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4 Atmospheric and Oceanic Circulation

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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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