Simple Mathematical, Dynamical Stochastic Models Capturing the Observed Diversity of the El Niño-Southern Oscillation

Transcription

1 Simple Mathematical, Dynamical Stochastic Models Capturing the Observed Diversity of the El Niño-Southern Oscillation Lectures 2 and 3: Background and simple ENSO models 14 september 2014, Courant Institute A.J. Majda, N. Chen and S. Thual 1

2 Outline 1. Introduction to The El Nino Southern Oscillation 2. Simple Model Overview 3. Ocean Dynamics 4. SST Thermodynamics 5. Atmosphere Dynamics 6. Coupled Dynamics 7. Model Linear Solutions 8. Model with Nonlinear Thermocline Feedback 9. ENSO Oscillation Mechanisms 2

3 Average Circulation of the Equatorial Pacific Introduction - Walker Circulation in the atmosphere: trade winds blowing from east to west, convection in the west and subsidence in the east - SST zonal gradient in the ocean: warm waters are advected westward to the warm pool region, and replaced by cold waters in the east region through upwelling - Thermocline depth : The thermocline surface that separates warm surface waters from cold deep waters is shallower in the east and deeper in the west 3 (KSU Web home)

4 Equatorial Pacific Conditions Introduction Normal Conditions: cold eastern Pacific and warm western Pacific, walker circulation and trade winds in the atmosphere, thermocline shallow in east and deep in west El Nino: warmer eastern Pacific, decreased walker circulation, deepened thermocline La Nina: colder eastern Pacific, enhanced walker circulation, shallower thermocline La Niña Trade winds Normal El Niño Trade winds Hereafter we define the sea surface temperatures (SSTs), winds and thermocline 4 (NOAA)

5 Sea Surface Temperatures (SST) Introduction - Sea Surface Temperatures change dramatically during El Nino and La Nina - We decompose SST (K or C) into mean T (average over all years) and anomalies T La Nina 1989 Total T+T ( C) El Nino 1998 Total T+T ( C) Deg latitude Deg longitude La Nina 1989 SST anomalies T ( C) El Nino 1998 SST anomalies T ( C) 5 (NOAA, computed over months of Jan-Mar)

6 El Niño SST anomalies 6 (NOAA)

7 Zonal Winds Introduction -On Normal conditions the zonal winds are westward (i.e. blow from east to west) in the equatorial Pacific. Those are the trade winds related to the Hadley, Walker circulations and the SST contrast between east and west. - We decompose zonal winds (m.s-1) into mean u (average over all years) and anomalies u. -During El Nino the trade winds are weakened or even reverse (u>0) Mean winds during Normal conditions Anomalies of winds (vectors) and SST (colors,c) during El Nino 7 (apollo.lsc.vsc.edu; McPhaden et al 2006)

8 Thermocline Depth Introduction - The thermocline depth is the area with maximal stratification separating the cold deep waters from the warm surface waters. It is usually estimated at the 20C isotherm. - We decompose thermocline (m) depth into mean H (average over all years) and anomalies H. Mean Ocean temperatures (C) thermocline depth Evolution of ocean temperatures during El Nino (NOAA)

9 The El Nino Southern-Oscillation Introduction Zonal winds u Thermocline SST T H Hovmollers of winds, SST and thermocline depth anomalies averaged at equator (5N-5S), as a function of longitude (deg E) and time (years): (anomalies) La Nina 1999 El Nino There is an alternance between El Nino and La Nina conditions (every 2 to 7 years) on all variables at equator. La Nina This interannual oscillation is called the El Nino Southern-Oscillation (ENSO) El Nino 1993 La Nina 1988 El Nino 1987 La Nina 1984 El Nino (Chen et al. 2017, averages 5N-5S)

10 El Nino indices Introduction -Several indices are used routinely to monitor the equatorial Pacific conditions, measuring either SST, thermocline depth or winds etc. - For example, the indice Nino3.4 SST>0.5C indicates an El Nino, while <-0.5 indicates a La Nina. The WWV is another indice that measures the heat content in the ocean. Niño3.4-SST: average of SST anomalies in east (170W-120W, 5N-5S) WWV: average of thermocline anomalies depth over basin (120E-90W, 5N-5S) Timeseries of Niño3.4-SST (C) and WWV (1014m3) 10 (NOAA)

11 Outline 1. Introduction to The El Nino Southern Oscillation 2. Simple Model Overview 3. Ocean Dynamics 4. SST Thermodynamics 5. Atmosphere Dynamics 6. Coupled Dynamics 7. Model Linear Solutions 8. Model with Nonlinear Thermocline Feedback 9. ENSO Oscillation Mechanisms 11

12 Hierarchy of ENSO Models A vast hierarchy of models of increasing complexity simulates the ENSO, for example: 1) ODE models. Empirical, reproduces the basic oscillation but no spatial structure. Includes statistical models for predictions. 2) Intermediate complexity models. Reproduces the main dynamical processes of the ENSO. Still runs on a personal laptop. E.g. the model Cane-Zebiak that made the first successful prediction in the 1980s.The simple ENSO model from the present lectures falls into this category, though it is simpler than the Cane-Zebiak. SST 3) General Circulation Models: Solve the circulation on the entire globe, including the ocean, atmosphere, sea ice etc. Very complex and expensive computationally. Typically have several bias in representing the ENSO. 12 (Jin 1997, NOAA, Zebiak and Cane 1986)

13 Simple ENSO Model We derive hereafter a simple ENSO model: -1) Includes atmosphere dynamics, ocean dynamics and an SST thermodynamic budget relevant to ENSO -2) simple coupling between the ocean, atmosphere and SST 13

14 Simple ENSO Model We derive hereafter a simple ENSO model: -1) Includes atmosphere dynamics, ocean dynamics and an SST thermodynamic budget relevant to ENSO -2) simple coupling between the ocean, atmosphere and SST: Couplings 14

15 Simple ENSO Model - The simple ENSO model Reproduces a basic ENSO oscillation with regular El Nino and La Nina events - During El Nino the eastern Pacific warms (T>0), with a deep thermocline (H>0), the trade winds weaken (u>0). La Nina is the exact opposite. - Hereafter we derive the entire model and detail those interactions. Solutions of the simple ENSO model. The Pacific extends from x=0 to x= km 15

16 Outline 1. Introduction to The El Nino Southern Oscillation 2. Simple Model Overview 3. Ocean Dynamics 4. SST Thermodynamics 5. Atmosphere Dynamics 6. Coupled Dynamics 7. Model Linear Solutions 8. Model with Nonlinear Thermocline Feedback 9. ENSO Oscillation Mechanisms 16

17 Simple ENSO Model We derive hereafter a simple ENSO model: -1) Includes atmosphere dynamics, ocean dynamics and an SST thermodynamic budget relevant to ENSO -2) simple coupling between the ocean, atmosphere and SST: 17

18 Thermocline Depth Introduction - Recall the thermocline separating the cold deep waters from the warm surface waters. Hereafter we implement a simple shallow-water model that represents its evolution. Ocean temperatures (C) thermocline depth 18 (NOAA, Clarke 2008)

19 Starting Shallow water model - Starting from momentum and continuity budget in the ocean and using common hypothesis we derive the well known shallow-water equations (see e.g. Majda , or Clarke ). - We assume a shallow upper layer of constant density on a denser resting layer: 19

20 Starting Shallow water model Coriolis force near equator Pressure due to gravity acting on layers of different densities Wind stresses at surface Momentum budget x-axis: Momentum budget Y-axis: Mass conservation:: Converge/divergence Time tendency Dissipation 20

21 Non-dimensionalization 1) We non-dimensionalize the shallow water model to units relevant to the ENSO: Dimensional First, choose units that match the scales of the ENSO: Non-dimensional Second, replace with new variables: We obtain desired orders of magnitude : U=1 corresponds to Udim=[u]=0.25 cm.s-1 21

22 Asymptotic Expansion 2) We now simplify the system removing small terms First, because we have desired orders of magnitude, we can identify small non-dimensional terms. Here: the Froude number is small other parameters are of order 1 Second, we perform an asymptotic expansion. Expand all terms in powers of the small parameters Here only retain the first order of the asymptotic expansion for simplicity, of order O(1). We obtain a simplified system with all terms of similar order: - Note that we have dropped the time derivative of V and meridional wind stress, which is typically called the long-wave approximation. This is because the Y-scale is much smaller than the x-scale. - We have also dropped small ocean dissipation (typically around 2.5 years) 22

23 Projection on Basis of Meridional Functions 3) We now further simplify the system using a projection on a suitable basis of functions on the Y-axis. See Majda 2002 for a detailed derivation. This is used in practice for solving the system. Parabolic Cylinder Functions: 23

24 Equatorial Waves 3) Using the projection on the parabolic cylinder functions, we obtain the equations for the equatorial (long-)waves. Start from the previous shallow-water model (non-dimensional, long-wave approximation): First, Project the system onto the first Parabolic Cylinder functions: Second, make a variable change with Riemann invariants: Finally, using the properties of parabolic cylinder functions find the equatorial long-waves equations: Those are characteristic equations: in the absence of wind stress forcing the Riemann invariants are conserved during the propagation: We can reconstruct variables from the initial shallow-water model afterwards: 24 (Note: Second parabolic cylinder function is optional in reconstruction)

25 Equatorial Waves The equatorial ocean waves have important properties for ENSO. Propagation: Recall the characteristic equations - The equatorial Kelvin wave of amplitude Ko, propagates east at c1=2.5 m.s-1. It takes around 2 months to cross the equatorial Pacific from west to east. -The equatorial Rossby wave of amplitude Ro, propagates west at -c1/3=0.8 m.s-1 It takes around 8 months to cross the equatorial Pacific from east to west. Reconstruction of variables: - A Kelvin wave Ko>0 or Rossby wave Ro>0 deepens the thermocline H>0: they are called downelling waves - Ko<0 and Ro<0 are upwelling waves with H<0 25

26 Equatorial Waves Examples in nature during El Nino 2006/2007: - Visually two Kelvin downelling waves, one Kelvin downelling wave (takes around 2 month to cross the basin eastward from 120E to 80W) - Visually one Rossby downelling wave, one Rossby upwelling wave (takes around 8 month to cross the basin westward) Kelvin Kelvin Rossby Rossby 26 (McPhaden 2006)

27 Reflections at boundaries We define boundary conditions at the edges of the equatorial Pacific (see e.g. Clarke ): - The eastern boundary (x=lo) of south America is approximated as a meridional wall - The western boundary (x=0) of Australia-Indonesia is a approximated as a zero mass exchange Reflection coefficients for equatorial waves: - Kelvin waves reflect into Rossby waves at the eastern boundary (losses are omitted) - Rossby waves reflect into Kelvin waves at the western boundary (with losses) 27

28 Equatorial Waves Examples in nature during El Nino 2006/2007: - Both downelling and upwelling Kelvin and Rossby waves reflections - There are also losses at the eastern boundary in practice (coastal Kelvin waves) Rossby Kelvin Rossby Kelvin Rossby Rossby 28 (McPhaden 2006)

29 Equatorial Waves Delayed Negative feedback: Recall the characteristic equation Positive wind stress forces both a 1) a downelling Kelvin Ko>0 and 2) an upwelling Ro<0, with opposite effects This has important implications for ENSO. - The eastern Pacific sees first 1) the downelling Kelvin wave, then around 8 months later 2) a reflected upwelling Kelvin wave with opposite effects. Reflections continue but the waves eventually weaken. - This provide a delayed negative feedback that is important for the ENSO to oscillate. It allows the system to switch from opposite phases (El Nino to La Nina and conversely). This will be discussed hereafter. 29

30 Equatorial Waves An illustration during El Nino 98: - Positive wind stress forces several: downelling Kelvin Ko>0 and upwelling Ro<0, with opposite effects - the delayed negative feedback helps terminates the event - Note that the continuum of equatorial waves (i.e. superposition ) is important, not only individual waves 30

31 Outline 1. Introduction to The El Nino Southern Oscillation 2. Simple Model Overview 3. Ocean Dynamics 4. SST Thermodynamics 5. Atmosphere Dynamics 6. Coupled Dynamics 7. Model Linear Solutions 8. Model with Nonlinear Thermocline Feedback 9. ENSO Oscillation Mechanisms 31

32 Simple ENSO Model We derive hereafter a simple ENSO model: -1) Includes atmosphere dynamics, ocean dynamics and an SST thermodynamic budget relevant to ENSO -2) simple coupling between the ocean, atmosphere and SST: -3) The model admits solutions consisting of equatorial waves in the ocean (Ko, Ro) Done! 32

33 Sea Surface Temperatures Introduction - Recall the profile of ocean temperatures with the thermocline. We now want to parametrize the evolution of Sea Surface Temperatures (or SST) near the surface. Sea surface temperatures (SST) Ocean temperatures (C) thermocline depth SST 33 (NOAA, Clarke 2008)

34 SST Thermodynamics - 1) We consider a mixed layer in the first 50m of the ocean, with homegeneous temperature T, akin to SST. The homogeneity is usually due to winds and vertical advection. TSUB -2) We start with a general thermodynamic budget for total SST T in the mixed layer (dimensional, T is in K): Temporal derivative dissipation Zonal, meridional advection Heat exchanges with the atmosphere Vertical advection 34

35 SST Thermodynamics 3) We start by linearizing and simplifying the SST budget. Starting SST budget (total SST): First, separate totals into mean and anomalies: Then linearize removing the background equilibrium and nonlinear terms: Finally, here we simplify by neglecting most processes. We obtain the simplest budget relevant to ENSO, latter on more processes can be retained - We have parametrized heat flux echanges with the atmosphere as latent heat losses: 35

36 Thermocline Feedback parametrization 4) Next, we parametrize subsurface temperatures as a function of the thermocline depth. We recall the previous linearized/simplified SST budget: We parametrize an empirical linear functional between Tsub and thermocline depth anomalies Then we obtain a SST budget with a thermocline feedback: T SUB H 36

37 Thermocline Feedback parametrization 5) Finally, the thermocline feedback is empirically parametrized as varying with x: The thermocline feedback is maximal in the eastern Pacific. This is because: 2) The mean upwelling is stronger 1) The mean thermocline is shallower. Ocean temperatures (C) H H 37 (Clarke 2008, KSU Web home)

38 SST Thermodynamics Examples in nature during El Nino 1997/1998: - The thermocline first deepened in the eastern Pacific - Shortly after the SST increased due to the thermocline feedback 38

39 SST Thermodynamics A few last steps for consistency with the ENSO model: We consider again the previous SST budget: 1) non-dimensionalize to the ENSO scales: 2) Project on the basis of parabolic cylinder functions for the y-axis and truncate. We obtain: SST structure 39

40 Outline 1. Introduction to The El Nino Southern Oscillation 2. Simple Model Overview 3. Ocean Dynamics 4. SST Thermodynamics 5. Atmosphere Dynamics 6. Coupled Dynamics 7. Model Linear Solutions 8. Model with Nonlinear Thermocline Feedback 9. ENSO Oscillation Mechanisms 40

41 Simple ENSO Model We derive hereafter a simple ENSO model: -1) Includes atmosphere dynamics, ocean dynamics and an SST thermodynamic budget relevant to ENSO -2) simple coupling between the ocean, atmosphere and SST: -3) The model admits solutions consisting of equatorial waves in the ocean (Ko, Ro) 41

42 Atmosphere Dynamics 1) We start with a skeleton model for the atmosphere (Majda and Stechmann, 2009). The starting skeleton model captures the Madden-Julian Oscillation and intraseasonal variability in general in the tropics. We will modify it here to capture the ENSO. Starting skeleton model ENSO model atmosphere In the starting skeleton model there is interaction between: (i) planetary dry dynamics: wind speed anomalies u,v, potential temperature anomalies (ii) planetary lower level moisture anomalies: q (iii) planetary envelope of synoptic/convective activity: a Parameters: moisture gradient, growth rate, source of cooling and moistening Units (different from the ENSO model!): x,y: 1500 km, t: 8 hours, u,v: 50 m/s,, q: 15 K, :,10 K/day 42

43 Atmosphere Dynamics The starting skeleton model has the following features: Starting skeleton model The variable a is the planetary envelope of synoptic convection. It parametrizes the effect of smallscale convection on the planetary scale. An analogue for a in observation is Outgoing Long Wave Radiations (OLR). See Majda and Stechmann (2015) The skeleton model has an associated vertical structure (first baroclinic mode). See book of Majda (2003). To reconstruct the flow with vertical structure use: convection 43

44 Atmosphere Dynamics The starting skeleton model is derived as follows from the momentum and mass continuity budgets: Starting equations: Galerkin Projection on the basis of functions of the vertical baroclinic modes: Associated vertical structure (first baroclinic mode) Truncation to the first baroclinic mode: (the subscript 1 is omitted afterwards) 44

45 Atmosphere Dynamics To obtain the ENSO model atmosphere, the starting skeleton model is modified as follows: 1) Non-dimensionalize to units relevant to the ENSO (identical the ones of the ocean): Starting skeleton model Recall the units that match the scales of the ENSO: Change the model units from the ones of the starting skeleton model to the ones of the ENSO. Starting skeleton model with ENSO units The skeleton model has now same units as the ocean 45

46 Atmosphere Dynamics 2) We now simplify the system removing small terms Starting skeleton model with ENSO units We identify small non-dimensional terms. Here: the Froude number is small other parameters are of order O(1) Removing small terms we obtain the ENSO atmosphere model: In particular, we have removed the time derivatives in the atmosphere. On ENSO timescales, the atmosphere is assumed to adjust instantly to the latent heat release Eq. We have also neglected dissipation. The ENSO atmosphere is a non-dissipative version of the Matsuno-Gill model. Also note that convection is in direct balance with latent heat release from the ocean and other sources: In addition, the zonal mean circulation is treated separately: 46

47 Atmosphere Dynamics The ENSO atmosphere reproduces a large-scale response to latent heating Eq, with delocalized winds. The wind anomalies are oriented towards the center of latent heat release. ENSO model atmosphere Latent heat release from the ocean Convection Delocalized zonal winds response 47

48 Atmosphere Dynamics Example during El Nino The increased SST in the eastern Pacific release latent heat Eq>0, creating delocalized zonal winds anomalies in the western Pacific u>0 oriented eastward. Note however that all winds fluctuations (for example small bursts) are not necessarily related to SST variations, as discussed in next lectures. ENSO model atmosphere 48

49 Atmosphere Dynamics Next, in order to find simple solutions for the ENSO atmosphere we project on the basis function of parabolic cylinder functions. Parabolic cylinder functions in the atmosphere: Note the functions are identical to the ones of the ocean except for a different meridional scaling: The strategy for projecting is therefore identical! 49

50 Atmosphere Dynamics Next, in order to find simple solutions for the ENSO atmosphere we project on the basis function of parabolic cylinder functions. 1) Start with the ENSO model atmophere 2) Project the system onto the first Parabolic Cylinder functions: 3) Find the expressions for the atmospheric equatoral waves: With reconstruction The atmosphere equatorial Kelvin wave Ka and Rossby wave Ra are derived in fashion identical to the ones of the ocean. However, they adjust instantly (no propagations) and there are periodic boundary conditions around the equatorial belt.. 4) In addition to this, the zonal mean circulation is treated separately simply as:. 50

51 Atmosphere Dynamics The atmosphere model with equatorial waves adjusting instantly is in very good agreement with observations (see Stechmann and Ogrosky 2015). There is no dissipation in the atmosphere. Observed OLR is used to determine ENSO model atmosphere Atmosphere equatorial waves solutions Kelvin wave during El Nino or La Nina (Stechmann Ogrosky 2015), from model and observations 51

52 Outline 1. Introduction to The El Nino Southern Oscillation 2. Simple Model Overview 3. Ocean Dynamics 4. SST Thermodynamics 5. Atmosphere Dynamics 6. Coupled Dynamics 7. Model Linear Solutions 8. Model with Nonlinear Thermocline Feedback 9. ENSO Oscillation Mechanisms 52

53 Simple ENSO Model We derive hereafter a simple ENSO model: -1) Includes atmosphere dynamics, ocean dynamics and an SST thermodynamic budget relevant to ENSO -2) simple coupling between the ocean, atmosphere and SST: -3) The model admits solutions consisting of equatorial waves in the ocean and atmosphere (Ko, Ro, Ka, Ra) 53

54 Coupled Dynamics We now express flux exchanges between the atmosphere and ocean to couple them: 1) Wind stresses on the ocean are related to wind speed using an empirical bulk formula: With the bulk coefficient: 2) Latent heat release from the ocean to the atmosphere is related to SST using an empirical formula: For increased SST T>0 there is increased release of latent heat towards the atmosphere. Note that this also appears now as a dissipation term in the SST budget: Note that this relation derives from an empirical formula for total latent heat release related to lower level moisture: with That is linearized and simplified. In particular the latent heat parameter reads: 54

55 Coupled Dynamics For coupling equatorial waves solutions however a few additional modifications must be taken into account. The ocean and atmosphere use different parabolic cylinder functions for equatorial waves, therefore to couple both we use projection coefficients from one basis to the other: Those projection coefficient appear in the complete model for its equatorial wave solutions: atmosphere Atmosphere projected on Except Eq projected on ocean Ocean and SST projected on Except projected on SST couplings 55

56 Coupled Dynamics For completeness we recall the simple ENSO model for the equatorial wave solutions. The simple ENSO model equations shown previously are used for physical insight, while the equatorial wave equations here are the ones we solve in practice from an applied mathematical perspective! Both views are important and complementary. ENSO model for equatorial wave solutions atmosphere ocean SST couplings Reconstructions of the variables: 56

57 Simple ENSO Model We have now completed the derivation of the simple ENSO model: -1) Includes atmosphere dynamics, ocean dynamics and an SST thermodynamic budget relevant to ENSO -2) simple coupling between the ocean, atmosphere and SST -3) The model admits solutions consisting of equatorial waves in the ocean and atmosphere (Ko, Ro, Ka, Ra) 57

58 Outline 1. Introduction to The El Nino Southern Oscillation 2. Simple Model Overview 3. Ocean Dynamics 4. SST Thermodynamics 5. Atmosphere Dynamics 6. Coupled Dynamics 7. Model Linear Solutions 8. Model with Nonlinear Thermocline Feedback 9. ENSO Oscillation Mechanisms 58

59 Simple ENSO Model We have derived a simple ENSO model: -1) Includes atmosphere dynamics, ocean dynamics and an SST thermodynamic budget relevant to ENSO -2) simple coupling between the ocean, atmosphere and SST: -3) The model admits solutions consisting of equatorial waves in the ocean and atmosphere (Ko, Ro, Ka, Ra) 59

60 Model Linear Solutions Hereafter we compute the linear solutions of the ENSO model. For this we solve the equatorial waves equations 1) First, the simple ENSO model equatorial wave equations are deterministic and linear: atmosphere ocean SST couplings and can be expressed in the more general matrix form: is a vector describing the entire state of the system at a given time is the evolution matrix 60

61 Model Linear Solutions Hereafter, as a first step we compute the linear solutions of the ENSO model: 2) The ENSO model in matrix form: admits linear solutions of the form: The eigenvalues are complex and computed by solving 3) The first linear solution We replace that satisfy (see next lectures) is analyzed. and express the linear solution as: Oscillatory part of the solution: is the oscillation frequency is the eigenvector (complex) Growth/decay with time. The g is the growth/decay rate. 4) The first linear solutions captures the basic ENSO oscillation. However g<0 therefore the solution decays over time. Other linear solutions have much weaker g<0 and therefore are secondary (they decay much faster). 61

62 Model Linear Solutions Recall the linear solution of the ENSO model. Oscillatory part of the solution: is the oscillation frequency is the eigenvector (complex) Growth/decay with time. The g is the growth/decay rate. 5) We now recompute the linear solution for a different parameter values (here ). Varying model parameters we can find both regimes where the first linear solution is either stable (g<0) or unstable (g>0). The frequency and eigenvectors are also modified. Changes of the linear solution with parameters g Going from a stable to unstable regime or inversely is called a Hopf Bifurcation. Hereafter we will make this bifurcation happen in the model. 62 Frequency

63 Outline 1. Introduction to The El Nino Southern Oscillation 2. Simple Model Overview 3. Ocean Dynamics 4. SST Thermodynamics 5. Atmosphere Dynamics 6. Coupled Dynamics 7. Model Linear Solutions 8. Model with Nonlinear Thermocline Feedback 9. ENSO Oscillation Mechanisms 63

64 Model with NonLinear Thermocline Feedback - Hereafter we will modify the model slightly in order to reproduces a maintained ENSO oscillation. - In order to obtain this regime we will combine two linear solutions of the model (one stable, one unstable) into a Hopf bifurcation by introducing a simple nonlinear thermocline feedback. - The modification is for this lecture only (next lectures will consider different modifications based on winds). Model Solutions With Nonlinear Thermocline Feedback Zonal winds Zonal currents Thermocline depth SST Averages of SST in eastern and western PAcific 64

65 Model with NonLinear Thermocline Feedback We now include in the ENSO model a nonlinear thermocline feedback allowing Hopf bifurcations (for this lecture only) Thermocline feedback is here nonlinear 65

66 Model with NonLinear Thermocline Feedback 1) We now combine both a stable and unstable regime of the ENSO model. For this we modify the thermocline feedback with a simple nonlinear switch. In the SST equation of the ENSO model: the thermocline paramter reads: modified with nonlinearity: Stable regime Unstable regime Average of thermocline depth anomalies H in eastern Pacific Threshold for switching between stable/unstable regimes 2) The ENSO model now experiences Hopf Bifurcations constantly as it switches back and forth between the stable and unstable regime. Temporal evolution of SST and H averaged in eastern Pacific This is a simple way to obtain a maintained ENSO oscillation without growth or decay of solutions. Unstable regime 66 Stable regime

67 Model with NonLinear Thermocline Feedback We now have an ENSO model that reproduces a basic ENSO oscillation with regular El Nino and La Nina events. The model has a nonlinear thermocline feedback allowing constant Hopf bifurcations between stable and unstable regimes. Hereafter we analyze the dynamics of this basic ENSO oscillation. Zonal winds Zonal currents Thermocline depth SST Averages of SST in eastern and western PAcific Stable regime Unstable regime 67

68 Model with NonLinear Thermocline Feedback A similar yet more complex nonlinear thermocline feedback is also used in the Cane-Zebiak model (1987), along with other nonlinearities allowing Hopf bifurcations. The model historically made the first accurate El Nino predictions in the 1980s. SST budget in the Cane-Zebiak model: Thermocline Feedback parametrization with SST timeseries in the Cane-Zebiak model (averages in east/west) Hopf bifurcations (?) 68 (Zebiak and Cane 1987)

69 Outline 1. Introduction to The El Nino Southern Oscillation 2. Simple Model Overview 3. Ocean Dynamics 4. SST Thermodynamics 5. Atmosphere Dynamics 6. Coupled Dynamics 7. Model Linear Solutions 8. Model with Nonlinear Thermocline Feedback 9. ENSO Oscillation Mechanisms 69

70 ENSO Oscillation Mechanisms We now review basic ENSO Oscillation Mechanisms, that allow the transitions between El Nina and La Nina. We will illustrate this with the linear solutions of the ENSO model (oscillatory part). Note that in this simple ENSO model that we will use in next lectures, the thermocline feedback is linear, therefore the nonlinear thermocline feedback modification from last section is now removed.. Zonal winds Zonal currents Thermocline depth SST El Nino La Nina 70

71 ENSO Oscillation Mechanisms We first recall couplings between the ocean and atmosphere in the model during El Nino. 1) Increased SST T>0 in the east releases latent heat Eq and forces an instant and delocalized zonal winds response u>0 in the center (and conversely). atmosphere Ocean SST Couplings 71

72 ENSO Oscillation Mechanisms 1) Increased SST T>0 in the east releases latent heat Eq and forces an instant and delocalized zonal winds response u>0 in the center (and conversely). atmosphere SST Ocean Couplings 72

73 ENSO Oscillation Mechanisms 2) Positive winds u>0 force ocean equatorial waves (both upwelling and downelling) that first increase the thermocline depth H>0 in the east then decrease it. This occurs through the downelling Kelvin waves (H>0) that reach the eastern Pacific first, and later the upwelling Rossby waves that reflect into upwelling Kelvin waves (H<0). atmosphere SST Ocean Couplings 73

74 ENSO Oscillation Mechanisms 2) Positive winds u>0 force ocean equatorial waves (both upwelling and downelling) that first increase the thermocline depth H>0 in the east then decrease it. This occurs through the downelling Kelvin waves that reach the eastern Pacific first, and later the upwelling Rossby waves that reflect into upwelling Kelvin waves. atmosphere SST Ocean Couplings 74

75 ENSO Oscillation Mechanisms 3) A deep thermocline H>0 in the east increases the SST through the thermocline feedback. Recall the thermocline feedback parameter is maximal in the eastern Pacific. atmosphere SST Ocean Couplings 75

76 ENSO Oscillation Mechanisms 3) A deep thermocline H>0 in the east increases the SST through the thermocline feedback. atmosphere SST Ocean Couplings 76

77 ENSO Oscillation Mechanisms For the equatorial Pacific to oscillate between El Nino and La Nina states, two feedbacks are necessary: 1) A positive feedback that reinforces initial anomalies and allows the growth of Nino (or Nina) events.. 2) A negative delayed feedback that eventually reverses the conditions (e.g. from El Nino to La Nina) atmosphere SST Ocean Couplings Delayed negative feedback Positive feedback 77

78 ENSO Oscillation Mechanisms The positive feedback reinforces SST anomalies in the eastern Pacific through coupling with zonal winds and the thermocline. This reinforces the initial SST anomalies and allows El Nino to develop as well as La Nina. This is also called the Bjerknes feedback after its discoverer. atmosphere SST Ocean Couplings Positive feedback 78

79 ENSO Oscillation Mechanisms During El Nino, SST anomalies T>0 in the eastern Pacific force zonal winds u>0 (trade winds are decreased). u>0 forces an ocean response H>0 in the eastern Pacific. This positive feedback reinforces T>0 and allows El Nino to further develop. atmosphere Ocean SST Couplings El Niño Normal SST cold in east (and warm in west) Trade winds Trade winds weaken SST increases in east Weak circulation Shallow in east 79in east deeper

80 ENSO Oscillation Mechanisms During La Nina, SST anomalies T<0 in the eastern Pacific force zonal winds u<0 (trade winds are increased). u<0 forces an ocean response H<0 in the eastern Pacific. This positive feedback reinforces T<0 and allows La Nina to further develop. atmosphere Ocean SST Couplings Normal SST cold in east (and warm in west) Trade winds La Niña SST colder in east Trade winds increase Strong circulation Shallow in east Shallower 80 in east

81 ENSO Oscillation Mechanisms The delayed negative feedback eventually decreases SST anomalies in the eastern Pacific through an opposite ocean thermocline response that eventually reaches the eastern Pacific. The delayed negative feedback comes from the slow adjustment of the ocean. It allows to revert conditions from El Nino to La Nina and conversely. atmosphere SST Ocean Couplings Delayed negative feedback 81

82 ENSO Oscillation Mechanisms The delayed negative feedback in the ocean is explained by wave propagations. Recall the ocean equatorial waves equations: During El Nino for example, Positive wind stress forces both a downelling Kelvin waves (Ko>0 and H>0) and upwelling Rossby waves (Ro<0 and H<0) with opposite effects on the thermocline depth. The downelling Kelvin waves quickly reaches the eastern Pacific and deepens the thermocline (H>0), which is part of the positive feedback described earlier. The delayed negative feedback comes in play later. It is due to the forced upwelling Rossby waves that reflect at the western boundary into upwelling Kelvin waves that eventually reach the eastern Pacific and shallow the thermocline (H<0). 82 (Schopf and Suarez 1988)

83 ENSO Oscillation Mechanisms The delayed negative feedback at work in the simple ENSO model, as seen for example with H<0 anomalies propagating from the western to the eastern Pacific between after El Nino.. This is due to the propagation of the upwelling Rossby waves reflecting into upwelling Kelvin waves. After La Nina the opposite happens. atmosphere SST Ocean Couplings Delayed negative feedback 83

84 ENSO Oscillation Mechanisms The slow adjustement of the ocean also changes the overall heat content in the Pacific. - In between La Nina and El Nino, the thermocline is deep overall in the Pacific: this is the recharged state - In between El Nino and La Nina, the thermocline is shallow overall in the Pacific: this is the discharged state This is due to the equatorial wave propagations. In particular, the Rossby waves create meridional advection v (towards the pole during El Nino and towards the equator during La Nina) that either recharge or discharge the overall heat content slowly. Although this has been presented as a potential delayed negative feedback that allows transition between El Nino and La Nina (Jin 1997), note that is only a side-effect of the equatorial wave propagations and reflections. Nino recharged Nina discharged 84

85 Outline 1. Introduction to The El Nino Southern Oscillation 2. Simple Model Overview 3. Ocean Dynamics 4. SST Thermodynamics 5. Atmosphere Dynamics 6. Coupled Dynamics 7. Model Linear Solutions 8. Model with Nonlinear Thermocline Feedback 9. ENSO Oscillation Mechanisms 85

86 Thank you! 86 (NOAA)

87 Lecture Summary: Simple ENSO Model We derive hereafter a simple ENSO model: -1) Includes atmosphere dynamics, ocean dynamics and an SST thermodynamic budget relevant to ENSO -2) simple coupling between the ocean, atmosphere and SST -3) The model admits solutions consisting of equatorial waves in the ocean and atmosphere (Ko, Ro, Ka, Ra) 87

88 Overview of next lectures 1) The present ENSO model will be modified in next lectures to capture more complex features of the ENSO. We will add stochatic wind bursts in the atmosphere. We will also use a linear thermocline feedback. 2) We will cover more complex features such as the irregularity and intermittency of ENSO, its diversity including rare extreme El Nino events, and its synchronization to the seasonal cycle. Addition of stochastic wind bursts Linear thermocline feedback 88