Turbulence in Climate Studies

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1 Turbulence in Climate Studies Ravi S Nanjundiah Centre For Atmospheric and Oceanic Sciences Indian Institute of Science Bangalore ICTS Turbulence Meeting Dec 2011

2 Climate: An Interactive System Climate Interactions Rotation Zonal Mean Circulation Assymetry Modelling Climate The Equations Equations Simplifications Resolution Numerical Techniques Turbulence Turbulence in Atmospehere Clouds Cumulus Parameterization Conditional Moist Static Instability Stable Condensation Kuo-Anthes Parameterization kuo scheme Arakawa Schubert Scheme Overview of Arakawa-Schubert Scheme The Cloud Work Function Buoyancy and Cloud Work Function Factors affecting Cloud Work Function Algorithm algoritm Conclusions Outline

3 source: Internet Climate: An Interactive System Climate is influenced by many factors: external and internal It is a coupled system in which atmosphere,ocean,land surface interact with each other Atmosphere drives ocean currents through winds Ocean impacts atmosphere through Sea Surface Temperature and moisture fluxes Atmosphere interacts with land-surface: transfer of sensible heat and frictional effects Precipitation leads to percolation into the deep soil and also to runoff which could flows into the sea through rivers Rivers can also influence oceans: fresh water discharge can change the salinity and hence stratification: such effects play a role in modulating Monsoons over Bay of Bengal.

4 Climate... Phase change between vapour, water and ice is important: latent heat release changes temperature and hence changes atmospheric circulation Atmospheric circulation through advection plays a major role in transporting moisture leads to a coupling between thermodynamics and fluid dynamics Multiple scales exist and interact continously with each other Water evaporates from ocean/land surface at extremely small scales Eddies carry them upward away from the surface Advection carries it over long distances Condensation occurs on cloud scales few km The heating in clouds induces large scale flow energy transfer from smaller to larger scales. Turbulence plays a major role in the boundary (where evaporation occurs) and in clouds (where condensation occurs)

5 Earth s Rotation source: wikipedia.org Earth s rotation also plays an important in atmospheric/oceanic circulation The effect of rotation is different at different latitudes Near the pole where the axis of rotation is close to the local vertical, the effects are more pronounced geostrophic balance As we move to the tropics the rotation vector moves more and more towards the horizontal plane (meridional-zonal plane) Geostrophic balance i.e. balance between Coriolis and horizontal pressure gradient is much weaker in tropics Exact role of rotation on tropical circulation (especially near equator) is still a matter of debate Heating (latent heat release in clouds) is more pronounced in tropics as compared to extra-tropics Turbulence and Non-linear interactions (within atmosphere and with ocean/land) play an important role in tropics

6 The Time Averaged Circulation Unequal heating between equator and poles causes an equator-to-pole heat transfer Near equator air rises and sinks near 30 0 the Hadley Cell It also gets deflected eastward due to earth s rotation (westerlies) However as one moves polewards, rotation becomes dominant Atmosphere is also more baroclinic (steeper horizontal temperature gradients) leads to eddy formations baroclinic instability Though the mid-latitude circulation is shown as the Ferrel Cell, it is actually large-scale baroclinic eddies and fronts which transfer energy towards the poles

7 Internet The Walker Circulation There is rising motion off the Indonesian coast (Western Pacific) Descending motion off the American coast This circulation is related to Sea Surface Temperature (SST) gradients in the East-West direction along the equator Water is warmer along the Indonesian Source: (West Pacific coast) and cooler off the American coast Warmer SST regions are more amiable for moist convection The relationship between SST and moist convection (rainfall) is non-linear Till about 26 o C rainfall is very low and increases rapidly between 26 o C o C Beyond 27.5 o - a high SST is necessary but not sufficient condition for moist convection

8 Ocean-Atmospheric Coupling Looking at the surface wind pattern we notice that the winds are generally westward (easterlies, the trade winds) The easterlies cause upwelling at the eastern end of a basin and along the equator - causes the thermocline (the interface across which there is cold water below and warmer water above) to be shallow. Shallower thermocline allows cold water to come up easily at the Eastern (American) end stays cold inhibits moist convection West (Indonesia) stays warm supports convection At the western end there is a piling up of water due to westward wind the thermocline is deep This coupled interaction leads to the Walker circulation The collapse of this circulation due to changes in coupled interaction causes El-Nino

9 The El-Nino In some years the easterly trades weaken Many theories about the cause of this: one of the more prevalent ones is about westerly wind bursts related to rainfall activity on intraseasonal scale (the Madden-Julian Oscillation) The weakening of easterlies reduces upwelling and the thermocline deepens over Central and Eastern Pacific These two regions now become warm and can also sustain convection Moist convection shifts from Western Pacific to Central and Eastern Pacific

10 Variations from the Mean Picture Time mean picture is not quite representative On a day-to-day basis there are large deviations from this mean picture (any animation of satellite pictures will show this!) There are large assymetries in rainfall e.g. Saudi Arabia (very dry) and North Eastern India (one of the wettest spots on earth) lie at the same latitude. Mountains complicate the picture So how do we model climate?

11 The Components We need to model Processes in the atmosphere Processes in the ocean Processes that govern land surface parameters Processes governing sea-ice

12 Modelling the Climate We have seperate models for atmosphere, ocean, land surface and sea-ice which couple to each other Most often these components are stand-alone models which can be run by themselves with appropriate forcing

13 The Model Equations We use equations for conservation of mass, momentum, energy and equation of state In addition if we advect additional tracers (such as green house gases, aerosol pollutants etc) we include conservation equations for them The mass conservation equation The momentum conservation equation: Energy Conservation Equation C v DT Dt 1 Dρ + U = 0 ρ Dt D U Dt + p Dα Dt = 2Ω U 1 p + g + Fr ρ The J term includes radiative heating, effect of cloud heating, transfer of sensible heat from the surface Radiative heating includes incoming solar radiation (mostly in visible part of spectrum) and out going terrestrial radiation (essentially in the infra-red) Computationally this could take upto 40% of the overall computing time Longwave emissivities and absorptivities computational can take 2-3 times the computational taken by rest of the atmospheric model Equation of state : P = ρrt Simplifications? = J

14 Simplifications Generally while looking at the global scale, we find that the aspect ratio (vertical/horizontal) is very small. Vertical scale is small about 10 km Horizontal scale could rangle from 100 km km (depending on the phenomena being studied When looking at larger scales we use the thin-shell approximation Radius r = a +z, where z is vertical distance from mean sea level, a the nominal radius of earth s surface All radial distances are replaced with a and radial derivatives by derivative wrt z While considering large scale flows, typical horizontal velocity is about 10m 1 s Vertical velocity is 1-10cm 1 s. To a first approximation we assume that vertical component of momentum equation can be replaced by hydrostatic balance p z = ρg This removes vertical acoustic modes Also removes the effect of Coriolis acceleration in the vertical (and hence for consistent energetics needs to be dropped from zonal momentum equation) Curvature terms are also sometimes dropped Flow looks quasi-two-dimensional (though clouds and boundary layer turbulence are more prominent in the vertical) Now vertical velocity needs to be calculated indirectly through continuity equation

15 Typical Resolutions used in Climate Models Climate models (using coupled Ocean-Atmospheric Models) need to be integrated for 100+ years to study climate change Coarse resolution 100km on a global scale Seasonal Forecasts (also with coupled models) are run at similar resolutions (typical duration is 9 months) Medium Range forecasts (upto 15days) are conducted with atmospheric global models resolution is about 30 kms Short range forecasts for short periods upto 3 days with regional models, resolution 5-10 km Very high resolution location specific forecasts could have resolution of 1 km for periods upto 12 hours

16 Numerical Techniques Used in Climate Models For weather prediction on local scale (high resolution, short range) Finite Difference methods are preferred (these models are usually non-hydrostatic) For Global medium range weather forecasts spectral methods (due to their higher accuracy) are preferred. Both NCEP & ECMWF use spectral models Climate models, also advect additional tracers (such as GHGs, aerosols) The concentration of these could vary substantially over space not smooth functions Sharp variations are resolved poorly (Gibbs oscillations) by spectral models hence not widely used Climate Models use finite volume/semi-lagrangian techniques as these techniques are also highly scalable (also large time-steps possible) Some models such as NICAM have used icosahedral technique for obtaining higher resolution and scalability Latest version of NCAR model has spectral elements.

17 Turbulence We have seen that most models have horizontal resolution of about km Vertical resolution is at best 100 m. Most models have higher resolution in the boundary layer and near tropopause but coarser everywhere. Surface fluxes of momentum, heat and moisture play a significant role in determing atmospheric circulation These processes occur at smaller scales which cannot be resolved (or likely to be resolved in the near future) Hence the effects of boundary layer are parameterized The common approaches are 1. Mixed Layer Models 2. Local closures based on eddy diffusivity 3. Nonlocal closures We are nowhere close to LES (let alone DNS!) for BL in a climate model Are simulations sensitive to Boundary Layer parameterization?

18 Impact of Boundary Layer Parameterization On Monsoon Rainfall Simulations are with WRF at 18 km resolution Rainfall during July is sensitive to boundary layer scheme Seems to improve using the Mellor-Yamada Janjic scheme (which uses a higher order closure)

19 What do Clouds Do? Clouds influence climate system by: 1. Coupling Dynamical and Hydorological processes through latent heat of condensation, redistributing sensible & latent heat and momentum 2. Coupling radiative and dynamical-hydrological processio in the atmosphere through reflection, absorption and emission of radiation 3. Influencing hydrological process in the ground through precipitation 4. Influencing the coupling between atmosphere and oceans (or ground) through modification of PBL and radiative processes All the interactions are two-way: Clouds influence the large-scale and vice-versa These interactions need to be included in addressing the cumulus parameterization problem

20 Schematic of A Cloud Updrafts, Downdrafts, Rainfall, Re-evaporation of rainfall Low level Convergence of moisture, Outflow at the top, lateral entrainment

21 What Should Cumulus Parameterization Calculate? The cumulus parameterization is formulation of statistical effects of moist convection to obtain a closed system for predicting weather and climate Arakawa (2004) categorizes the computations in a cumulus parameterization into Classical and Non-Classical Objectives Classical Objectives include: 1. Vertically integrated cumulus heating: This is the most basic objective. It is also related to the total convective precipitation falling to the ground 2. Vertical distribution of Cumulus heating and Drying of the atmosphere Non-Classical Objectives: 1. Mass Transfer by Cumulus convection: this would involve moving of lower level air to higher levels (and in the process advection of tracers), entrainment and detrainment and mixing with environmental air. 2. Generation of Liquid and Ice Phases of Water 3. Interaction with Planetary Boundary Layer : all the low level air that causes clouds is from the Planetary Boundary Layer. Diurnal cycle and variation of PBL with it could have an significant impact on formation of clouds 4. Interaction with radiation: clouds influence radation which in turn influences formation of clouds 5. Transport of Momentum: Momentum is transported in cumulus convection but Arakawa considers this the most difficult part of parameterization

22 Cumulus Parameterization In cumulus parameterization we do not model every individual cloud (we cannot resolve every cloud) However the effect of a statistical ensemble of clouds on the environment is incorporated This effect needs to be incorporated in terms of the large-scale variables (we call this the closure problem similar to the one in PBL) Clouds may not be present at every grid point or at all levels at a single grid point We use a set of necessary and sufficient conditions to check for the possible occurrence of clouds These conditions could be Presence of large-scale low level moisture convergence Presence of conditional moist static instability

23 Conditional Moist Static Instability Instability implies buoyancy of a parcel of air vis-a-vis environment If unstable the parcel will move up further in the direction of displacement At the surface a moist parcel is not generally saturated However on raising above the lifting condensation the moisture condenses, heats the parcel and makes it more bouyant > making the parcel unstable This is unknown as conditional moist static instability We now look at parameterizations that use either moist static stability or large-scale forcing to incorporate the effect of clouds Sometimes precipitation can also occur just by supersaturation with little turbulence

24 Stable Condensation This occurs when the lapse rate is stable but the atmosphere becomes saturated We can represent this as γ < γ m and q > q s The excess moisture is condensed, the latent heat released warms the air We could consider it the reverse of the wet-bulb process. We need to change the temperature and specific humidity in a layer such that it is stable and saturated (or sometimes slightly less than saturated) We can write this as and Lδq = c pδt δq = q q s(t + δt, p) We get an implicit system of equations in q and T This can be solved iteratively Sometimes we take into consideration re-evaporation of rainfall i.e. δq at a higher lower partly makes it saturated and partly falls down.

25 Kuo-Anthes Parameterization Scheme Used essentially for hurricane models. This scheme essentially depends on the occurrence of large-scale convergence to model cumulus convection. The vertically integrated moisture convergence M t is sum of convergence and surface evaporation and is given by: M t = 1 g Po 0 (q V dp + Evap Inside the cloud it is assumed that the temperature profile follows a moist adiabat [ ] κ Lv qs p θ e = T e Cp T (p e s) It is assumed that this vapour is used to make cloud column with temperature and moisture (T c, q c) from environmental air at (T, q) Part of this vapour will condense, raising the temperature from T e to T c W 1 = Cp L pb p t (T c T e) dp g

26 Kuo-Anthes... Reminder will increase the specific humidity of the cloud column from q e to q c pb W 2 = (q c q) dp g p t Here p t is pressure at the top of a cloud and p b pressure at the bottom. For deep clouds we can write p b p o and p t 0 Total vapour needed to create cloud over unit area is W = W 1 + W 2 We can write rate of cloud production C is assumed to be proportional to the convergence plus evaportion divided by amount of vapour necessary to produce the model cloud: C = Mt W = C p L 1 g pb p t Po 0 (q V )dp + Evap (T c T e) dp g + p b (q p c q) dp t g This in other words this is ratio of available moisture to reqd moisture and thus gives the fraction of area covered by clouds in a grid

27 Kuo-Anthes... Rate of latent heating Q is given by Q(p) = Cc p[t c(p) T (p)] if T < T c andm t > 0 = 0 if T T c or M t 0 Now assume that the large-scale forecast is made for a time step t, the temperature without cumulus effect be T Let temperature after cumulus parameterization be T (t + t) given by T (t + t) = T + C t(t c T ) Corresponding equation for q is gievn by q(t + t) = q + C t(q c q ) We can write rate of precipitation as R(p) = C cp L (Tc T ) Total rainfall at the surface would be R = p o o C cp L (Tc T )dp

28 Kuo-Anthes... Not all moisture that converges precipitates out. Some part moistens the atmosphere, the rest precipitates out. This partitioning is done on the basis of a parameter b b = 1 q q s where q = 1 0 qdσ and qs = 1 0 qsdσ. Here, qs is the saturated specific humidity. The total moistening is bm t Actual clouding will be given by C = (1 b)m t W Heating will be given as before by T (t + t) = T + C t(t c T ) We can write rate of precipitation as R(p) = C cp L (Tc T ) Total rainfall at the surface would be R = p o o Corresponding equation for q is given by C cp L (Tc T )dp q(t + t) = q + C t(q c q ) Kuo-Scheme generally overestimates rainfall over oceans It assumes that there is one type of cloud A More complex scheme is the Arakawa-Schubert Scheme

29 Arakawa-Schubert Scheme This scheme is based on theory of interaction between a cumulus ensemble (a set of clouds) and the large-scale environment The scheme operates on the presence of static instability rather than on occurrence of large-scale low-level moisture convergence scheme Most modern day schemes such as the Simplified Arakwa-Schubert (SAS) and Relaxed Arakawa Schubert (RAS), the Zhang-McFarlane mass flux scheme are off-shoots of the Arakawa Schubert Scheme

30 Overview of Arakawa Schubert Scheme In the Kuo-Anthes, a single cloud type, here a set of clouds each set (sub-ensemble) with a distinct cloud top and an entrainment rate Works on the principal that large-scale environment becomes unstable due to various processes And Occurrence of clouds tend to remove this instability Instability determines cloud base mass flux Which is related to A concept called Cloud Work Function Works on the principle of quasi-equilibrium i.e. at the end of the application of the cumulus scheme, the instability is completely removed This scheme considers three regions for energy and moisture budget equations 1. Lower most layer : sub-cloud layer or mixed layer 2. Clouds 3. Large-Scale environment

31 Buoyancy and Cloud Work Function Cloud Work Function represents the rate of Kinetic Energy generation by buoyancy force Buoyancy Force depends on properties of cloud properties of environment The cloud work function (essentially Convective Available Potential Energy) is given by: Zt [ ] gcp A(λ) = η(z, λ)[s vc(z, λ) S ve(z)]dz T e(z) Z b

32 Cloud Work Function... Cloud work function is an integral of bouyancy force that governs the K.E generation in a ensemble Bouyancy affected by entrainment rate λ If the cloud ensemble does not dissipate within the disspative time scale τ adj (typically few hours) then A(λ) > 0 for that particular λ

33 Factors Affecting Cloud Work Function Clouds form due to instability of the environment static control When clouds form they transport heat and moisture flux upwards Reduces the instability A(λ) dynamic feedback We can divide rate of change of A into two parts [ ] [ ] da(λ) da(λ) da(λ) = + dt dt dt LS represents environment, c represents cloud ] We write in a symbolic form as [ da(λ) dt c [ ] da(λ) = dt c λmax 0 c LS K (λ, λ )m b (λ)dλ This essentially represents how clouds influence each other K (λ, λ ) (usually <0) represents how clouds influence each other. Presence of other clouds will reduce the buoyancy in a cloud. K (λ, λ ) is determined by vertical structure of the environment

34 The Algorithm 1. First do a large scale forecast without cloud effects. This will non-zero cloud work function A(λ) 2. Make cumulus adjustment to large scale parameters so that cloud work function is near zero. 3. Difference between above two give da t use this for F(λ) dt 4. Calculate m b (λ) from λ max o K (λ, λ ) + F(λ) 0 5. Calculate changes in T and q using s t = V s ω s p + g p F s Ll + LR + Q R q t = V q ω q p + g p F q+l R 6. Total rainfall is integral of R over the entire column where R(p) is given by: R(p) = λd (p) 0 η(p, λ)r(p, λ)m b (λ)dλ

35 Algorithm... Arakawa-Schubert find that using λ for integration is cumbersome. Since for a given entrainment rate and environment conditions, the height of detrainment is fixed, the replace λ by Z D (p) this is the height of detrainment.

36 Sensitivity to Cumulus Parameterization Simulations are very sensitive to prescription of cumulus parameterization Does monsoon simulation improve in very high resolution model?

37 Simulations With A Very High Resolution Model A 7 km icosahedral non-hydrostatic model (NICAM) was used for a season s simulation This does not use any cumulus parameterization scheme. Tries to explicitly model the clouds Higher resolution appears to help though some problems remain source: Oouchi et al, 2004, GRL L11815

38 Concluding Remarks Climate is a result of multi-scale, multi-component system interacting continously Turbulence plays an important role in boundary layers and clouds Particularly in tropics, clouds and turbulence could be more important These appear to be the weakest links in climate system modelling Long way before we can think of LES in a climate system model However results of LES and DNS could be used for improving the boundary layer parameterizations Additionally in the tropics, free convection at low wind speeds could have a significant impact on turbulent fluxes (as shown by Rao and Narasimha,2006) this is missing in almost all models. Similar exercises with clouds and cloud-like flows could help in improving cumulus parameterization Perhaps a scheme that addresses both boundary layer and cloud convection simultaneously would improve simulations ( Unified Parameterization )

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