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C6.4a Topics in Fluid Mechanics 4–1 4 Atmospheric and Oceanic Circulation If we had to deﬁne what environmental ﬂuid dynamics was about, we might be tempted to limit ourselves to physical oceanography and numerical weather prediction. The wind and the sea are the most obvious examples of ﬂuids in motion around us, and the nightly weather forecast is a commonplace in our perception of our surroundings. Certainly, groundwater levels and river ﬂood forecasting are other environmental ﬂuid ﬂows of concern, but they are more often associated with stochastic behaviour and uncertainty, whereas we all know that ocean currents and weather systems are described, however inexactly, by partial diﬀerential equations. The general idea (which may or may not be correct) is that we know, at least in principle, the governing equations. The diﬃculty with weather prediction is then that the solutions are chaotic. Oceanography and atmospheric sciences, together tagged with the epithet of geo- physical ﬂuid dynamics (GFD), are huge and related subjects which each can and do have whole books devoted to them. In this section we describe brieﬂy some of the prin- cipal phenomena of GFD with a view to making sense of how the Earth’s oceans and winds operate. The atmosphere is a layer of thin ﬂuid draped around the Earth. The Earth has a radius of some 6,370 kilometres, but the bulk of the atmosphere lies in a ﬁlm only 10 kilometres deep. This layer is called the troposphere. The atmosphere extends above this, into the stratosphere and then the mesosphere, but the ﬂuid density is very small in these upper layers (though not inconsequential), and we will simplify the discussion by conceiving of atmospheric ﬂuid motion as being (largely) conﬁned to the troposphere. Atmospheric winds (and thus weather) are driven by heating from the sun. The sun heats the Earth non-uniformly, because of the curvature of the Earth’s surface, but the outgoing long wave radiation is much more uniform. Consequently, there is an energy imbalance between the equator and the poles. The equator is diﬀerentially heated, and the poles are diﬀerentially cooled. It is important to realise that the primary climatic energy balance (which determines the mean temperature of the Earth) is between net incoming short wave radiation and outgoing long wave radiation; the Earth’s weather systems and general circulation arise as a consequence of spatial variation in this balance, and as such are a perturbation to the basic energy balance. Weather is a detail. The oceans are similar. The ﬂuid is water and not air, but the oceans also lie in a thin layer on the Earth. For various reasons, their motion is more complicated and less well understood. For a start, their motion is baulked by continents. The great oceans lie in basins, and their global circulation is dictated to some extent by the topography of these basins. The atmosphere may have to ﬂow over mountains, but it can do so: oceans have to ﬂow round continents. 4–2 OCIAM Mathematical Institute University of Oxford In addition, the oceans are driven not only by the same diﬀerential heating which drives the atmosphere, but also by the atmospheric winds themselves; this is the wind driven circulation. It is not even clear whether this is the primary driving force. A ﬁnal complication is that the density of ocean water depends on salinity as well as tem- perature, so that oceanic convection is double-diﬀusive in nature. (One might say in compensation that cloud formation in the atmosphere means that atmospheric convec- tion is multi-phase convection, but this is not conceived of as being fundamental to the nature of atmospheric motion.) The basic nature of the atmospheric general circulation is thus that it is a convecting ﬂuid. Hot air rises, and so the equatorial air will rise at the expense of the cold polar air. In the simplest situation, the Earth’s diﬀerential heating would drive a convection cell with warm air rising in the tropics and sinking at the poles; this circulation is called the Hadley circulation. In reality, the hemispheric circulation consists of three cells rather than one. The tropical cell (terminating at about 30◦ latitude) is still called the Hadley cell, then there is a mid-latitude cell and a polar cell. This basic circulation is strongly distorted by the rotation of the Earth, which as we shall see is rapid, so that the north/south Hadley type circulations are ﬂung to the east (at mid-latitudes): hence the prevailing westerly winds of common European experience.1 This eastwards wind is called the zonal wind. And it is unstable: a phenomenon called baroclinic instability causes the uniform zonal wind to form north to south waves, and these meandering waves form the weather systems which can be seen on television weather forecast charts. At a smaller scale, such instabilities lead to weather fronts, essentially like shocks, and in the tropics these lead to cyclones and hurricanes. In order to begin to understand how this all works, we need a mathematical model, and this is essentially a model of shallow water theory (or shallow air theory) on a rapidly rotating sphere. 4.1 Basic equations The basic equations describing atmospheric (or indeed, oceanic) motion are those of mass, momentum and energy in a rotating frame, and can be written in the form Dρ +ρ · u = 0, (4.1) Dt Du ρ + 2Ω × u = − p − ρ Φ + F, (4.2) Dt DT p Dρ ρcv − =− · q + Q. (4.3) Dt ρ Dt In these equations D/Dt = ∂/∂t + u · is the material derivative following a ﬂuid element, Ω is the angular velocity of the Earth, Q is a heating rate and the equations 1 A westerly wind is one coming from the west. It will be less confusing to call such a wind eastwards, and vice versa for easterlies, i. e., westwards winds. C6.4a Topics in Fluid Mechanics 4–3 have been written with respect to a set of coordinates ﬁxed in the (rotating) Earth.2 Equation (4.3) is the heat equation used in Waves & Compressible Flow, (1.8) of the Introduction, after the application of the conservation of mass. We therefore have a term that dependent on variations in the density of the atmosphere. This term is typically negligible for small scale laboratory experiments on incompressible materials, but plays an important role in the atmosphere: atmospheric ﬂows are not Boussinesq. The heat ﬂux q is principally due to turbulent thermal conduction (molecular ther- mal conductivity is negligible) and radiative transport. The latter is awkward to quan- tify, but in the limit of an opaque, grey atmosphere, it can be approximated by a Fourier heat conduction law, with an eﬀective radiative thermal conductivity 16σT 3 kR = , (4.4) 3κT ρ where σ is the Stefan-Boltzmann constant and κT is the radiative absorption coeﬃcient (independent of wavelength in a grey atmosphere). If the eddy thermal conductivity is kT , then we suppose ¯ q = −k T, (4.5) where ¯ k = kR + kT . (4.6) The geopotential Φ is the gravitational potential corrected for the eﬀect of centrifugal force, and is deﬁned by 1 Φ = Φg − |Ω × r|2 , (4.7) 2 where Φg is the gravitational potential. The surface Φ = 0 is called sea level; the surface of the oceans would be this geopotential surface in the absence of motion. We take z to be the coordinate normal to Φ = 0; essentially it is in the radial direction, and to a good approximation we can take Φ = gz, where g is called the gravitational acceleration (although in fact it includes a small component due to centrifugal force). Equation (4.3) must be supplemented by an equation of state. In the atmosphere, we take the perfect gas law p = ρRT, (4.8) where R = R∗ /Ma , R∗ is the universal gas constant and Ma is the molecular weight of dry air. 4.1.1 Eddy viscosity The force F represents the eﬀects of friction. Molecular viscosity is insigniﬁcant in the atmosphere and oceans, but the ﬂows are turbulent, and the result of this is that momentum transport by small scale eddying motion is often modelled by a diﬀusive 2 The eﬀect of the rotating coordinate system is that time derivatives of vectors a are transformed as da/dt|f ix = da/dt|rot + Ω × a, because in diﬀerentiating a = ai ei , both the components ai and the unit ˙ vectors ei change with time, and ei = Ω × ei . 4–4 OCIAM Mathematical Institute University of Oxford frictional term of the form ρ T 2 u, where T is an eddy (kinematic) viscosity, and u is an averge velocity on a scale much larger than the eddies.3 More generally, the eddy viscosity varies with distance from rough boundaries.4 A complication in the atmosphere is that the vertical motion is much smaller than the horizontal, and this leads to the idea that diﬀerent eddy viscosities are appropriate for horizontal and vertical momentum transport. We denote these coeﬃcients as H and V , and take them as constants. To be precise, we then represent the frictional terms in the form 2 ∂2u F =ρ H Hu +ρ V , (4.9) ∂z 2 where 2 = ∂ 2 /∂x2 + ∂ 2 /∂y 2 . Frictional eﬀects are generally relatively small. In the H atmosphere, they are conﬁned to a “boundary layer” adjoining the surface, having a typical depth of 1000 metres, and bulk motion above this layer is eﬀectively inviscid. 4.2 Atmospheric vorticity and the Taylor-Proudman the- orem The Coriolis acceleration term 2Ω × u in the equation of motion (4.2) has important eﬀects on ﬂow that are not seen in the absence of rotation. A particularly interesting example is the Taylor-Proudman eﬀect, which can be studied by obtaining an equation 1 for vorticity. Using the vector identity ( × u) × u = u · u − 2 |u|2 and taking the curl of the momentum equation (4.2) yields ∂ω ρ× p F + × [(2Ω + ω) × u] = + × . (4.10) ∂t ρ2 ρ Using the identity ×(A×B) = A ·B+B· A−B ·A−A· B with A = 2Ω+ω and B = u and assuming that Ω is a constant vector gives the absolute vorticity equation Dωa ρ× p F = −ωa · u + ωa · u+ + × , (4.11) Dt ρ2 ρ where the absolute vorticity ωa = ω + 2Ω is the sum of the local vorticity ω = × u and the planetary vorticity 2Ω. Equation (4.11) indicates that the planetary vorticity can induce local vorticity in a ﬂuid. Consider a frictionless (F = 0), incompressible ( · u = 0) ﬂuid in a system such that Dω/Dt = 0, ω 2Ω and ρ × p = 0 (constant density surfaces parallel to constant pressure surfaces). These assumptions are approximately satisﬁed in many laboratory and atmospheric ﬂows. The vorticity equation in this case reduces to the Taylor–Proudman theorem Ω · u = 0. (4.12) 3 In the atmosphere and oceans eddies tend to appear on all length scales, from millimetres to thou- sands of kilometres, and hence a unique eddy viscosity cannot always be deﬁned. This is one of the reasons weather predicion is a challenge. 4 And then, we write F = · τT , where τT = 2 T ( u + uT ). 1 C6.4a Topics in Fluid Mechanics 4–5 Figure 4.1: A vortex street in the wake of the Guadalupe Islands. Despite the fact that these clouds are well above the topography of the islands, the ﬂow is essentially two-dimensional around the islands leading to the formation of the vortices. For a ﬂuid rotating about a vertical axis Ω = Ωk and (4.12) implies ∂u/∂z = 0. Thus if w is zero at some point in the ﬂow (eg a rigid boundary) then it will remain zero everywhere. The ﬂow is purely 2 dimensional and the ﬂuid will appear to move in columns, referred to as Taylor columns, oriented parallel to the rotation axis. Remarkably, if an object such as a cylinder is towed through the ﬂuid, the ﬂow must remain 2D. Fluid above and below the object is forced to mimic the ﬂuid parted by the body allowing a phantom body, composed of the ﬂuid in the Taylor column made from the projection of the body along the rotation axis, to move with the body as if it too were solid. An atmospheric example of this phenomenon is illustrated in ﬁgure 4.1, where clouds high above an island are observed to form vortex streets, as if a ‘phantom’ island were present at the level of the clouds. 4.3 Conservation of potential vorticity Now, the continuity equation (4.1) can be combined with the absolute vorticity equation (4.11) to give Dωa /ρ 1 ρ× p 1 F = ωa · u + 3 + × . (4.13) Dt ρ ρ ρ ρ Consider a scalar ﬂuid property λ (e.g. the temperature T , or potential tempera- ture θ, see later sections). We note that ∂i (uj ∂j λ) = uj ∂j ∂i λ + (∂j λ)(∂i uj ) (using the summation convention) and so we can write D(∂i λ) Dλ = ∂i − (∂j λ)(∂i uj ). (4.14) Dt Dt Taking the scalar product of λ with (4.13) and of ωa /ρ with (4.14) and summing 4–6 OCIAM Mathematical Institute University of Oxford Figure 4.2: Latitude φ and longitude λ in spherical polar coordinates. gives an equation governing the potential vorticity Π = ωa /ρ · λ: DΠ ωa Dλ ρ× p λ F = · + λ· + · × . (4.15) Dt ρ Dt ρ3 ρ ρ End of lecture 13 (21/11/11) If the property λ is conserved following the ﬂow then Dλ/Dt = 0. Furthermore, if λ obeys an equation of state such that λ = λ(ρ, p) then λ · ( ρ × p) = 0. Finally if viscous friction is negligible then the potential vorticity is conserved for each element, i.e. DΠ = 0. (4.16) Dt This result is very important in the atmosphere and oceans, and provides the governing equation for large scale nondissipative motions. For example, the result shows that for a ﬂow with initially zero vorticity, changes in λ (for example by air becoming compressed as it passes over a mountain range) must be balanced by inducing vorticity ω in order to keep Π constant (see problem sheets). 4.4 Hydrostatic and geostrophic approximations The natural coordinates in which to express equations (4.1)–(4.3), when they are applied to the Earth, are spherical coordinates (r, φ, λ), where r is the distance from the centre of the Earth, φ is latitude and λ is longitude (ﬁgure 4.2). It is convenient to introduce small incremental distances dx = r cos φ dλ in the eastward (or zonal) direction and dy = r dφ in the northward (or meridional) direction. We also introduce the vertical distance z from the Earth’s surface, so that r = a + z, where a is the Earth’s radius and dz = dr. Some simpliﬁcations render the momentum equation (4.2) more tractable. The ﬁrst follows from the fact that the atmosphere is relatively shallow: the length scale of major C6.4a Topics in Fluid Mechanics 4–7 weather systems (the synoptic scale) is of the order of thousands of kilometres, whereas the depth scale of the troposphere is about ten kilometres. Thus the aspect ratio of the ﬂow is large. A typical consequence of this large aspect ratio is reminiscent of lubrication theory: vertical velocities are small, and the vertical pressure ﬁeld is almost hydrostatic 1 ∂p = −g. (4.17) ρ ∂z Even in the presence of atmospheric motion, (4.17) is very accurate on large scales and is used in many weather forecasting and climate models. Using (4.17) to replace the vertical equation of motion is known as the hydrostatic approximation. In addition the zonal wind tends to be small compared to the rotation rate of the Earth U 1, aΩ where U is a typical horizontal atmospheric wind speed. We introduce the Coriolis parameter, f = 2Ω sin φ (4.18) so that 2Ω = f k. In this case the horizontal momentum equations can be written as Du 1 ∂p − fv + = F (x) , (4.19) Dt ρ ∂x Dv 1 ∂p + fu + = F (y) , (4.20) Dt ρ ∂y where F (x) and F (y) are components of the frictional force in the eastward and north- ward directions, respectively. Equations (4.17)–(4.20) are commonly referred to as the hydrostatic primitive equations. 4.4.1 Tangent plane approximation Equations (4.17)–(4.20) are expressed in spherical coordinates. However, the use of dx and dy as small eastward and northward distances suggests a further useful sim- pliﬁcation, which is valid when we are considering a comparatively small region near a point P at latitude φ0 and longitude λ0 . In this case we can introduce Cartesian coordinates (x, y, z) on the tangent plane at the point P : clearly there will be little dif- ference between distances (x , y , say) on the surface of the Earth and distances (x, y) on the tangent plane in the neighbourhood of P . We can therefore re-interpret equations (4.17)–(4.20) as applying to the Cartesian coordinates on the tangent plane and avoid complications due to spherical geometry. We must, however, replace f (which varies with latitude) by the constant value f0 = 2Ω sin φ0 . (4.21) Note that equations (4.19) and (4.20) then become identical5 to those for a system that 1 is rotating about the z-axis with angular velocity 2 f0 . This approximation is called the f-plane approximation and the analogous system is the f-plane. 5 Except that centripetal accelerations are again neglected. 4–8 OCIAM Mathematical Institute University of Oxford 4.4.2 Geostrophic equations A further simpliﬁcation arises from the observation that on the Earth, one form of the Rossby number U Ro = , (4.22) f0 l is small; here l is the synoptic length scale.6 A typical value is Ro ≈ 0.1. In this case terms like ∂u/∂t and u∂u/∂x are small and the horizontal momentum equations can be written as 1 ∂p f0 v = (4.23) ρ ∂x 1 ∂p −f0 u = . (4.24) ρ ∂y Thus, for synoptic-scale systems, the horizontal momentum equations reduce to geo- strophic balance, in which the horizontal pressure gradients are balanced by Coriolis forces associated with the horizontal winds. A consequence of this is that u · H p = 0, which implies that horizontal winds are approximately along isobars. This odd behaviour (we normally expect ﬂuid ﬂow to be down pressure gradients) is entirely due to the rapid rotation of the planet. This kind of ﬂow is called geostrophic ﬂow and the velocity ﬁeld ug = (u, v, 0) is called the geostrophic wind. 4.4.3 Thermal windshear equations From the geostrophic equation (4.23) and the ideal gas law (4.8) we have RT ∂p ∂ ln p f0 v = = RT , (4.25) p ∂x ∂x while from equations (4.17) and (4.8) we have g ∂ ln p − = . (4.26) RT ∂z Neglecting vertical (but not horizontal) variations7 in T , cross-diﬀerentiation of these gives ∂v g ∂T f0 ≈ (4.27) ∂z T ∂x and similarly, from equations (4.23), (4.17) and (4.8), we obtain ∂u g ∂T f0 ≈− . (4.28) ∂z T ∂y 6 The synoptic scale corresponds to a horizontal scale typical of mid-latitude cyclonic motion (∼ 3 10 km). Most high and low pressure areas seen on weather maps are synoptic-scale systems. 7 Vertical variations of T need not be neglected if we use pressure p, rather than height z, as a vertical coordinate (Andrews, Introduction to Atmospheric Physics, Section 4.9). C6.4a Topics in Fluid Mechanics 4–9 (a) (b) Figure 4.3: (a) Zonal-mean zonal wind (m s−1 ) for January, 1990. Thin solid lines: eastward winds; thick solid line: zero winds; dashed lines: westward winds. (b) Zonal mean temperature (K) for January, 1990. From Andrews, Introduction to Atmospheric Physics, 2000. These are called the thermal wind equations or, more correctly, the thermal windshear equations; they give a very useful relation between horizontal temperature gradients and vertical gradients of the horizontal wind, when both geostrophic balance and hydrostatic balance apply. An example of thermal windshear is the zonally averaged (ie, longitudinally aver- aged) zonal (ie, east-west) winds in the lower and middle atmosphere, shown in ﬁgure 4.3a, which tend to be nearly in thermal-windshear balance with the zonally averaged temperature, shown in ﬁgure 4.3b. It can readily be checked that the signs, at least, of ∂u/∂z and ∂T /∂y in these ﬁgures are consistent with equation (4.28). Another exam- ple is known as the baroclinic instability and will be discussed later. Equation (4.28) suggests that if ∂u/∂z > 0 then a negative temperature gradient exists in the poleward direction, as illustrated in ﬁgure 4.3b. The temperature gradient induces a density gradi- ent and, if the poleward y density gradient is suﬃciently large, it can induce a convective instability, even if the vertical z density proﬁle is stably stratiﬁed. This instability gov- erns the onset of high and low pressure systems observed in weather forecasts, and will be explored in more detail in section 4.9. 4–10 OCIAM Mathematical Institute University of Oxford 4.5 Thermodynamics of the atmosphere 4.5.1 Local equilibrium Owing to large variations in temperature and pressure in the atmosphere, thermody- namic considerations play an important role. One of the most useful thermodynamic principles is the concept of local equilibrium, in which certain quantities (such as the density) are assumed to be completely determined by local values of the temperature and pressure. Furthermore measurements on a ﬂuid at rest can be used to obtain the function ρ = ρ(T, P ), called an equation of state, which is assumed to characterize the local state of the ﬂowing nonequilibrium ﬂuid. Diﬀerentiating the density equation of state gives dρ = −ραT dT + ραp dp, (4.29) where 1 ∂ρ 1 ∂ρ αT = − and αp = (4.30) ρ ∂T p ρ ∂p T are the thermal expansion coeﬃcient and isothermal compressibility, respectively. In general αT and αp are functions of temperature and pressure. Assuming they are con- stant over a small range in T and p from a reference state (eg room temperature and atmospheric pressure) we can integrate to obtain ρ = ρ0 [1 − αT (T − T0 ) + αp (p − p0 )] . (4.31) For water, which is nearly incompressible, the pressure term in (4.31) is usually negligible and the equation reduces to (3.14). If signiﬁcant variations in salinity are present then End of lecture 14 ρ = ρ(T, p, C) and an approximation of (4.31) leads to equation (3.41). (22/11/11) 4.5.2 Potential temperature Recall the energy equation (4.3). In ﬂuids αT and αp given by (4.30) are typically very small and so the variatiation with density in (4.3) may be neglected to yield the heat equation used elsewhere. For an ideal gas, however, αT = 1/T (see problem sheet) and so the Dρ/Dt term cannot typically be neglected. This term, however, makes the energy equation cumbersome to deal with. Hence in the atmosphere it is often useful to deﬁne the potential temperature p0 R/cp θ=T , (4.32) p where T is absolute temperature, p0 is a reference pressure,8 R is the gas constant for dry air and cp is speciﬁc heat at constant pressure. (Bear in mind that cp = R + cv ). Potential temperature is the temperature an air parcel would have if it were expanded or compressed adiabatically (with no heat transfer to its surroundings) from its existing 8 We shall always take p0 to be the pressure at sea-level, which we deﬁne to be 105 Pa.

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