2 Dynamical basics, part 1

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1 Contents 1 Dynamical basics, art What drives the atmosheric circulation? Conservation laws Large-scale circulation: Basic force balances Vertical momentum balance: hydostaticity Horizontal momentum balance on raidly rotating lanets: geostrohy and the imortance of rotation

2 2 Dynamical basics, art 1

3 Chater 1 Dynamical basics, art 1 Here we survey the basic dynamical regimes and mechanisms that occur in lanetary atmosheres. Before resenting any theory, let s describe qualitatively the kind of regimes that exist. This shows the atmosheric circulation of a generic terrestrial exolanet: Showman et al. (2013, Atmosheric circulation of terrestrial exolanets, in Comarative Climatology of Terrestrial Planets, Univ. Arizona Press) There are two basic regimes: Troics: In the troics is the Hadley circulation, whereby air rises near the equator, moves oleward in the uer trooshere, sinks at latitude, and returns to the equator near the surface. This is an efficient means of transorting heat and contributes to the relative constant temeratures with latitude in the troics. However, Earth s rotation rohibits the Hadley cell from extending to the oles. The oleward movement of air in the uer branch of the Hadley cell causes an eastward deflection due to the Coriolis force, which leads to the so-called subtroical jets. Extratroics: Poleward of 30, dynamical instabilities lead to warm

4 4 Dynamical basics, art 1 and cold eddies ( km across in the case of Earth) that transort heat oleward. This is called the baroclinic zone and the instability resonsible for roducing these eddies is called baroclinic instability. This is not such an efficient heat-transort mechanism and so the latitudinal temerature gradient is larger in midlatitudes. The eddies converge momentum into their latitude region, generating the so-called eddy-driven jet. Let s look at a few observations for Earth. This shows the annual-mean surface and 200 mbar ( trooause) winds. Note weak equatorial winds, moving slightly westward at the surface. Midlatitude winds are eastward and get very strong in the uer trooshere. This is the signature of jet streams. Peixoto & Oort (1992)

5 Showman notes 5 From Peixoto & Oort (1992). Pressure Pressure Pressure This shows vertical structure of east-west ( zonal ) wind, the streamfunction of the so-called meridional circulation (i.e., the circulation in the latitude-height lane), and the structure of the longitudinal-mean isentroes: Peixoto & Oort (1983). This circulation attern controls the geograhic distribution of vegetation. The abundant rainfall associated with the rising branch of the Hadley cell leads to the troical rainforests (Brazil, central Africa, southeast Asia). The dearth of rainfall associated with the descending branch of the Hadley cell leads to the abundant deserts at latitude (the Sahara, central Asia, the American Southwest, Australia, and South Africa). The midlatitude eak in cloudiness and reciitation associated with the baroclinic zone leads to abundant temerate forests in midlatitudes.

6 6 Dynamical basics, art 1 Poleward heat transort in the baroclinic zone involves continual growth and decay of eddies. These eddies cause the fronts and multi-thousand-km-long arc-shaed clouds seen in weather mas and satellite images at high latitudes and cause most winter weather in the U.S., Euroe, and other midlatitude locations. They also cause the meanders on the jet stream seen in the geootential heights of the 500-mbar surface: Salby (1996) Now let s look at the circulation of the giant lanets. Showman, Menou & Cho (2010). Juiter, Saturn, Uranus, and Netune have radii of 11, 9, 4, and 4 times that of Earth. They are comletely fluid lanets with no solid surfaces and

7 Showman notes 7 thus challenge our understanding of how atmosheric circulation should behave in the absence of a surface. They are also raid rotators, with rotation eriods ranging from 10 hours on Juiter to 17 hours for Uranus. Cloud tracking shows that the circulation is dominated by zonal jets with seeds eaking at 200 m sec 1 on Juiter and 400 m sec 1 or more on Saturn and Netune. These clouds are at ressures of 1 bar within a factor of a few. How winds can be so fast desite the weak solar forcing is not well understood. 1.1 What drives the atmosheric circulation? Atmosheric circulation is fundamentally a couled radiation-dynamics roblem. In the absence of circulation, the vertical temerature rofiles would relax into a local radiative equilibrium, which would be very hot on the dayside and cold at the oles/nightside. The radiative heating rate would everywhere be zero. When a circulation oerates, however, horizontal temerature contrasts imly horizontal ressure contrasts, which drive winds. The winds in turn drive the atmoshere away from radiative equilibrium by transorting heat from hot regions to cold regions (e.g., from the equator to the oles on Earth). This deviation from radiative equilibrium allows net radiative heating and cooling to occur, thus maintaining the horizontal temerature and ressure contrasts that drive the winds. Satial contrasts in thermodynamic heating/cooling therefore drive the circulation, yet it is the existence of the circulation that gives rise to these heating/cooling atterns in the first lace. The couled nature of the roblem can be illustrated with this flow chart, which reresents an infinite loo of cause-and-effect: 1 1 See Showman, Menou & Cho (2010), Atmosheric circulation of exolanets, in Exolanets [S. Seager, Ed.], Univ. Arizona Press; online at arxiv

8 8 Dynamical basics, art 1 On the Earth, for examle, the equator and oles are not in radiative equilibrium. The equator is subject to net heating, the oles to net cooling, and it is the mean latitudinal heat transort that is both resonsible for and driven by these net imbalances. Figure 1.1: Earth s energy balance vs latitude, averaged over the year. Blue is absorbed sunlight and red is OLR. Because the circulation mutes the latitudinal temerature contrasts, the OLR exhibits weaker latitude variation than absorbed sunlight, and this is what allows the net heating at the equator and net cooling at the oles which in turn drives the circulation. Data are an annual-average for 1987 obtained from the NASA Earth Radiation Budget Exeriment (ERBE) roject. M. Pidwirny, (Source: 1.2 Conservation laws Atmosheric circulation is fundamentally controlled by the laws governing the conservation of momentum, mass, and energy. Conservation of momentum: This is essentially F = ma alied to a fluid. The acceleration of a fluid arcel, which is just the rate of change of the arcel s velocity with time, dv dt, is the sum of the forces er mass that act on the arcel: dv dt = 1 + g 2Ω v + F ρ where is ressure, ρ is density, g is gravity, Ω is rotation rate, v is velocity, and F is friction. On the right-side, the forces that act to accelerate the arcel are the ressure-gradient force, gravity, Coriolis force, and friction. The Coriolis force acts erendicularly to both the lanetary rotation vector, Ω, and to the velocity, v. An imortant rule to remember is that, in the horizontal force balance, the Coriolis force acts to the right of the fluid motion in the northern hemishere and to

9 Showman notes 9 the left of the fluid motion in the southern hemishere. 2 3 Friction is generally small excet within a few cm of a lanetary surface. However, small-scale turbulent mixing can cause accelerations of the large-scale flow that in some cases exhibit qualitative similarities to friction. In models that do not exlicitly resolve such small-scale turbulence, this is often reresented with a frictional term. In a Cartesian frame, where x is eastward, y northward, and z uward, the momentum equations are du dt = 1 ρ x + 2Ωv sin φ 2Ωw cos φ + F x dv dt = 1 ρ y 2Ωu sin φ + F y dw dt = 1 ρ z g + 2Ωu cos φ + F z where u, v, and w are the eastward, northward, and uward velocity comonents, φ is latitude, and F x, F y, and F z are the eastward, northward, and vertical frictional forces. Conservation of mass: Also called the continuity equation, this is 1 dρ ρ dt + v = 0 which simly states that fractional changes in the volume of a fluid arcel (i.e. nonzero divergence) must be accomanied by changes in the density. For order of magnitude considerations, it often suffices to treat the fluid as incomressible, leading to v = 0 Conservation of thermal energy: This is just the first law of thermodynamics, which is dt c dt = Q + 1 d ρ dt 2 Northern and southern hemisheres are here defined as those where Ω k > 0 and < 0, resectively, where k is the local vertical (uward) unit vector. 3 On a sherical lanet, the Coriolis force can intuitively be understood as reresenting two hysical effects. When a fluid arcel moves oleward (equatorward), conservation of angular momentum about the lanetary rotation axis tends to deflect the arcel eastward (westward). When a fluid arcel moves eastward (westward) relative to the lanetary surface, there is an acceleration away from (toward) the rotation axis. It can be seen that, in both cases, the force is to the right (left) of the motion in the northern (southern) hemishere.

10 10 Dynamical basics, art 1 or alternately c dθ dt = θ T Q where θ is otential temerature, θ = T ( 0 /) R/c, where 0 is a reference ressure. Q is thermodynamic heating due to radiation, conduction, latent heating, etc. Together, this set constitutes five equations (the east-west, north-south, and vertical momentum equations, continuity equation, and thermodynamic energy equation) in six unknowns: the three velocity comonents (u, v, w), ρ,, and T (or alternately θ). The system is closed by secifying an equation of state, tyically ideal gas. In all of the above equations, the time derivatives on the left-hand sides give the time rate of change following an individual air arcel as it moves around. This is called the material or total derivative and, for any scalar quantity ψ, is given by dψ dt = ψ t + v ψ The term v ψ is called the advection term, because it reresents the advection (transort) of ψ by the flow. 1.3 Large-scale circulation: Basic force balances We now consider some basic asects of the large-scale circulation. In this section we consider the dominant force balances, in the vertical and horizontal momentum equations, for tyical large-scale circulations. We do this by estimating the magnitudes of the terms in the equations, allowing small terms to be thrown out and exosing the dominant balances Vertical momentum balance: hydostaticity Atmosheres are shallow, with horizontal scales greatly exceeding vertical ones. Consider the magnitudes of terms in the vertical momentum equation for a tyical terrestrial lanet. Let U, W, L, H, τ, and P be the characteristic magnitudes of horizontal and vertical velocity, horizontal and vertical length scale, evolution time scale for the circulation, and ressure change across vertical scale H. Terms in the vertical momentum equations have magnitudes

11 Showman notes 11 w t + v w = 1 g + 2Ωu cos φ ρ z W UW P g ΩU τ L ρh where the numerical estimates are in m sec 2 and use H 10 km, L 10 3 km, τ 10 5 sec, U 10 m sec 1, W 10 2 m sec 1, and P 1 bar, relevant to Earth. Clearly, hydrostatic balance dominates. But much of this is a static balance that has nothing to do with meteorological motions. What we want to do is subtract off the static, background state and look at the balances associated solely with meteorological motions. Define = 0 (z) + and ρ = ρ 0 (z) + ρ, where 0 (z) and ρ 0 (z) are the time-indeendent basic-state ressure and density and, by construction, 0 z ρ 0g. Primed quantities are the deviations from this basic state caused by dynamics. Substituting these exressions into the momentum equation, we obtain ( ) w t + v w = 1 ρ z + ρ g + 2Ωu cos φ W UW P ρ τ L ρh ρ g ΩU where the basic-state hydrostatic balance has been subtracted off. The terms in arentheses give only the flow-induced contributions to the vertical ressure gradient and weight (they go to zero in a static atmoshere). In the order-of-magnitude estimate, 1000 Pa and ρ ρ 10 2 are the characteristic ressure and fractional density erturbations associated with midlatitude large-scale meteorology on Earth, and the numerical values are again in m sec 2. The terms in the arentheses on the right side still dominate by several orders of magnitude. Thus, even after we cancel out the mean (hydrostatic) ressure gradient and gravity, we find that the meteorologically induced ressure erturbations are in hydrostatic balance with the density erturbations: z + ρ g 0 Therefore, vertical accelerations are negligible in the force balance and, for the large-scale circulation, we can relace the vertical momentum equation with local hydrostatic balance: z = ρg

12 12 Dynamical basics, art 1 This analysis can be formalized, and doing so shows that local hydrostatic balance is a good assumtion rovided that (i) L 2 /H 2 1, and (ii) the fluid is stably stratified (see Vallis 2006, Atmosheric and oceanic fluid dynamics, ). Note that this does NOT mean vertical motions do not occur: they still occur, but the accelerations are simly so small that we do not need to kee track of them in the vertical momentum equation Horizontal momentum balance on raidly rotating lanets: geostrohy and the imortance of rotation A similar scale analysis of the horizontal momentum equation can be erformed as follows: v t + v v = 1 2Ωu sin φ ρ y U U 2 P ΩU τ L ρl where again the numbers are m sec 2. The scaling analysis indicates that the time-derivative and advection terms are significantly smaller than the ressure gradient and Coriolis terms, which aroximately balance. This can be formalized by defining the Rossby number, which is just the ratio of horizontal advection to Coriolis accelerations: Ro = U fl where f 2Ω sin φ is called the Coriolis arameter. Thus, whenver Ro 1, the dominant horizontal momentum balance is between ressure-gradient and Coriolis accelerations. This is called geostrohic balance.

13 Showman notes 13 The dynamics at Ro << 1 are significantly different from those at Ro 1. So we can use this to define regimes: Extratroics: Regime of Ro 1. Rotationally dominated (geostrohically balanced to leading order). In the resence of latitudinally varying stellar forcing, can develo significant temerature differences and baroclinic instabilities. Troics: Regime of Ro 1. Dynamics are ageostrohic, though rotation still usually lays a role. Hadley circulation and wave adjustment mechanisms tend to maintain small temerature differences. This gives estimates for the midlatitudes: Rotation Ω L U (m/sec) eriod (sec 1 ) (km) Ro Venus days Earth hours Mars hours Titan days Juiter hours Saturn hours Uranus hours Netune hours HD b 1000? 2.2 days ,000? 0.5 Brown dwarf ? 3-5 hours ? Earth-sized lanet in 10? 30 days ? 2? habitable zone of M star We can see that Earth, Mars, Juiter, Saturn, Uranus, Netune, and brown dwarfs exhibit geostrohic balance at mid-to-high latitudes, but of course f 0 at the equator so geostrohy breaks down there. Slowly rotating lanets like Titan and Venus have Ro 1 across the whole lanet, at least in the uer trooshere and above where winds are fast. Tidally locked hot Juiters, due to faster wind seeds and modestly slower rotation than Earth,

14 14 Dynamical basics, art 1 may have Ro 1, while tidally locked terrestrial lanets in the habitable zones of M dwarfs also have Ro 1 due to their slow rotation eriods. When geostrohy holds, we can write the aroximate horizontal momentum balance as fu 1 ρ y fv 1 ρ x What does this mean? Geostrohy imlies that the horizontal winds tend to flow arallel to the intersection of isobars with a horizontal lane. This is why the ressure field is so useful on a midlatitude weather ma: the winds simly flow along isobars, so one knows the aroximate wind field even if only the isobars, but not winds, are lotted. Imortantly, this henomenon differs drastically from common intuition, in which air flows from high to low ressure. In the midlatitudes of raidly rotating lanets, nature does not abhor a vacuum but rather organizes a cyclone around it. Isobars (black), isotherms (red) and winds (arrows) near 850 mbar level, in the bottom art of the trooshere above the boundary layer.

15 Showman notes 15 Another examle, this one from the 500-mbar level. This shows the uer troosheric jet stream and the baroclinic eddies (associated with baroclinic instabilities) as seen from over the north ole. For March 4, 1984 (Salby 1996). Juiter s Great Red Sot, from Choi et al. (2007, Icarus 188, 35-46).

16 16 Dynamical basics, art 1 Geostrohy also exlains why currents generally flow arallel to the contours of time-mean ocean surface toograhy (when waves and tides are subtracted off): the horizontal ressure gradients just below the ocean surface are roortional to the toograhic gradients. Satellite measurements of global ocean toograhy therefore determine the surface currents. Measurements from NASA Toex-Poseidon sacecraft Vertical Structure of geostrohy: The thermal wind When Ro 1, geostrohy generally holds at all heights excet near the surface, where friction becomes strong. This leads to a tight couling between the winds and temeratures, or more recisely between horizontal temerature gradients and vertical gradients of the horizontal wind. To illustrate, imagine a geostrohically balanced atmoshere in the resence of a horizontal temerature gradient. In the warm air column, the air is less dense than in the cold air column. Because the mass er area contained between isobars 1 and 2 is 1 2 g in hydrostatic balance, the isobars are more widely saced in the warm air column than the cold air column. As the diagram makes evident, this changes the magnitude, and otentially even the sign, of the horizontal ressure-gradient force as you consider different altitudes. When these ressure gradients are geostrohically balanced, this leads to strong variations in the horizontal wind with altitude. Thus, in a geostrohic atmoshere, horizontal temerature gradients imly vertical shear of the horizontal wind. This is called the thermal-wind relationshi. As shown in the diagram, this allows a weather system to be a cyclone at the surface and an anticyclone at the trooause (or vice versa).

17 Showman notes 17 Doing this for an atmoshere is comlicated by the fact that density is not constant, but the situation becomes simle if we convert from height to ressure as a vertical coordinate. To do so, exress a ressure increment d = which imlies that ( ) ( ) = x z z ( ) ( ) = y z z ( ) dx + x z x,y x,y ( ) dy + y z ( ) ( ) z z = ρg x x ( ) ( ) z z = ρg y y ( ) dz z x ( ) Φ = ρ x ( ) Φ = ρ y where hydrostatic balance has been used between the second and third stes and Φ is the gravitational otential (often called geootential in the dynamics literature). This imlies that, in the horizontal momentum equations, the ressure gradient force can simly be reresented as gradients of geootential on isobars rather than gradients of ressure on constant-height surfaces. Geostrohy is then just fu = ( ) Φ y and fv = ( ) Φ x. Differentiating this with ressure, f u ( ) ( ) Φ RT = y f v = x ( Φ ) = y = x ( RT ) (1.1) where we have used the fact that, in ressure coordinates, hydrostatic balance is Φ = 1 ρ, and we have exressed ρ = RT using ideal gas. Absorbing the ressure on the righthand side into the derivative, and assuming R is constant, yields the thermal-wind equations for an ideal-gas atmoshere: f f u ( T ln = R y v ln = R ( T x Here are some examles. Dee ocean currents are generally weak, and the fast surface currents exist in a thermal-wind balance with horizontal temerature gradients. This shows the oceanic Gulf Stream and rings ( 100-km sized vortices that sin off the Gulf Stream): ) )

18 18 Dynamical basics, art 1 Olson (1991), Ann. Rev. Earth Planet. Sci. 19, Figure 1.2: A warm-core ring showing isotherms (dashed) and contours of azimuthal current seed (solid), with direction indicated. Mars: To anels show zonal- and seasonal-mean temerature (K), measured by the Mars Global Surveyor sacecraft (Smith 2008). Bottom anels show zonal- and seasonal-mean zonal winds (m/sec) derived from the temeratures assuming thermal-wind balance and zero wind at the surface. Left column is equinox and right column is northern winter solstice. Notice (i) that greatest vertical wind shear occurs at latitudes where temerature gradients are strongest, and (ii) that redominant wind seed in midlatitudes is eastward.

19 19 Zonal-mean zonal winds ( m sec 1 ) Tem (K) Showman notes June-July-Aug Dec-Jan-Feb Annual average Earth: Left anels show zonal- and time-mean temeratures ( C) and right anels show zonal- and time-mean zonal winds (m/sec) from observations (Peixoto & Oort 1992). Again notice that (i) eak wind shears occur at latitudes with eak temerature gradients, and (ii) redominant wind direction is eastward.

20 20 Dynamical basics, art 1 On the giant lanets, temeratures above the clouds generally exhibit a correlation with jet shear (cold in anticyclonic regions and warm in cyclonic regions), imlying through thermal wind that, in the uer trooshere and lower stratoshere, the jets tend to decay with altitude above the clouds. Some excetions exist. This shows Saturn, but a similar icture holds for the other three giant lanets. The details of why this occurs are not well understood. Read et al. (2009), Planet. Sace Sci. 57, It is imortant to emhasize that thermal-wind balance is not a mechanism to exlain the generation and maintenance of articular wind and temerature atterns; rather, it is simly a statement of force balance. All we really know from thermal-wind balance is that, wherever there are large horizontal temerature gradients, there will be large vertical wind shears. There exist an infinite number of ossible configurations for the wind and temeratures in such a balance, so one needs additional hysics to determine what controls the articular temerature/wind attern on any given lanet. Still, insofar as the time-mean horizontal temerature gradients are redominantly north-south, with the equator being hotter than the ole and the greatest temerature gradients in midlatitudes, thermal-wind balance makes several useful statements: It shows why the time-mean zonal (east-west) winds are stronger than meridional (north-south) winds. In other words, given the temerature distribution, thermal-wind balance rovides a artial exlanation for the zonally banded aearance of the winds on lanets. Given that temerature gradients eak in midlatitudes, thermal-wind hels exlain why the zonal wind shear hence the uer-troosheric zonal winds themselves are greater at midlatitudes than in the troics or at the oles on Earth and Mars. This is the latitude of the jet streams, so thermal-wind balance would aear to lay a role in showing how the winds increase with height in the jet streams.

21 Showman notes 21 Since the equator is generally hotter than the oles on terrestrial lanets, thermal-wind balance imlies that the zonal wind increases (i.e., becomes more eastward) with altitude. Assuming the surface winds are weak, this exlains the redominantly eastward nature of the troosheric winds, esecially in midlatitudes. It is also worth emhasizing that, since thermal-wind balance only constrains the vertical wind shear, and not the wind itself, it says nothing about what controls the wind at the surface.

22 Reference Choi, D.S., Banfield, D., Gierasch, P., Showman, A.P., 2007: Velocity and vorticity measurements of Juiter s Great Red Sot using automated cloud feature tracking, Icarus, 188, Olson, D.B., 1991: Rings in the ocean, Annu. Rev. Earth Planet. Sci., 19, Peixóto, J.P., Oort, A.H., 1983: The atmosheric branch of the hydrological cycle and climate, Variations in the Global Water Budget, Peixóto, J.P., Oort, A.H., 1992: Physics of Climate, American Institute of Physics,.520. Read, P.L., Conrath, B.J., Fletcher, L.N., Gierasch, P.J., Simon-Miller, A.A., Zuchowski, L.C., 2009: Maing otential vorticity dynamics on saturn: Zonal mean circulation from Cassini and Voyager data, Planet. Sace Sci., 57, Salby, M.L., 1996: Fundamentals of Atmosheric Physics, Academic Press,.627. Showman, A.P., Cho, J.Y-K., Menou, K., 2010: Atmosheric circulation of exolanets, Exolanets, University of Arizona Press, Showman, A.P., Wordsworth, R.D., Merlis, T.M., Kasi, Y., 2013: Atmosheric circulation of terrestrial exolanets, Comarative Climatology of Terrestrial Planets, University of Arizona Press, Smith, M.D., 2008: Sacecraft observations of the Martian Atmoshere, Annu. Rev. Earth Planet. Sci., 36, (.5) htt://eoimages.gsfc.nasa.gov/images/imagerecords/57000/ 57752/land_shallow_too_2048.jg (.8) htt:// balance_erbe_1987.jg (.14) htts://commons.wikimedia.org/wiki/file:gfs_850_mb.png (.16) htt://sealevel.jl.nasa.gov/images/ostm/newsroom/ress-releases/ images/too_arrows.gif