impulse. represents the mean (zonal) x-momentum induced by the relative vorticity in the fluid. The physical interpretation will be clear after our discussion below. For 0, the standard choice of the conserved potential enstrophy, G q = 1 2 q2, yields the conserved quantity 1 + h 2 + I (1.117) 2 since q = +h+ y. The difference of this conserved quantity and the conserved large-scale enstrophy Q yields the fact that for 0 V t I t is conserved in time (1.118) Thus, the production of mean x-momentum (horizontal flux rate) in time is exactly associated with the changes of impulse in time. In particular, for a zonal The topography, impulse represents the impulse the mean x-momentum (horizontal flux rate) induced by the relative vorticity in the fluid. I is conserved in time, if h = h y (1.119) Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

30 The impulse and conserved quantities The 48 impulse, I Barotropic, defined geophysical by flows and two-dimensional fluid flows Note that I has the units of I = velocity. yω, Physically, where ω= ψ the impulse. represents the mean (zonal) x-momentum induced by the relative vorticity in the fluid. The physical interpretation will be clear after our discussion below. For 0, the standard choice of the conserved potential enstrophy, G q = 1 2 q2, yields the conserved quantity 1 + h 2 + I (1.117) 2 since q = +h+ y. The difference of this conserved quantity and the conserved large-scale enstrophy Q yields the fact that for 0 V t I t is conserved in time (1.118) Thus, the production of mean x-momentum (horizontal flux rate) in time is In exactly particular, associated for a zonal with topography, the changes of impulse in time. In particular, for a zonal topography, the impulse I, is conserved in time, if h = h (y). I is conserved in time, if h = h y (1.119) Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

31 topography. Conservation of circulation Conservation of circulation The circulation, Γ, defined by Another interesting quantity forˆ the channel is the so-called circulation, which is defined as Γ = ω, where ω= ψ. = where = (1.120) This measures the total vorticity (or the global rotation of the fluid) in the domain. It is called circulation since ( = v 1 y + v ) 2 = n x 2 v 1 + v 2 n 1 ds = v ds = top + bottom where is the unit counter-clockwise tangent vector at the boundary of the channel and top = y= v 1 dx, bottom = y=0 v 1 dx. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

32 Outline 1 Review: Some special exact solutions 2 Conserved quantities Conservation of energy and enstrophy Barotropic geophysical flows in a channel domain 3 Variational derivatives and an optimization principle 4 More equations for geophysical flows Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

33 Variational Derivatives For an ordinary differential function F from R N to R, the directional derivatives satisfy F (x + ɛy) F (x) def. lim = F (x) y. ɛ 0 ɛ Given a functional, F (u), from a Hilbert space H to R, the variational derivative of F, denoted as δf δu, is the function satisfying F (u + ɛδu) F (u) def. δf lim = ɛ 0 ɛ δu, δu, for any δu. The inner product space will be ordinary integrable functions on a two-dimensional domain f, g = (f, g) 0 = fg. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

34 the vorticity. Recall that for periodic functions,, the (small-scale) mean kinetic Some energy important is the quadratic variational functional derivatives There are some elementary variational derivatives which play an important role. E = 1 v 2 = 1 2 = 1 (1.133) Kinetic where = Energy. Given a directional variation E (ω) = 1, define v 2 = 1 by ψ 2 = 1 ψω. 2 2 = 2 (1.134) We calculate explicitly that E + E = (1.135) 2 2 so that E + E lim = 1 + = = = 0 Other important functionals, where it is easy to calculate the variational deriva- with the inner product in (1.131). Thus, with the definition in (1.132), we have the formula δe δω E = ψ, ψ = ω = = (1.136) Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

35 with the inner product in (1.131). Thus, with the definition in (1.132), we have the formula E = = (1.136) 2. Generalized Enstrophy Other important functionals, where it is easy to calculate the variational derivatives, are given by = G (1.137) for a differentiable function G. Here it is easy to see that + lim = G (1.138) 0 so that = G (1.139) The two elementary formulas of functional derivatives calculated above in (1.136) and (1.139) will occur throughout the book. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

36 An optimization principle for elementary geophysical solutions We consider the averaged enstrophy E (ω) = 1 2 (ω + h) 2, and ask the question: Which functions, ω, minimize the enstrophy for a fixed value of the energy, E (ω) = E 0? According to the Lagrange multiplier principle, the function ω should satisfy the Euler-Lagrange equation δe where µ is a Lagrangian multiplier. The Euler-Lagrange equation becomes δω = µδe δω, ψ + h = µψ. Note that, as in ordinary calculus, not every solution is necessarily a minimize of the enstrophy, and some of the solutions are saddle points. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

37 An optimization principle for elementary geophysical solutions We consider the averaged enstrophy E (ω) = 1 2 (ω + h) 2, and ask the question: Which functions, ω, minimize the enstrophy for a fixed value of the energy, E (ω) = E 0? According to the Lagrange multiplier principle, the function ω should satisfy the Euler-Lagrange equation δe where µ is a Lagrangian multiplier. The Euler-Lagrange equation becomes δω = µδe δω, ψ + h = µψ. Note that, as in ordinary calculus, not every solution is necessarily a minimize of the enstrophy, and some of the solutions are saddle points. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

38 An optimization principle for elementary geophysical solutions We consider the averaged enstrophy E (ω) = 1 2 (ω + h) 2, and ask the question: Which functions, ω, minimize the enstrophy for a fixed value of the energy, E (ω) = E 0? According to the Lagrange multiplier principle, the function ω should satisfy the Euler-Lagrange equation δe where µ is a Lagrangian multiplier. The Euler-Lagrange equation becomes δω = µδe δω, ψ + h = µψ. Note that, as in ordinary calculus, not every solution is necessarily a minimize of the enstrophy, and some of the solutions are saddle points. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

39 Outline 1 Review: Some special exact solutions 2 Conserved quantities Conservation of energy and enstrophy Barotropic geophysical flows in a channel domain 3 Variational derivatives and an optimization principle 4 More equations for geophysical flows Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

40 The models F -plane equations q t + J (ψ, q) = D ( ) ψ + F, q = ψ F 2 ψ + h + βy, where F = L/L R with L the characteristic horizontal length, L R = gh 0 /f 0 the Rossby deformation radius. Continuously stratified quasi-geostrophic equations ( ) ( ) ψ + F 2 2 ψ t z 2 + J ψ, ψ + F 2 2 ψ z 2 + β ψ x = Dψ + F, where ψ (x, y, z, t) is defined in the domain [0, 2π] [0, 2π] [0, 2πΘ]. Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

41 The two-layer model Two-layer model ψ1 ( ψ1 F1 (ψ1 ψ2 )) + J (ψ1, ψ1 F1 (ψ1 ψ2 )) + β = F, t x ψ2 ( ψ2 + F2 (ψ1 ψ2 )) + J (ψ2, ψ2 + F2 (ψ1 ψ2 )) + β = r ψ2 + F. t x Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

42 in the derivation below. Without loss of generality, we assume = 1 and we approximate the stream function with the first two modes in the vertical Fourier expansion Relationships between various models x y z t = b x y t + t x y t 2 sin z (1.151) Substitute this into the continuously stratified model (1.146) and notice for the Jacobian we have ( ) J + F 2 2 z 2 = J b + t 2 sin z b + t F 2 t 2 sin z = J b b + 2 sin z J t b +J b t F 2 t + 2 sin 2 zj t t F 2 t The projection of the right-hand side on to the first two vertical Fourier bases 1 2 sin z yields, following components for each modes t b + J b b + J t t F 2 t + x b = 0 (1.152) t t F 2 t + J t b + J b t F 2 t + x t = 0 (1.153) Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40

43 F + J F Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / Barotropic geophysical flows and two-dimensional fluid flows These two equations can be re-arranged (addition and subtraction) into the following The well-known two-layer model ten follows: form t b + t F 2 t + J b + t b + t F 2 t + x b + t = 0 (1.154) t b t + F 2 t + J b t b t + F 2 t + x b t = 0 (1.155) The well-known (inviscid) two-layer model then follows, once we define the stream function i for each layer as follows and the two-layer model takes the form 1 = b + t 2 = b t (1.156)

44 Questions & Discussions Di Qi, and Andrew J. Majda (CIMS) Conserved Quantities Sept. 22, / 40