Modeling Large-Scale Atmospheric and Oceanic Flows 2

Transcription

1 Modeling Large-Scale Atmospheric and Oceanic Flows V. Zeitlin known s of the Laboratoire de Météorologie Dynamique, Univ. P. and M. Curie, Paris Mathematics of the Oceans, Fields Institute, Toronto, 3 jet

2 Plan known s of the jet of dry and moist instabilities known s of the jet

3 General problematics Importance of moisture in the atmosphere: obvious. Influences large-scale dynamics via the latent heat release, due to condensation and precipitation. Atmospheric circulation ing: equation of state of the moist air extremely complex. Discretization/averaging: problematic. Current parametrizations of precipitations and latent heat release: relaxation to the equilibrium (saturation) profile of humidity threshold effect essential nonlinearity Consequences: no linear limit; linear thinking: modal decomposition, linear stability analysis, etc impossible problems in quantifying predictability of moist - convective dynamical systems. known s of the jet

4 Aims and method I Aim: Understanding the influence of condensation and latent heat release upon large-scale dynamical processes Reminder: Simplest for large-scale motions: rotating shallow water. Link with primitive equations: vertical averaging Baroclinic effects: (or more) layers. Problem with this approach for moist air: averaging of essentially nonlinear equation of state. known s of the jet

5 Aims and method II Our approach Combine (standard) vertical averaging of primitive equations between the isobaric surfaces with that of Lagrangian conservation of moist enthalpy Allow for convective fluxes (extra vertical velocity) across the isobars Link these fluxes to condensation Use relaxation parametrization in terms of bulk moisture in the layer for the condensation/precipitation known s of the jet

6 Aims and method III Advantages: Simplicity, qualitative analysis of basic phenomena straightforward Fully nonlinear in the hydrodynamic sector Well-adapted for studying discontinuities, in particular precipitation s Efficient numerical tools available (finite-volume codes for shallow water) Various limits giving known s Inclusion of topography (gentle or steep) straightforward known s of the jet

7 Primitive equations in pseudo-height coordinates d dt v + f k v = φ d dt θ = v + z w = known s of the z φ = g θ θ v = (u, v) and w - horizontal and vertical velocities, d dt = t + v + w z, f - Coriolis parameter, θ - potential temperature, φ - geopotential. jet

8 Moisture and moist enthalpy Condensation turned off: conservation of specific humidity of the air parcel: d dt q =. Condensation turned on: θ and q equations acquire source and sink. Yet the moist enthalpy θ + L c p q, where L - latent heat release, c p - specific heat, is conserved for any air parcel on isobaric surfaces: d dt ( θ + L c p q ) =, known s of the jet

9 Vertical averaging with convective fluxes 3 material surfaces: w = dz dt, w = dz + W, w = dz + W. dt dt W known s z of the θ θ W z z jet Mean-field + constant mean θ

10 Averaged momentum and mass conservation equations: { t v + (v )v + f k v = φ(z ) + g θ θ z, t v + (v )v + f k v = φ(z ) + g θ { t h + (h v ) = W, t h + (h v ) = +W W, known s of the θ z + v v h W, jet

11 Linking convective fluxes to precipitation I Bulk humidity: Q i = z i z i qdz. Precipitation sink: t Q i + (Q i v i ) = P i. In precipitating regions (P i > ), moisture is saturated q(z i ) = q s (z i ) and the temperature of the air-mass W i dtdxdy convected due to the latent heat release θ(z i ) + L c p q s (z i ), is the one of the upper layer: θ i+. We assume "dry" stable background stratification: θ i+ = θ(z i ) + L c p q(z i ) θ i + L c p q(z i ) > θ i, with constant θ(z i ) and q(z i ). known s of the jet

12 Integrating the moist enthalpy we get W i = β i P i with a positive-definite coefficient β i = L c p (θ i+ θ i ) q(z i ) >. Last step: relaxation formula with relaxation time τ. known s of the P i = Q i Q s i τ H(Q i Q s i ) where H(.) is the Heaviside (step) function. jet

13 -layer with a dry upper layer Vertical boundary conditions: upper surface isobaric z = const, geopotential at the bottom constant (ground) φ(z ) = const, Q =, Q = Q: t v + (v )v + f k v = g (h + h ), t v + (v )v + f k v = g (h + αh ) + v v h βp, t h + (h v ) = βp, t h + (h v ) = +βp, t Q + (Qv ) = P, P = Q Qs τ H(Q Q s ) α = θ θ - stratification parameter. known s of the jet

14 Sketch of the θ W h known s of the θ P> h jet

15 Immediate relaxation limit τ, P = Q s v (Gill, 98), and t v + (v )v + f k v = g (h + h ), t v + (v )v + f k v = g (h + αh ) v v h βq s v, t h + (h v ) = +βq s v, t h + (h v ) = βq s v, humidity staying at the saturation value: Q = Q s. known s of the jet

16 Baroclinic reduction Rewriting the in terms of and barotropic velocities: v bt = h v + h v h + h, v bc = v v, and linearizing in the hydrodynamic sector gives: t v bc + f k v bc = g e η, t η + H e v bc = βp, t Q + Q e v bc = P, where g e = g(α ), Q e = He H Q s, η - perturbation of the interface, H e - equivalent height. Model first proposed by Gill (98) and studied by Majda et al (4, 6, 8)., known s of the jet

17 Quasigeostrophic limit In the small Rossby number limit on the β-plane Lapeyre & Held (4) follows: d () ( ψ + y η ) = βp, dt D D d () ( ψ + y η ) = βp, dt D D Here d () i dt = t + (v () i ), k v () i = ψ i, D i = H i H, and ψ, (geostrophic streamfunctions) are related to the free-surface (η ) and interface (η ) perturbations as: ψ = η + η, ψ = η + αη. known s of the jet

18 -layer moist-convective RSW In the limit H /(H + H ) the reduced-gravity one-layer moist-convective shallow water follows (Bouchut, Lambaerts, Lapeyre & Zeitlin, 9): t v + (v )v + f k v = η, t η + {v ( + η)} = βp, t Q + (Qv ) = P, (Nondimensional equations, η - free-surface perturbation) known s of the jet

19 Horizontal momentum ( t + v )(v h ) + v h v + f k (v h ) = g h gh h v βp, ( t + v )(v h ) + v h v + f k (v h ) = αg h gh h + v βp, Red: moist convection drag. Total momentum: v h + v h is not affected by convection. known s of the jet

20 Energy Energy densities of the layers: { e = h v + g h, e = h v + gh h + αg h, For the total energy E = dxdy(e + e ) we get: t E = dx βp (gh ( α) + (v v ) ). st term: production of PE (for stable stratification); nd term destruction of KE. known s of the jet

21 Potential vorticity ( t + v ) ζ + f h ( t + v ) ζ + f h = ζ + f h = ζ + f h βp, βp + k { ( v v βp h h where ζ i = k ( v i ) = x v i y u i (i =, )- relative vorticity. PV in each layer is not a Lagrangian invariant in precipitating regions. known s )}, of the jet

22 Moist enthalpy and moist PV Moist enthalpy in the lower layer: m = h βq and is always locally conserved: t m + (m v ) =. Conservation of the moist enthalpy in the lower layer allows to derive a new Lagrangian invariant, the moist PV: ( t + v ) ζ + f m =. known s of the jet

23 Quasilinear form and characteristic equations -d reduction: y (...) =, quasilinear system: t f + A(f ) x f = b(f ). Characteristic equation: det(a ci) = "Dry" characteristic equation } } F(c) = {(u c) gh {(u c) αgh gh gh =, "Moist" characteristic equation (τ ) F m (c) = F(c) + ((u u ) (α )gh )gβq s =. known s of the jet

24 Characteristic velocities about the rest state "Dry" characteristics: "Moist" characteristics: Here C = c and C ± = g(h + αh ) ±, C± m = g(h + αh ) ± m. = 4H H (α ) (H + αh ) = (H αh ) + 4H H (H + αh ). m = + 4(α )βqs H (H + αh ). known s of the jet

25 characteristic velocities c m is real for positive moist enthalpy of the lower layer in the state of rest : M = H βq s >, and C m < C < g(h + αh ) < C + < C m +, for < M < H moist internal (mainly ) mode propagates slower than the dry one, consistent with observations. known s of the jet

26 Discontinuities in dependent variables (no rotation) Rankine-Hugoniot (RH) conditions (immediate relaxation): s[v h + v h ] + [u v h + u v h ] =, s[m ] + [m u ] =, s[h ] + [h u + βq s u ] =. s - propagation speed of the discontinuity. Remark: mass conservation moist enthalpy conservation in the lower layer. b Due to lim xs a lim b xs a P =, P does not enter RH conditions for u, v, h. known s of the jet

27 Discontinuities in derivatives RH conditions linearized about the rest state: { (s C + )(s C )[ x u ] = (α )gh gβ[p], (s C m + )(s C m )[ x u ] = s(α )gh gβ[ x Q]. For a configuration where it rains at the right side of the discontinuity, P = and P + = Q s x u + >, there exist five types of precipitation s: known s of the jet

28 Precipitation s. the dry external s, C + < s < C m +,. the dry internal subsonic s, C m < s < C, 3. the moist internal subsonic s, C m < s <, 4. the moist internal supersonic s, C + < s < C, 5. the moist external s, s < C m +. This result confirms previous studies within a linear (Frierson et al, 4). known s of the jet

29 Wave scattering on a moisture : setting Localized internal simple wave centred at x P = and moving eastward: { u (x, ) = σ(x x P ) U + U if σ x x U P otherwise, U =., σ = () Stationary moisture at x M = 5, saturated air at the east, unsaturated at the west: Q(x, ) = Q s { + q tanh(x x M )H( x + x M )}, q =.5. () Strong downflow convergence in the lower layer P > near the moisture. σ, known s of the jet

30 t t Wave scattering on a moisture : velocity and moisture x known s of the u bc x Q x jet

31 t t t t Wave scattering on a moisture : condensation zone I s s r i x r m i x x r i x x Dry and moist internal Riemann invariants. s, - precipitation s (dry subsonic and moist supersonic) r m i x x known s of the jet

32 Characteristics and s in the condensation zone t Q<Qs Q=Qs known s S S3 of the P= P> P= Cd Cm Q<Qs Cq Q=Qs S x jet

33 Evaporation and its parametrizations In the presence of evaporation source E t Q + (Qv ) = E P Hence: t m + (m v ) = βe Simple parametrizations of E (may be combined): Relaxational: E = ˆQ Q τ E H(m ), where ˆQ - equilibrium value. Dynamic: E = α E v H(m ) m should stay positive (plays a role of static stability) known s of the jet

34 Baroclinic Bickley jet Geostrophically balanced upper-layer jet on the f -plane. non-dimensional profiles of velocity and thikness perturbations: ū =, η = α tanh(y), ū = sech (y), η = α tanh(y). No deviation of the free surface: η + η =. Parameters: Ro =., Bu = - typical for atmospheric jets. known s of the jet

35 u i /U η i /H PV i /fh y/r d 3 3 y/r d 3 3 y/r d Bickley jet: zonal velocity ū i, thickness deviation η i and PV anomaly. Lower (upper) layer: solid black (dashed gray). known s of the jet

36 Linear stability diagram c p σ k k Phase velocity (top) and growth rate (bottom) known s of the jet

37 The most unstable mode y/r d y/r d x/r d x/r d Most unstable mode of the upper-layer Bickley jet. Upper(top) and lower (bottom) layer- geostrophic streamfunctions and velocity (arrows) fields. ψ ψ known s of the jet

38 Early stages: evolution of moisture.5 (a) (c) T i T i x (b) (d) T i 3T i Evolution of the moisture anomaly Q Q with superimposed lower-layer velocity. Black contour: condensation zones x known s of the jet

39 Early stages: growth rates σ/ro Dry Moist known s of the t/t i Red: moist, blue: dry simulations. Transient increase in the growth rate due to condensation. jet

40 Cyclone-anticyclone asymmetry <ζ 3 >/<ζ > 3/ <ζ 3 >/<ζ > 3/ t/t i 6 4 UPPER LAYER LOWER LAYER 5 5 t/t i Skewness of relative vorticity. Red: moist, blue: dry simulations. known s of the jet

41 How condensation enhances cyclones: -layer For Ro and Bu O(), close to saturation ψ q << : [ ] ( t + v () ) ψ ψ = βp, (3) [ ] ( t + v () ) q Q s ψ = P, (4) v () = ( y ψ, x ψ) - geostrophic velocity, ψ = η + η, and q is moisture anomaly with respect to Q s. PV of the fluid columns which pass through the precipitating regions increases. For τ q, and: [ ] Q s ( t + v () ) ψ P τ >, (5) increase of geostrophic vorticity in the precipitation regions. known s of the jet

42 Dry vs moist simulations: evolution of relative vorticity (a) Dry (b) Moist (c) T i (d) T i known s of the (e) 5T i (f) 5T i (g) 3T i 35T i (h) 3T i 35T i jet Lower layer: colors, upper layer: contours. Condensation: solid black.

43 Dry vs moist simulations: formation of secondary zonal jets at late stages Dry.5 Moist known s of the 4T i T i jet 8T i T i.5.5.5

44 x Unbalanced (aheostrophic) motions: divergence (a) Dry (b) Moist (c) T i x 3 (d) T i known s of the 5T i 5T i (e) (f) (g) 3T i (h) 3T i 35T i 35T i jet

45 Moist in Nature known s of the jet

46 The Physically and mathematically consistent Simple, physics transparent Efficient high-resolution numerical schemes available Benchmarks: good known s of the jet

47 local enhancement of the growth rate of the moist-convective at the precipitation onset, significant increase in intensity of ageostrophic motions during the evolution of the moist, substantial cyclone - anticyclone asymmetry, which develops due to the moist convection effects. substantial differences in the structure of zonal jets resulting at the late stage of saturation. known s of the jet

48 References Presentation based on: Lambaerts J.; Lapeyre G.; Zeitlin V. and Bouchut F. "Simplified two-layer s of precipitating atmosphere and their properties" Phys. Fluids 3, 4663,. Lambaerts J.; Lapeyre G. and Zeitlin V. "Moist versus Dry Baroclinic Instability in a Simplified Two-Layer Atmospheric Model with Condensation and Latent Heat Release" J. Atmos. Sci. 69, 45-46,. known s of the jet