A Three-Dimensional Wave Activity Flux Applicable to Inertio-Gravity Waves. Saburo Miyahara Kyushu University 2 March 2011 Honolulu, Hawaii

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1 A Three-Dimensional Wave Activity Flux Applicable to Inertio-Gravity Waves Saburo Miyahara Kyushu University March 011 Honolulu, Hawaii

2 Outline 1. Motivation. Theoretical derivation of 3-D flux and 3-D Transformed-Eulerian-Mean (TEM) Equation system, and physical interpretations 3. An example of application of the 3-D flux to non-geostrophic wave disturbances 4. An extension of the theory

3 1. Motivation of this study An Example of global distributions of gravity waves; GPS/MET and GCM Kawatani et al., 003, GRL. Non-uniformly distributed in zonal direction E P flux analysis is not suitable. A three-dimensional analysis is appropriate, and it is desirable to define a 3-D flux applicable to IGWs.

4 . Derivation of 3-D flux and 3-D TEM equation system Miyahara, 006: A three dimensional wave activity flux applicable to inertio-gravity waves, SOLA,. # Three dimensional non-geostrophic Boussinesq equation system # Define a 3-D wave activity flux and a 3-D Transformed Eulerian-Mean (3-D TEM) equation system that are applicable to non-geostrophic disturbances We consider non-geostrophic disturbances in a time mean flow; u, v, and w.

5 Time mean hydrostatic Boussinesq equation system Du Dt Dv Dt fv = u u u v y u w + fu = y u v v v y v w = r u + v y + w = 0 Dr Dt w = r u r v y r w D Dt = t + u + v y + w r = ρ ρ m ρ m g : the buoyancy = Reynolds stress terms can be rearranged by introducing partial differentials of S 1 u + v + 1 u + v + w w r : r 1 r u v w + u v y u w = y 1 r u + v + w y u v y 1 v u w + r (Kinetic energy) (Available potential energy). v w

6 Rearranged time mean hydrostatic Boussinesq equation system Du Dt Dv Dt fv = S 1 + fu = y S y u v = r u v y Dr Dt w = r u r v y r w u + v y + w = 0 S = 1 u + v + w r 1 w + v u r u v y u w w + : (K. E.) (A. P. E.) r v w

7 3-D residual circulation 3-D TEM equation system u * = u + u r + 1 f v * = v + v r 1 f w * = w S y S u r v r y Du Dt Dv Dt fv * = F ( ) x + fu * = y F ( ) y Dr Dt w * = r w Flux Tensor F: F 1 u v w + u v r u v 1 v u w + r u w + f v w f v r u r

8 3. Physical interpretations of the flux F and the 3-D residual circulation Weak shear limit: local plane wave solution of an inertio-gravity wave a ( x, y,z,t) = A X,Y,Z,T { ( )} ( )exp i kx + ly + mz ωt X, Y, Z, T: slowly varying variables D Dt = t + u + v y + w = iω + iku + ilv + imw i ω ˆ ˆ ω = ω ku lv mw : local intrinsic frequency Shears of the time mean flow are neglected, and only the Doppler shift effect is taken into account: A local plane wave solution and a local dispersion relation will be derived from disturbance equations.

9 Disturbance equations Simultaneous equations for the plane wave Du Dt fv = φ Dv Dt + fu = φ y Dw Dt Dr = φ r Dt w = 0 u + v y + w = 0 The dispersion relation of IGW ( ω ˆ = k + l ) + f m k + l + m iω ˆ U fv = ikφ iω ˆ V + fu = ilφ iω ˆ W = imφ R i ˆ ω R W = 0 iku + ilv + imw = 0 The local plane wave solution U = k ω ˆ + ilf ω ˆ f Φ, V = l ω ˆ ikf ω ˆ f Φ W = mω ˆ ω ˆ N Φ, R = im ω ˆ N Φ

10 Intrinsic Group velocity c ˆ gx ω ˆ k = c ˆ x c ˆ gy ω ˆ = c ˆ l y c ˆ gz ω ˆ m = c ˆ z ω ˆ k + l + m = c u gx ω ˆ k + l + m = c gy v f ˆ ω k + l + m = c gz w Intrinsic phase velocity ω c ˆ x ˆ k = c u l x k v m k w ω c ˆ y ˆ l = c v k y l u m l w ω c ˆ z ˆ m = c z w k m u l m v Wave energy E = 1 u + v + w + r = 1 ( k + l )( f ) ω ˆ ( ω ˆ f ) ω ˆ ( ) Φ

11 3.1. Physical interpretation of the 3-D flux F F 1 u v u v w + r u v 1 v u w + u w + f F denotes zonal and meridional fluxes of eastward and northward pseudo-momentum relative to the local time mean flow. r v w f v r u r E E E c ˆ gx c ˆ c ˆ gy c ˆ x c ˆ gz x c ˆ ( c gx u ) Ḙ c x c gy v x E E E = c ˆ gx c ˆ c ˆ gy c ˆ y c ˆ gz y c ˆ = c gx u y ( ) Ḙ ( c c gz w ) Ḙ x c x ( ) Ḙ ( c c gy v ) Ḙ ( c y c gz w ) Ḙ y c y

12 3.. Physical interpretation of 3-D residual circulation 3-D residual circulation u * = u + u r + 1 f v * = v + v r 1 f w * = w where S y S u r v r y S = 1 u + v + w r Singular at the equator? 3-D TEM Du Dt Dv Dt fv * = F ( ) x + fu * = y F ( ) y Dr Dt w * = r w Does it have same meaning as -D case? In case of plane waves of Inertio Gravity vaves S = 1 ( k + l ) f ω ˆ f 1 lim f 0 f ( ) Φ, S y = k + l ω ˆ f y Φ 1, lim f 0 f S = 0 Not singular at the equator at least for Inertio-Gravity Waves

13 ξ = ( ξ η ς ) ( ξ η ς ) :Particle displacement vector from a mean position : Cartesian components of the particle displacement vector, respectively Dξ Dt = ul = u + ξ u + η u y + ς u Dη = v L = v + ξ v Dt + η v y + ς v Dς Dt = w L = w + ξ w + η w y + ς w Weak shear limit Dξ Dt u Dη Dt v Dς Dt w ξ iˆ u ω η iˆ v ω ς iˆ w ω The Stokes drift (weak shear limit) u S ξ u + η u y + ς u = ξ u + η u y + ς u v S ξ v + η v y + ς v = ξ v + η v y + ς v w S ξ w + η w y + ς w = ξ w + η w y + ς w

14 u S ξ u + η u y + ς u = 1 f ( k + l ) Φ ( ω ˆ f ) y + 1 mlf ω ˆ ω f ( )( ˆ ) Φ u * = u + 1 f S y + u r u + u S 1 f S y + u r = 1 f ( k + l ) ( ω ˆ f ) y + 1 mlf ω ˆ ω f ( )( ˆ ) u * = u + 1 f v * = v 1 f w * = w S y + S + u r u + u S v r v + v S u r v r w + w S y 3-D residual circulation v * v + v S (Eulerian-time-mean) + (Stokes drift) Approximately equal to a 3-D time mean tracer transport velocity for non-dissipative, non-transient, and adiabatic linear plane wave disturbances, similar to -D TEM case.

15 3. 3. lnp-spherical coordinate system 1 θ u v + α u v u w Ωsinϕ v θ θ θ 1 θ F u v v u + α v w + Ωsinϕ u θ pacosϕ θ θ E E E c ˆ gλ c ˆ c ˆ gϕ c ˆ λ c ˆ gz λ c ˆ λ E = E E c ˆ gλ c ˆ c ˆ gϕ c ˆ ϕ c ˆ gz pacosϕ ϕ c ˆ ϕ α = R H exp( κz / H)

16 u t + v t + θ t + u u acosϕ λ + u v acosϕ λ + v a v acosϕ ϕ u cosϕ ( ) + w u Ωsinϕv * = 1 φ acosϕ λ pacosϕ ( ) 1 F v ϕ + w u + u a tanϕ + Ωsinϕu * = 1 φ a ϕ ( pacosϕ ) 1 ( F) ϕ u θ acosϕ λ + v θ a ϕ + w * θ = G ( ) λ ( ) = 1 ( ) acosϕ λ + 1 acosϕ ( cosϕ) + 1 ϕ p ( p) 1 1 u * = u + Ωsinϕ ϕ ( u + v ) 1 cos ϕ 1 ϕ α θ cosϕ θ 1 pu θ p θ v * = v θ u + v α Ωsinϕ acos ϕ λ θ cosϕ 1 pv θ p θ 1 u θ w * = w + acosϕ λ θ + 1 pv θ cosϕ acosϕ ϕ θ

17 4. Application of the 3-D flux to QBO analysis by a high resolution GCM (T13L56) Kawatani et al., 010, JAS r f u v + uv uw + vr N N 1 r f F = uv v u + vw ur N N ( F ) x Averaged over 10º S 10º N

18 Gravity waves Zonal wavenumber s 1 Averaged over 10º S 10º N

19 5. An extension of the 3-D flux TEM equation system u t + v u * y + w u * fv * = y 3-D TEM equation system u t + u u + v u y + w u u v + u fv * = v r u w + f u v r y 1 u v w + r y [ u v ] u w + f Kinoshita et al., 010: On the three-dimensional residual mean circulation and wave-activity flux of the primitive equations. JMSJ, 88. u t + u u * + v u * y + w u * fv * = where S 1 u + v + w r 1 u v w + u v u y S f + u r + u y S f + u v r u w u v r v r v r N + f u v r y Unsolved issue: The flux and the residual circulation become singular in case of S 0 at the equator, for example anti-symmetric equatorial modes. (Horinouchi, personal communication)

20 6. Conclusions A 3-D flux and a 3-D Transformed-Eulerian-Mean (3-D TEM) equation system are derived. These are applicable to non-geostrophic disturbances. Physical meanings are also discussed, and it is shown that they have similar meanings to the EP flux and the TEM equation system.

21 Miyahara, 006: A three dimensional wave activity flux applicable to inertio-gravity waves, SOLA, , and Errata. Free PDF file is available.