environmental and geophysical flows · 2004-01-29 · environmental and geophysical flows...

G F Lane-Serff 1 29-Jan-04

Environmental and Geophysical Flows Organisation Module coordinator: Dr G F Lane-Serff (Room P/B20, extn 64602, [email protected]) The material covered in the module is summarised in these notes. As a rough guide, each numbered sub-section will be covered in approximately one one-hour lecture slot, though the tutorial sessions will generally be longer.

A Introduction and equations of motion A1 Fluid flows in the natural environment A2 Equations of motion A3 Buoyancy frequency and internal waves A4 Properties of natural fluids A5 Tutorial: Internal waves

B Dense and buoyant sources B1 Gravity currents B2 Turbidity currents B3 Tutorial: Gravity currents and box models B4 Buoyant plumes B5 Plumes in a stratified environment B6 Tutorial: Integral plume models

C Shallow flows and hydraulic control C1 Single-layer flows and hydraulic control C2 Controlled exchange flows C3 Tutorial: Weirs and windows

D Rotation D1 Equations of motion and geostrophy D2 Rossby adjustment and deformation radius D3 Further approximations and viscous effects D4 Tutorial: Rotating flows D5 Appendix: Waves in rotating flows

E Mixing and turbulence E1 Convection, Rayleigh and Nusselt numbers E2 Instabilities in stratified shear flows (Richardson number) E3 Turbulent length-scales, Monin-Obukhov theory E4 Modelling flows in the natural environment E5 Tutorial: Instabilities and turbulence

Assessment Coursework, consisting of problem sheets handed out at intervals through the module, will make up 20% of the final mark. The remaining 80% is from a conventional unseen, closed-book examination. The examination will have a section of short, compulsory questions covering the basic material in the module followed by a section with a choice of longer questions, testing your ability to apply your knowledge in more complicated cases. Books Tritton, D. J. Physical Fluid Dynamics. OUP. Library classmark 532 TRI Turner, J. S. Buoyancy Effects in Fluids. CUP. Library classmark 532.2 TUR Pedlosky, J. Geophysical Fluid Dynamics. Springer. Library classmark 551 PED Relevant information may also be found in the books referred to in core, first semester modules, and references to scientific journal papers will be given throughout the module.


A Introduction and equations of motion

A1 Fluid flows in the natural environment The natural environment is essentially a fluid environment, contained in or submerged in water or air. This course deals with the types of flow found in these environments (atmosphere, oceans and freshwater). Examples will cover a range of scales, from flows around buildings, in channels and buoyant plumes, through larger flows such as estuary outflows, sea breezes and pollutant dispersion to the largest oceanic and atmospheric flows, in which the effect of the earth's rotation is important. In many engineering fluid dynamics applications we tend to be interested in the effect of the fluid flow on a structure, machine or vehicle. Here we are much more interested in the fluid flow itself, and its ability to transport scalars (e.g. pollution). The style is more of a science course than either a maths or engineering course. We will derive equations, and consider practical applications, but will be particularly interested in the physical understanding of fluid flows and the characterization of the flows in terms of dominant features and scales (which are important in deciding how to model them). The subject is a vast one so the content of the course is selective and influenced by my own interests and background. Furthermore, since boundary layers, turbulence modelling and CFD are covered in other MSc courses, we will cover numerical modelling more briefly in this course. The term Environmental Fluid Dynamics is often applied in a narrow sense to the study of (turbulent) flow around buildings or hills, or in rivers and lakes and its effect on pollutant dispersion. In this course we employ a broader definition to include a wide range of buoyancy-driven flows of moderate scale. For larger scale oceanic and atmospheric flows (and also other planetary and stellar flows) rotation becomes important and equations based on the dominance of rotation can be derived. This is generally referred to as Geophysical Fluid Dynamics (GFD), though the term can be used to encompass a broader range of flows. The flows studied in this course are driven by naturally occurring pressure differences, often resulting from density differences (buoyancy) within the fluid. There is an important natural flow driven by external mechanical forcing, namely tides, but this is not dealt with here. Fluid density is not uniform in the natural environment: sea water is denser than fresh water, hot air rises. Tracer transport is important, not just for, for example, tracking pollution, chemistry, biology, etc, but also for the dynamics of the flow. There are some extra complications (which are only touched on here) connected with changes of state, such as ice formation in sea water, and water condensing in a moist atmosphere. We begin by deriving equations and approximate equations for fluid flow with (small) density differences. We will consider typical properties of natural fluids, the effects of density stratification and internal waves. Next we deal with gravity currents and plumes, which are simple buoyancy-driven flows produced by sources of dense or light fluid. Then we consider shallow flows and hydraulic control theory, which has applications to a range of environments from open windows to ocean straits. The main effects of rotation on fluid flow will be described, identifying important scales and processes. Finally we consider mixing and turbulence in natural flows, and various approaches to the modelling of these flows based on the underlying physics.

A2 Equations of motion Equations of motion For the flows in the natural environment studied in this course we will assume that the fluid can be regarded as incompressible so that the continuity equation becomes

0 u ,

where the fluid velocity is u(x, t). In practice, treating the flow as incompressible is a good approximation except for very deep atmospheric convection (vertical scales ~10 km) and for flow speeds close to the speed of sound (unusual in the natural environment). While we will treat the flow field as effectively incompressible, the variation of the density (and other properties) of fluid elements by changes in pressure and the (slow) diffusion of heat and solutes are important in natural flows.


Conservation of momentum for a fluid with gravitational acceleration g = (0, 0, -g) is given by

guu 2p

DtD

,

where is the fluid viscosity and p the pressure. It is important to note that the density, is not a constant, but is (in general) a function of position and time, (x, t). If diffusive processes are much slower than advective processes (usually the case) then the density of fluid elements is conserved,

0

DtD

.

Hydrostatic pressure For a fluid in equilibrium at rest (u = Du/Dt = 0), the equations of motion reduce to

yp

xp

0 and g

zp

0 .

This implies that the pressure is a function only of z (constant on horizontal planes), with

z

ref dzgpzp0

)( ,

where pref is the pressure at z = 0. The pressure at a point in the fluid is simply equal to the weight of fluid above it with the vertical pressure gradient opposing the gravitational force. Note that we must also have density a function of z alone (i.e. constant on horizontal planes but possibly varying in the vertical). The pressure in a fluid in equilibrium at rest is known as the hydrostatic pressure. Equilibrium does not imply stability. Broadly speaking, a stationary fluid of uniform density is neutrally stable, a fluid whose density decreases with increasing height, z, is stable, while a fluid with density increasing with z (e.g. as a result of heating from below) is unstable. In this course we will use the convention that z always increases upwards, but note that oceanographers sometimes use z to denote depth below sea-level (i.e. increasing downwards). In most natural flows, density contrasts are small and we can use this to manipulate the equations of motion and to introduce useful approximations. It is often useful to write the density as the sum of a constant mean or reference density and a varying part:

),(0 tx . We can then expand the pressure in a similar way, about a reference state of hydrostatic equilibrium such that,

g0 00 p , i.e. refpgzp 00 ,

where pref is a reference pressure at z = 0. Thus the pressure can be written as a steady part (function of z alone) and variations around this:

),()(0 tpzpp x . Substituting this into the momentum equation gives

gguu 0

200 pp

DtD

,


with the terms due to the hydrostatic balance cancelling leaving only the fluctuations in density and pressure (often p is used to denote the pressure fluctuation). The mass conservation equation is simply

0

DtD

.

It is also possible to use fluctuations about a reference state in which the reference density is a function of height, i.e. 0(z). In that case care must be taken as, while the momentum equation has the same form, the mass equation must take account of the fact that the reference density varies in the vertical:

0

DtD

zw o ,

where w is the vertical component of the velocity. Boussinesq approximation If density fluctuations are small then ′ << 0 and so we can ignore the effect of density fluctuations on the inertia of the fluid (though not on the buoyancy force). This is known as the Boussinesq approximation. Dividing the momentum equation by 0 then gives,

guu

0

20

0

pDtD

.

This approximation is valid for most natural flows under conditions where the incompressibility assumption is valid. Reduced gravity Under the Boussinesq approximation, we are expecting accelerations of the order of (′/0)g. This is the net acceleration experienced by a fluid element of density (0 + ′) surrounded by fluid of density 0. (The weight of the fluid element is largely opposed by the buoyancy force due to the surrounding fluid.) This acceleration is known as the "reduced gravity" and is usually denoted by g′ ("g-prime"). Hydrostatic approximation In many environmental flows, horizontal scales are much larger than vertical scales, and vertical accelerations are relatively small. In such cases, the vertical momentum equation can be approximated by the form valid for stationary fluid,

gzp

0 .

The pressure at a point is then given purely by the weight of fluid above it (as if the fluid were stationary). The pressure can then be easily found from the density distribution, and the horizontal gradient used in the horizontal momentum equations. The vertical velocity is then found from the continuity equation. The hydrostatic approximation is used in many numerical ocean models. The approximation is not valid where there are strong vertical motions, e.g. deep convection, buoyant plumes, solitary internal waves, and so care must be taken in interpreting model results. Further approximations Much of the study of fluid dynamics consists of identifying the important physical scales appropriate to the problem under consideration and using these to simplify or approximate the equations of motion. These approximate versions of the equations are usually easier to solve than the original equations and so we can make further progress in understanding the flow. Flows in the natural environment, which have scalings related to buoyancy forcing and the earth's rotation, are particularly rich in the range of scalings, with the corresponding non-dimensional numbers and approximations, that may be made.


It is always important to understand the physical basis on which the approximations are made and to take this into account when interpreting the results of simpler models, and to be aware of the boundaries of the regimes for which the approximations are valid.

A3 Buoyancy frequency and internal waves Buoyancy frequency If we consider a stationary fluid with a density that varies linearly with height, fluid elements displaced from their initial position will oscillate about this position. The restoring force is proportional to the vertical displacement, so that (in the absence of viscous effects), the fluid element would undergo simple harmonic motion.

zNz 2 , where

zgN

2 .

The frequency N is known as the Brunt-Väisälä or buoyancy frequency. Periods of oscillation (T = 2/N) range from a few minutes in the atmosphere and upper ocean to many hours in the deep ocean. Internal waves If fluid elements are constrained to move in a plane at some angle to the vertical, then the component of the gravitational restoring force along the plane will again give oscillations but with a reduced frequency

cosNn . This motion is the basis for internal waves. Seeking wave-like solutions to the (linearised) equations of motion, we find waves with wave-number vector k satisfy the dispersion relation

22

22

tan1nN

kk

H

V ,

where kV and kH are the vertical and horizontal components of k. An important feature of internal waves is that the frequency of the wave depends on the direction of wave propagation but not on the wavelength. The phase velocity and group velocity are perpendicular to each other:

kk

c n and

22

2 cos,sin VHH

kkkn

gc .

group velocity and energy flux

phase velocity

fluid motion

oscillating source


Internal waves have an upper limit on their frequency of the buoyancy frequency, N. Higher frequency disturbances simply generate local turbulence (see §E). (In rotating flows we also find a lower limit of f.) The internal wave energy is transmitted along the group velocity directions, at fixed angles to the vertical. Internal waves reflected at boundaries are constrained to have a fixed angle to the vertical, so the angle of incidence and reflection are not necessarily equal. Where the boundary has a slope equal to the angle of the group velocity (critical slopes), the reflected wave attempts to propagate along the slope. In practice this leads to strong dissipation. If the buoyancy frequency changes slowly, then the direction of propagation of internal waves can be calculated from the local angle made with the vertical (ray tracing). For a wave of fixed frequency, a reduction in the value of N will lead to the wave propagating more towards the vertical until the wave frequency and N are matched. At this point the wave is reflected (though with some loss of energy). Thus waves can be trapped in a region of strong stratification bounded by weaker stratification above and below (known as a "wave guide"). So far we have only considered a stationary ambient fluid. If there is a velocity shear in the fluid it is possible that the Doppler-shifted frequency will vanish (the phase speed matches the local velocity). These regions are known as "critical layers." At the critical layer, large vertical displacements are produced leading to wave breaking and dissipation. This provides a mechanism for transferring momentum to the mean flow and producing mixing at points far from wave generation locations. Lee Waves Lee waves are internal waves formed in the lee of obstacles in stratified flow. If we look for wave-like solutions to the linearised inviscid flow equations in a uniformly stratified fluid (uniform N) with a constant uniform flow U, we can reduce them to a single equation for the vertical velocity component w:

ikxezww )(ˆ , 0ˆˆ 22

2

2

2

wkUN

zw

.

In principle we could use Fourier transforms to solve for flow over arbitrary topography, but it is useful to consider solutions in a simple channel of height H (free modes). This gives solutions of the form

Hznw

sinˆ , where n is an integer (nth mode).

The wave-number corresponding to the nth mode, kn, satisfies

222

22 1 n

FkH n , where F = U/NH is known as the internal Froude number.

The behaviour of the system depends on the value of F. For large F the flow is supercritical and all waves are swept downstream. As F is reduced it is possible to find waves of stationary phase when F < 1/n (least restrictive for n = 1). The phase speed of waves is given by (relative to the fluid),

2/1

2

222

Hnk

Nc

n

.

These results are not substantially altered if we put a shallow obstacle in the flow. The amplitude of forced waves will be greatest when the inverse wavenumber kn

-1 is close to the horizontal lengthscale of the hill, i.e. the length of the hill, L, matches the wavelength of the free mode. Critical flow speed (for exciting the first mode) is thus given by

2/1

2

22

1

HL

NLUcrit .


Lee waves are observed in atmospheric and oceanic flows: see Wurtele et al. (1996: Ann. Rev. Fluid Mech., 28, 429) for a review of atmospheric lee waves.

A4 Properties of natural fluids Potential density and potential temperature The comment on stability (light over heavy => stable, heavy over light => unstable) in an earlier section requires some qualification. A more precise test of stability is to consider the motion of a fluid element if it is displaced upward or downward from its initial position. If there is no force on the displaced fluid element the fluid is neutrally stable, if there is a net force returning the displaced fluid element to its original position the fluid is stable, while if there is a net force tending to increase the displacement the fluid is unstable. If a fluid element is moved up adiabatically (without losing or gaining heat) within a stationary fluid the pressure (and temperature) will decrease. This in turn will alter the density of the fluid element. The stability of the system depends on the new density of the displaced fluid element relative to its new surroundings. It is convenient to define a potential density: this is the density a fluid element would have if moved adiabtically to a reference pressure (typically standard sea level atmospheric pressure). Stability depends on whether the potential density (rather than the actual density) decreases or increases with height, z. We can also define a potential temperature: the temperature a fluid element would have if moved adiabtically to a reference pressure. The usual symbol for potential temperature is , while T is used for the actual "in situ" temperature. Henceforth we will generally work in terms of potential temperature and density for simplicity. In a neutrally stable atmosphere, the potential temperature is constant with height and the actual temperature decreases with height. This rate of decrease in a dry atmosphere (dry adiabatic lapse rate) is approximately 1 C per 100 m. If the actual temperature increases with height (as often happens during still, cool nights) there is said to be an "inversion" (since the temperature gradient is the reverse of the usual case). The potential temperature must increase with height and so the potential density decreases with height, giving a stable system. Equations of state The density of the atmosphere varies with variations in absolute temperature (T in K), total pressure (p in N m-2) and specific humidity (q, mass of water vapour per unit mass of moist air) as follows,

)6078.01( qRTp

,

where R is the gas constant for dry air: R = 287.04 J kg-1 K-1. The specific humidity is limited by the saturation value qS, which is strongly temperature dependent (e.g. qS = 0.0038 at 0 C, 0.04 at 37 C). Pure water is an unusual fluid, with a density maximum of 999.975 kg m-3 at 3.9839 C (at standard pressure: 1013.25 Pa). Water cooler or warmer than this is less dense, dropping to 999.840 kg m-3 at 0 C and approximately 958.4 kg m-3 at 100 C. Even freshwater lakes generally have a small salt content which depresses the temperature of maximum density slightly. With increasing salinity, the overall density increases and both the temperature of the density maximum and the freezing point decreases. Oceanographers usually measure salinity in "practical salinity units" (psu, approximately equal to parts per thousand). For typical seawater with a salinity of 35 to 36 psu, the density (at sea level) is approximately 1026 kg m-3, the freezing point is approximately -1.95 C and the temperature of the density maximum is lower than the freezing point, so that the density of seawater decreases monotonically with temperature (but significantly non-linearly). Though water is even less compressible than air, the density of water (and seawater) is dependent on pressure. For example, seawater in the deepest parts of the ocean (depths of ~10 km) increases in density by approximately 40 kg m-3 compared with its value at sea level. The variation of the density of sea water with temperature, salinity and pressure is described in detail by the generally used UNESCO formulae: these have 14 terms for finding densities at atmospheric pressure, with a further 26 terms for finding the density at other pressures. For many applications and models simpler approximations are used with a smaller number of terms, though it is often wise to retain some of the non-linearity.


Complicated thermodynamics We will not study the detailed effects of the complex nature of equations of state and other thermodynamic properties in this course, but just list a few examples of the extra effects that can arise as a warning when interpreting simple models of natural flows - reality may be more complicated! The non-linearity in the equation of state of water and seawater means that if two water masses of the same density but different temperatures and salinities mix, the resulting water will be denser than the initial density. This results in the phenomenon known as cabbelling in ocean waters and lakes. The presence of a density maximum in fresh water is responsible for thermal bars in freshwater lakes and for the possibility of two overturning phases (during both cooling and heating). The freezing point of water depends on the pressure (being lower at higher pressures) and this has important implications for the melting of the underside of the large Antarctic ice shelves.

A5 Tutorial: Internal waves

B Dense and buoyant sources

B1 Gravity currents Gravity currents (also known as density currents) are the flow of fluid of one density into fluid of a different density primarily in the horizontal direction. The flow may be along an upper or lower boundary or surface (possibly sloping), or occur as an intrusion at some intermediate depth within a stratified fluid. Examples include accidental dense gas releases, the flow of cool air through a doorway into a warm room, sea breezes, river water flowing into the sea at the surface and dense salty Mediterranean water flowing down into the Atlantic. The density contrast can be provided by suspended material, as in avalanches, turbidity currents and pyroclastic flows. See books and papers by JE Simpson for good reviews of the topic (e.g. 1982: Ann. Rev. Fluid Mech., 14, 213). Basic features of gravity currents The flows we consider here will mostly have large Reynolds numbers and miscible fluids. Viscous effects can be important even at large scales in the natural environment, for example for lava flows and mudflows (the latter can also exhibit non-newtonian flow behaviour). Typical features for a dense current on a solid lower boundary: Gravity currents typically have a deeper "head" at the front of the flow, with a shallower tail behind. If the current is flowing over a solid surface, there is a raised "nose" with the dense gravity current fluid overrunning the lighter ambient fluid. This nose is absent when the boundary is a free surface (e.g. fresh water flowing into seawater). The overrun fluid rises up through the head in sheets roughly parallel to the flow direction, giving a plan view of a gravity current its characteristic "lobe and cleft" structure. Immediately behind the head, the shear between the current and the ambient fluid generates billows. These billows and associated turbulence dissipate energy and leave a mixed layer above the tail. Main results for steady gravity currents in channels In a channel of finite total height with no net flow, there must be a counter-flow in the ambient fluid in the opposite direction to the gravity current. Both the character and the speed of the gravity current vary with the relative depth of the gravity current to the total depth of the channel.

head

billows

tail nose


The depth of the current is usually given in terms of the depth of the tail (here denoted by h), rather than the head. We can define a Froude number for the gravity current,

Fr = u/√(g'h). Gravity currents were considered in detail theoretically by Benjamin (1968: JFM, 31: 209), ignoring mixing and viscosity, and treating the system as having two, discrete layers. There is only one solution if there is no energy loss in the system: that with h = H/2. For the half-height gravity current, there is no raised head and no mixing behind the head. The speed of the flow is given by Fr = 1/√2. If it is assumed that there is some energy loss (with dissipation occurring uniformly in the ambient fluid), then there are solutions with h < H/2. The theoretical value of the Froude number increases, with Fr → √2 as h/H → 0. In practice, experiments have shown that the while the lower limit is accurate (for the half height gravity current), the actual upper limit for the Froude number is approximately 1.1 for a shallow current in a deep channel (the error in the theoretical calculation is perhaps due to the assumption about where the energy is dissipated). Lock exchange experiments: finite releases If a dense body of fluid is released from a channel section of finite extent ("lock") by the sudden removal of a vertical barrier, the initial motion (after a short acceleration phase) is a half-height gravity current of constant speed. There is effectively an inverted less dense gravity current flowing in the opposite direction in the upper half of the channel. This upper current reflects from the end of the channel and then propagates in the same direction as the dense current as a bore, eventually catching up with the front of the gravity current after about ten lock lengths. The gravity current now "knows" that it is finite and the current begins to slow down and become shallower. It is possible to construct a simple box model as follows. If the cross-sectional area of the current is A (fixed), and the length is x, then the average height h = A/x. We further assume that the speed of the current is controlled by some Froude number condition at the front, with the Froude number constant (= Fr0, say). This is a reasonable assumption once the height is significantly lower than the channel depth, or for a release of a finite volume of fluid in a deep ambient fluid. Then

xAgFrhgFrux /00 , and so,

x ~ t2/3, u ~ t-1/3. Similar equations can be derived for an axisymmetric flow and also for later phases of the flow when viscous forces dominate. Gravity currents and obstacles A gravity current reaching an obstacle may be completely or partly reflected. The reflected flow takes the form of a bore travelling upstream (communicating the presence of the obstacle upstream and possibly setting up a new steady flow). The initial run-up will reach a greater height than the following flow, which may allow a finite volume (splash) to overtop the obstacle even if there is no steady flow over the obstacle. An obstacle of height greater than approximately twice the incoming gravity current (tail) depth will completely block the steady flow (see Lane-Serff et al. 1995 for further details - JFM, 292, 39).

H

h

u


B2 Turbidity currents An important class of flows are those where the fluid contains a suspension of particles. Despite the different density of the particles compared to the fluid, the particles may remain well-mixed within the flow provided the turbulence levels are high enough. In some cases the particles are simply passive tracers carried by the flow, but often presence of the particles has an important contribution to the relative density of the mixture compared to the ambient fluid. Examples included avalanches (the air/snow mixture is denser than the surrounding air, resulting in a rapidly advancing gravity current), turbidity currents (mixtures of sediment and sea water flowing, for example, down continental slopes into the deep ocean) and pyroclastic flows (mixtures of volcanic particles and air flowing down the sides of volcanoes). During the flow particles may be deposited (as they fall out of the flow) or sometimes eroded and swept into the flow. Consider a two-dimensional turbidity current flow as sketched above, with the density contrast is given by = C0 where C is a non-dimensional particle concentration (with C = 1 at t = 0 and C → 0 as t → ∞). The particles drop out, having a fall speed WS. We will assume that those that drop out are not re-entrained but those remaining in the current are well-mixed. The cross-sectional area A = HL will be assumed to be constant (though we could include entrainment in the model if we thought it appropriate). The speed is given by a simple gravity current model, U = Fr √(g′H). The particles drop from the bottom of the current:

LWt

AS

,

which rearranges to give

CLA

WtC S

.

The velocity condition can be written as

2/12/1

LCAgFrUtL

o .

Non-dimensionalisation C is already a non-dimensional quantity. We introduce scales for length and time, writing L = LSl and t = TS. Substituting these into the equations, we find

5/10

223 gWFrAL SS and 5/11

0322

gWFrAT SS reduces the equations to

ClC

and 2/12/1

lCl

.

And thus

L

H

U


2/32/1 lC

lC

.

This can be integrated to give a relation between the particle concentration and distance. If the initial length is small compared to LS (i.e. l0 ≈ 0), then the integration yields the result,

151 2/52/1

lC .

Thus the maximum length (which occurs when C = 0) is

5/25

l or 5/10

22325 gWFrAL S

.

The total amount of sediment (per unit length) deposited at a given position X can be found by integrating the sedimentation rate (proportional to C) throughout the time after the turbidity current has passed X. We can also find C and l as functions of time (though the resulting integrals can only be evaluated numerically).

B3 Tutorial: Gravity currents and box models

B4 Buoyant plumes Buoyant plumes and the entrainment assumption Here we consider an isolated source of buoyancy (e.g. heat) at the bottom of a body of fluid (see §E for heating from below spread uniformly over a lower boundary). The buoyant fluid rises in a turbulent flow, engulfing ambient fluid as it rises. These types of flows are visualised in the form of smoke rising from a chimney or a bonfire. Measurements show that the mean profiles of vertical velocity and density contrast (in many cases proportional to temperature) fit a Gaussian profile, with the spread increasing linearly with distance from the point source.

2)( br

ezUu

, 2)( br

m ezTT

, where zb

56

(constants = 0.085 and = 1.16).

Note that the measurements suggest a slightly broader spread in the temperature profile to the velocity profile. It possible to attempt to model these flows using turbulence models but a much simpler approach has found to be successful. The mixing of ambient fluid into the rising plume is modelled using an "entrainment assumption" assuming that the flow of ambient fluid into the plume is equivalent to a steady flow proportional to the mean vertical velocity (or other velocity scale). The classic paper is Morton, Taylor and Turner (1956) Proc Roy Soc Lond A 234: 1.

U

U


Integral equations are then derived (integrating in horizontal planes) for conservation of mass, momentum and density contrast. Starting from the Gaussian profiles given above, the entrainment is represented by a velocity proportional to the centre-line mean value, with constant = 0.085 (this value is not precisely certain: e.g. 0.082 has been suggested as a better value). A simpler approach is to use so-called "top-hat" profiles, where properties (such as velocity, temperature) are assumed to have a uniform value within the plume (at a given height Z) and revert to ambient values outside the cone defined by the increasing plume radius. The entrainment velocity is then proportional to the mean plume velocity. The top-hat version of the equations lead to essentially the same results, though with slightly different scalings: under the top-hat approach the constants = 0.116 and = 1.108 (instead of 0.082 and 1.16). Top-hat integral plume equations (uniform ambient) For simplicity we will ignore the difference in the spread between velocity and density (i.e. take = 1). Volume flow rate URQ 2

)(2 URdZdQ

,

"Momentum flux" 22URM

gRdZdM

2 ,

Buoyancy flux gQgURB 2

0

dZdB

.

We use the convention that g′ is positive (acting upwards) with

gg0

, where the ambient density is 0 and the plume density is 0 - (a function of Z).

We assume that the density contrasts are relatively small, so that a Boussinesq approximation is valid. The volume flow rate increases by entrainment around the edge of the plume. The momentum flux is increased by the buoyancy force - M is actually the momentum flux divided by the density (sometimes called the specific momentum flux). The buoyancy flux is conserved as the total mass flux (actually a deficit here) remains constant. A heat flux of 1 kW in air at typical room temperatures is approximately equal to a buoyancy flux of 0.028 m4s-3. We can rearrange the equations to give,

2/12/12 MdZdQ

,

MBQ

dZdM

.

At Z = 0, M = Q = 0, so we can integrate these equations to give

22/52/1

54 BQM

, … ZR

56

, 3/13/13/1

22425

ZBU .

B5 Plumes in a stratified environment Uniform Stratification If the ambient fluid has a uniform stratification with buoyancy frequency N, then the buoyancy flux is no longer conserved:

2QNdZdB

.


With two dimensional parameters, N and B0 (having dimensions T-1 and L4 T-3, respectively), it is now possible to define characteristic scales for the problem. In particular, we can define a length-scale

4/34/10

NBLN , and also scales for the momentum and volume flux,

10

NBM N , 4/54/30

NBQN . We introduce non-dimensional variables (lower case) such that

Z = zLN, M = mMN, Q = qQN and B = bB0. The equations then become

2/12/12 mdzdq

, mbq

dzdm

and qdzdb

,

with initial conditions m = q = 0 and b = 1 at z = 0. The plume entrains denser fluid from low levels and eventually reaches a point where the plume density is equal to the local ambient density (integrating the equations gives b = 0 at z = 1.04 -1/2). The upward momentum carries the plume up above this level (to z = 1.37 -1/2), before it falls back and spreads out at its final level (experiments suggest this is at z = 3.76).

B6 Tutorial: Integral plume models

C Shallow flows and hydraulic control

C1 Single-layer flows and hydraulic control Many flows are much larger in their horizontal extent then in their depth. In this section we will consider such shallow flows, where the fluid can be regarded as having a uniform density (single layer flows) or made of shallow layers, with each layer having a distinct, uniform density (multi-layered flows). The term "shallow water equations" is used for various sets of equations describing shallow flows with one horizontal dimension (flows in rivers, canals, etc), two horizontal directions (flow in estuaries, lakes) and two horizontal directions with rotation (oceans, atmosphere). Here we will deal with the simplest case of flow with one horizontal dimension, also known as open channel flow since it describes the flow of water down simple channels with a free surface (as opposed to enclosed pipe flow). Assumptions

z

x

(x)

u(x)

h(x)

d(x)


The height of the channel bottom (relative to some arbitrary level z = 0) is given by d(x), where x is the downstream distance. The flowing fluid occupies a depth of h(x), so that the height of the free surface is = h + d. It is assumed that the horizontal length scales over which the channel geometry (i.e. d) and the fluid depth (h) vary are much larger than the typical fluid depth h. This is known as the "slowly varying assumption" and effectively means that the slope of the free surface and the slope of the channel bottom are small. Here we will assume that the channel has rectangular cross-section and constant width b, but the equations are very similar for channels with, for example, a slowly varying width or for non-rectangular cross-sections (varying slowly with x). The result of the slowly varying assumption is that vertical velocities and accelerations are small and are thus ignored. We will also ignore the effects of friction on the flow. The velocity of the fluid layer is described by a single velocity u(x), assumed to be constant throughout the layer (it can be taken to be the average velocity). Steady flow equations For steady flows, the continuity equation becomes a simple statement that the flux (per unit width) given by Q = uh is constant along the channel:

0)(

xQ

xuh

.

The x-momentum equation can be integrated to give the Bernoulli equation for the free-surface (an energy equation),

gHgu 2

21

, where H is a constant ("head").

(Flow is steady and frictionless.) Unsteady flow equations For unsteady flows the continuity equation gives the rate of change of the fluid depth caused by the gradient of the flux,

0)(

xuh

t.

The depth-integrated, unsteady momentum equation can be written as,

xgh

xhu

tuh

)()( 2

.

(Detailed derivation not given here.) One and a half layer flows If we have a dense fluid flowing in a channel beneath a deep, stationary layer of slightly less dense fluid (such as relatively cool or saline waters flowing through a channel beneath a deep ocean), then we find very similar equations to those given above. If the upper (stationary) fluid has density 0 and the moving fluid layer has uniform density ′ + 0, then the change to the equations consists of simply replacing the gravitational acceleration g with the reduced gravity g′. Since these flows have one active, moving layer and one passive, stationary layer, the flows are known as "one and a half layer flows" and also as "reduced gravity flows." Returning to the steady equations for flow in a rectangular cross-section channel of constant width ("1D flow"). Substituting for = h + d and h = Q/u, the Bernoulli equation becomes an equation to be solved to find u,


dHgu

gQu 2

21

.

Alternatively, a similar equation can be derived in terms of the fluid depth h. The downstream distance x effectively only appears in the equation as a parameter, through d(x). The RHS depends on the geometry of the channel (here characterised by the height of the channel bottom) and a constant, and is thus independent of the flow variables. If d has a maximum value, say dmax at x = 0, then the RHS has a minimum value there. The LHS depends purely on the flow, with LHS → ∞ as u → 0 and as u → ∞ and has a minimum when

3/1gQu or 2/1ghu , with LHS = 1½ (gQ)2/3 at the minimum. In general, the Bernoulli equation has two solutions for u (and thus for h) for a given value of the RHS, with one value less than √(gh) and one value greater. The ratio of the velocity to this critical value is known as the Froude number:

Fr = u/√(gh).

The speed √(gh) is the speed of small amplitude waves of "long" wavelength on a fluid of depth h, so the Froude number is the ratio of the fluid velocity to a wave speed (cf Mach number). Flows where Fr < 1 are said to be subcritical, while flows with Fr > 1 are said to be supercritical. The Bernoulli equation is satisfied when LHS = RHS, i.e. where the two lines intersect. For a given flow rate Q and head H, the LHS is an unchanging function of u, while the RHS changes with x (as d changes). The form of the graph sketched above gives us some useful information about the behaviour of steady channel flows. Consider a flow where we know the velocity and depth at some point, with Fr < 1. If the height of the channel bottom increases as x increases, then the RHS decreases and so the solution to the Bernoulli equation has larger u (and smaller h). Conversely, if we know that Fr > 1 at some point, then an increase in d results in a reduction u (and an increase in h). In both cases, if d reduces back to its original value as we move downstream, then the flow speed (and depth) also returns to its original value. What happens if d increases sufficiently that the RHS is below the minimum value of the LHS curve, so that there is no solution for u?

u √(gh)

1½ (gQ)2/3

RHS = g(H - d)

LHS


Flow over a weir We now consider flow through a channel with a single maximum value of d at x = 0, with a high water level upstream and a low water level downstream. However, in the solutions we considered earlier, the flow depth always returned to the same level if the channel bottom returned to the same level. In order to move from one solution branch (with low velocities and high depths) to the other (with high velocities and low depths), the flow must be such that when the RHS is at its lowest (when d = dmax), the LHS must be at its minimum. We are still free to set the total energy of the flow given by H (in practice this is the height of the water surface upstream where the fluid depth is so deep that u ≈ 0), but the flow rate Q must take a particular value. Writing

max32 dHhcrit ,

we find

critcrit ghhQ . At the crest the flow is critical (with Fr = 1), with

crithh and critghu . The flow is said to be "controlled" in that the flow rate is forced to take a particular value. The approach can be extended to channels with more complicated geometries, provided there is some "constriction" in terms of the channel width and/or depth. Mathematically the concept of hydraulic control involves solving a matching problem between asymmetric end conditions by finding special cases which allow a smooth transfer from one solution branch to another. Hydraulic control theory has been extended to multi-layer flows and to include the effects of rotation (to apply to large ocean straits).

C2 Controlled exchange flows Examples Multi-layered flows are those flows in which the fluid can be regarded as made up of a discrete number of (two or more) layers of fluid, with each layer having a uniform density. Both the ocean and the atmosphere can be treated in this way to some extent. For example, it is common to define "water masses" in the ocean, which have particular temperature and salinity properties and occupy well-defined layers that may spread of hundreds or thousands of kilometres in the horizontal while occupying a narrow depth range of the order of a kilometre. Even where the ocean or atmosphere is smoothly stratified, treating the fluid as made up of discrete layers still gives useful insights into the likely flow behaviours. In this section we will concentrate on multi-layer flows in channels. An example of this is the exchange flow at the Strait of Gibraltar between dense Mediterranean water and lighter Atlantic water. Similar flows at other ocean straits (sometimes with several layers), but very similar flows can be found at much smaller scales, for example in the exchange of warm and cool air through an open door.

z = 0

dmax

x = 0


Rigid Lid approximation For the gravity current in a channel of finite depth we assumed the upper surface was a fixed lid. In practice the upper surface may be a free surface or other density interface. If the density contrast at this upper interface is much larger than the any of the density contrasts below the interface, then the vertical movement of the upper interface will be relatively small. We can then treat the upper surface as horizontal when considering conservation of mass, which simplifies the equations. However, the pressure at this fixed horizontal level cannot be regarded as constant and we must allow for variations in the pressure at the upper surface. This approximation is known as the rigid lid approximation. It is often used in theoretical models of multi-layer flows in channels and in many numerical ocean models. An advantage in using a rigid lid approximation in numerical models is that the total depth at any point remains constant. (See the §E4 for further details about numerical models.) Hydraulic control The concept of hydraulic control introduced for single layer flows can be extended to multi-layer flows and to rotating flows. The development of hydraulic control theory is covered in papers by Gill (1977: JFM, 80, 641): basic concepts in hydraulic control theory, Dalziel (1991: JFM, 223, 135 and 1992: JPO, 22, 1188): two-layer flows and rotating flows, and Lane-Serff et al. (2000: JFM, 416, 269): multi-layer (non-rotating) flows. A brief summary is given here. For the single layer flow over a weir described above, we had (for each location x) an equation in u (or h) to solve. This equation had two solutions (one subcritical and one supercritical). To match between different upstream and downstream conditions we had to find a flow that allowed us to move from one solution branch to another. In more general terms, we have a problem that can be expressed in terms of a functional G(h; x), satisfying G(h; x) = 0. For the single layer flow G = 0 is just a rearranged version of the Bernoulli equation (in which x appears only as a parameter via the geometry of the channel). For a hydraulic problem, it is necessary that G = 0 has multiple solutions and that the geometry has some extremum (in the single layer flow d had a maximum value). For two-layer flows we find the functional is a Bernoulli equation for the interface between the two layers. In this case there are, in general, three solutions to the equation, two of which are supercritical and one of which is subcritical. For an exchange flow between two reservoirs of fluid of different density over a sill, we find it is necessary to pass through two control points, with the flow passing from supercritical to subcritical and back again. One of the control points is at the sill, while the other (known as a virtual control) is towards the reservoir with the denser fluid. A composite Froude number can be defined for two-layer flows, given by (under the Boussinesq approximation)

22

21 FrFrFr ,

where Fri = ui/√(g′hi) are the layer Froude numbers. It is possible to demonstrate relationships between the Froude number, the functional and the phase speed of interfacial waves. Similar approaches and relationships exist for flows with three or more layers, except now the functional is of the form G(h; x) = 0, with as many equations as there are interfaces (one less than the number of layers). Two-layer exchange flows For these flows (the simplest two-layer flows with no net flow along the channel), we can find the exchange flow by using the condition Fr = 1 at the control, provided we know the height of the interface there. Finding the interface height at the sill is not trivial for general channels which vary in both depth and width (even for rectangular channels). For a channel of constant depth, with a pure contraction in width, the channel is symmetric and the interface is at half the channel depth (h = H/2). For a pure sill, the interface is lowered to approximately h = 0.375H. Other cases lie between these limits. The flux in each layer is proportional to HW√(g′H), with the constant of proportionality being 1/4 for h = H/2 and 0.208 for h = 0.375H. Internal bores and solitons Unsteadiness in the conditions, for example through tidal variations in the net flow, may generate internal bores on the interfaces between fluid layers. A particularly striking example of this occurs at the Strait of Gibraltar when the tides are at their strongest. As with bores on single layer flows, they may be turbulent or undular (with waves behind the jump in interface depth). It is also possible to generate solitons on the interfaces between layers. For two layer flows, the soliton always has the


interface deflecting towards the thicker of the two layers (equivalent to the observation that solitons on single layer flows are always increases in layer depth, not waves of depression).

C3 Tutorial: Weirs and windows

D Rotation

D1 Equations of motion and geostrophy In studying natural flows in the oceans and atmosphere it is convenient to work in a frame of reference fixed with respect to the rotating earth. However, a rotating reference frame is an accelerating reference frame and thus there are extra contributions to the acceleration terms in the equations of motion (Coriolis 1835, Laplace 1778 & 1779):

ugxΩΩuΩu 22

pDtD

,

where velocity, u, and position, x, are measured with respect to axes rotating (about the origin) with rotation vector . The extra terms can be put on the RHS and regarded as extra forces (Coriolis force and centrifugal force). The centrifugal force can be incorporated into the gravitational term as follows. For the conventional gravitational term we can write g = -g where g is the gravitational potential (proportional to -1/r). The centrifugal force term can also be written as the gradient of a potential since

rx2

21xΩΩ , where xr is the distance from the rotation axis.

Thus if we write rg x2

21 , then the centrifugal force can be absorbed into the gravitational

term with g = -, where is known as the geopotential. Surfaces of constant are oblate spheroids and we now define "vertical" to mean in the direction of the vector g. Strictly speaking, we should then work in a coordinate system fitted to the geopotential surfaces, though we will ignore the differences between this and ordinary spherical coordinates here. f-plane approximation and Rossby number In looking at flow near a particular point on the earth's surface, it is often convenient to work in a local frame equivalent to a plane tangent to the earth's surface at the point in question (or, more accurately, a projection from the surface of a sphere onto this plane). It is conventional to have the y-axis point north and the x-axis east, with the z-axis corresponding to the local vertical direction. Writing out the Coriolis terms in full gives,

uvwuvw yxxzzy 22,22,222 uΩ , where zyx ,,Ω in the local reference frame.

x

xr


For shallow flows where vertical velocities are small and vertical accelerations (including that due to the Coriolis terms) are small compared to gravity (i.e. where the hydrostatic approximation is valid), then

0,,2 fufv uΩ , where f = 2z = 2sin(latitude) is the Coriolis parameter. The dynamics are then identical to those on a flat-plane rotating at a rate f/2 about a vertical axis. This is known as the f-plane approximation and is valid for shallow flows over distances where the value of f does not change significantly. The ratio of the non-linear acceleration term to the Coriolis term is given by (for flows of typical speed U and length scale L)

RofLU

~2 uΩ

uu,

which is known as the Rossby number. Small values of Ro imply that rotation dominates the dynamics, whereas large Ro implies that rotation effects are negligible. Geostrophic flow Consider a shallow layer of fluid of uniform density and mean depth H, with small variations in surface elevation . For small Ro, the steady, inviscid equations reduce to

xgfv

and

ygfu

.

The flow is everywhere along lines of constant , with no horizontal divergence. We can relate the surface elevation to the stream-function by the simple relation f = -g . The fact that the mass conservation equation gives no extra information than the momentum equation shows that these equations are degenerate: we will find another conserved quantity to resolve this degeneracy (see below). Flows where there is a balance between the buoyancy forces and the Coriolis forces are called geostrophic flows. For large scale flows in the natural environment this is the primary balance and as a result large scale natural flows are generally close to geostrophic. Thermal wind In a steady, stratified geostrophic flow, where the pressure is hydrostatic, the vertical gradient of the momentum equations gives expressions for the vertical shear in terms of the density gradients,

xg

zvf

and

yg

zuf

.

(More accurately the horizontal gradients should be on levels of constant total pressure, but constant z is usually sufficiently accurate.) Treating the atmosphere as a perfect gas, we can write,

xT

Tg

xg

(hence the term thermal wind).

For a two layer system (densities and + ) separated by an interface z = h(x,y), the change in velocity across the interface is related to the interface slope (subscript 1 denotes the upper layer),

xhgvvf

12 and

yhguuf

12 .

D2 Rossby adjustment and deformation radius Unsteady flow and relative vorticity The linearised unsteady equations for a single-layer flow (mean depth H, surface elevation h) are


xgfv

tu

,

ygfu

tv

,

0

yv

xuH

t.

We can combine these equations to give

02

2

2

22

2

2

fH

yxc

t, where c2 = gH and

yu

xv

is the relative vorticity. For steady flow

2

2

2

2

yxfg

.

The curl of the momentum equations gives a vorticity equation:

00

Hftyv

xuf

t.

Thus the perturbation potential vorticity Q′ is a constant:

2Hf

HQ

,

giving

QfHfyx

ct

222

2

2

22

2

2

.

The Rossby adjustment problem and Rossby radius Consider an initial state of no motion but a step in interface height:

at t = 0,

00

0

0

xx

and so

00

02

02

2

xfxf

QfH

There will be a complicated unsteady motion, but we can find the steady final state since it must satisfy

022

2

22

ff

xc (for x < 0, x > 0).

The solution is

0101

/

/

0 xexe

ax

ax

, where fgH

a is the Rossby radius of deformation.

H


The velocity is u = 0 and axefa

gv /0

.

The Rossby radius is the most important length scale in rotating flows. For flows driven by density contrasts within the fluid, a Rossby radius defined in terms of g′ (rather than g) is used. Instability in rotating flows If we release a dense fluid in a non-rotating system then it spreads out indefinitely. In a rotating system, however, the fluid only spreads out a distance of the order of a Rossby radius before the flow turns and flows perpendicular to the driving density gradient. If the initial body of fluid is a cylinder, released on a horizontal surface then the flow remains axisymmetric and circular in plan view. The shear flow between the dense fluid and the ambient fluid may be unstable, generating a series of vortices (eddies). In general, in flows where the shear is purely in the horizontal direction the instability is known as a barotropic instability (baratropic flow is flow which is uniform with depth or pressure). In flows where the shear is in the vertical (e.g.. between layers of different density) the instability is known as a baroclinic instability. In practice, instabilities are often a mixture, with energy coming both from the shear and from the potential energy. For a radially spreading gravity current, the "front" is unstable, forming waves with a wavelength approximately equal to 2a (where a is the Rossby Radius defined on the initial depth of the current, h0). See Griffiths & Linden (1982, Geophys. Astrophys. Fluid Dynmaics, 19, 159-187), Holford & Dalziel (1996, Applied Scientific Research, 56, 191-207). In non-rotating 2D turbulent flows, an inverse cascade is observed. Small-scale motions are subject ot greater dissipation, so that the observed length scale tends towards large-scale motion (typically a single, large eddy occupying most of the domain). In rotating flows, motion at scales larger than the Rossby radius tends to be unstable, breaking up into eddies that scale with the Rossby radius. As a result, flows in rotating systems are dominated by motion at the Rossby radius scale. For atmospheric flows the Rossby radius is typically several hundred kilometres (weather systems), while in the ocean the scale is around 50 km (oceanic eddies).

D3 Further approximations and viscous effects Quasi-geostrophy The geostrophic equations had no horizontal divergence and thus no vertical motions. For slow changes at time scales greater than 1/f, and to calculate vertical motion, we need to consider (small) departures from geostrophy. In the f-plane approximation it can be shown that the next level of approximation to the inviscid linear equations gives equations of the form,

ytp

fyp

ftv

fuu g

g

2

02

0

111

Departures of similar magnitude also occur because of the non-constant Coriolis parameter. Mid-latitude -plane If we fix local axes around a point of latitude 0 so that the y-axis points north and the x-axis points east, then

yff 0 , where 0cos2

Ryf

.

Approximations based on this linear variation of the Coriolis parameter are known as -plane approximations. A column of water moved southwards (in water of constant depth, H) would have to spin cyclonically to conserve its total vorticity. This is similar to the effect on a column of water moved into deeper water in a simply rotating system (f-plane), since the column would stretch and thus spin up (relative the rotating frame). Columns of fluid moving such that f/H remains constant will not


increase or decrease their relative vorticity, and thus contours of constant f/H are sometimes useful in analysing fluid flow. Equatorial -plane At the equator the value of f changes sign (and there is no useful depth analogy). The equatorial value is = 2/R = 2.310-11 m-1 s-1, where R is the earth's radius.

We can define an equatorial Rossby radius, 2/1

2

cae , where gHc .

Viscous effects: Ekman layer So far we have ignored viscous effects. The ratio between the viscous and Coriolis terms in the momentum equation is

fHfU

HU

22

.

The Ekman number is a non-dimensional number based on this ratio

2

2fH

E .

Viscous Ekman layer: laminar boundary layer at a bottom boundary. It is convenient to split the velocity into a geostrophic and ageostrophic component:

u = ug + uE

As with non-rotating boundary layers, the pressure changes across the boundary layer are small. The geostrophic component satisfies:

xpfvg

1,

ypfug

1

while the viscous part satisfies:

2

2

zufvE

, 2

2

zvfuE

The sum must satisfy the no-slip condition at the boundary, while the viscous part tends to zero away from the boundary. The result (for the viscous part) is

zfi

ggEE eivuivu2/1

21

The boundary layer thickness is HEfE

2/12

. The stress on the bottom is,

ggggs vuvuf

,

2

2/1

.

Similar results can be obtained for flow at a free surface. Integrating the Ekman velocity gives the "Ekman transport" (UE, VE). In steady conditions the Ekman transport in the ocean surface layer is given by (-f VE, f UE) = (1/), where is the wind stress on the ocean. In typical oceanic and atmospheric flows the Ekman layer is turbulent. The precise nature can be complicated (partly because of stratification effects) but the magnitude of the stress is related to the square of the geostrophic velocity,

2*

2 uuC gDg ,


from which we can define a boundary layer thickness zb = u*/f = ()1/2/f. If the turbulence can be represented by a turbulent eddy viscosity T, further matching gives,

avDgT uCf 221 2/1

E.g. at mid-latitude (f = 10-4 s-1) in the atmosphere u ~ 10 m s –1, CDg ~ 0.002 gives T = 30 m2 s-1; in the ocean u ~ 0.1 m s –1 gives T = 310-3 m2 s-1.

D4 Tutorial: Rotating flows

D5 Appendix: Waves in rotating flows Poincare waves Surface waves with frequency and wavenumber kH satisfy the relation

Hg tanh2 , where is related to the wavenumber by

HaHk

H

coth2

22

The depth H << a, so we can use limiting forms: If kH

-1 << a, then ~ kH, and we recover the result for non-rotating surface waves. If kH

-1 >> a, then ~ kH + a-, rotation is important and the fluid moves in anticyclonic circles, r ~ u/f,

~ f. This circular motion is known as inertial motion. The equation for simple progressive wave (x-direction, wavenumber k) is

tkxηη cos0 , tkxkH

u

cos0 , tkxkHfv

sin0 .

Internal waves For rotating flows, the internal wave frequency has both an upper and lower bound N > > f. For ~ N we get the non-rotating internal wave solution as before. If ~ f the fluid motion is still on tilted planes (as for non-rotating internal waves) but now forms circles (ellipses at intermediate frequencies). The wave direction is again independent of wavelength, with 2 = f2 sin2 + N2 cos2, where is the angle of the group velocity from vertical (or phase velocity from horizontal). Coastal Kelvin wave If we look for wave-like solutions which have no normal flow at a boundary along the x-axis, we find v = 0 everywhere. In planes of constant y, the linear equations of motion are identical to those for non-rotating (linear) surface waves. The solution is

tkxe ay cos/0 , tkxe

Hgu ay

cos/

0 ,

with the wave propagating with the coast on the right. = kc, where the wave (phase) velocity c = af = (gH). (For stratified flows we can have g′ for g.)


Equatorial Kelvin wave Here the y-axis points north, with the equator at y = 0. The result is a wave that looks like a pair of Kelvin waves propagating eastward with c = (gH).

tkxecy

cos/

21

0

2

, tkxeHgu

cy

cos/

21

0

2

, v = 0.

There are a series of other equatorially trapped waves. Rossby waves Very low frequency planetary waves, found at mid-latitude (f ~ f0). There is a maximum frequency given by = c/2f. With x-axis eastward, and waves given by = 0 cos(kx + ly - t), the dispersion relation is

220

22 cflkk

.

All have westward phase velocity, but the boundary between westward and eastward group velocity is k2 = l2 + f0

2/c2. The first order geostrophic velocity field is

(ug, vg) = (l, -k) (g0/f0)sin(kx + ly - t)

(but the ageostrophic part is important).

E Mixing and turbulence

E1 Convection, Rayleigh and Nusselt numbers Heat diffusion So far we have assumed that the density of fluid elements remains constant and used this to give continuity equations. In practice heat (and solutes such as salt) diffuse, described by the equation

TDtDT 2

, where the thermal diffusivity is pc

k

.

A typical value for thermal diffusivity in water is = 1.410-7 m2 s-1, for air 210-5 m2 s-1, while the diffusivity of salt (NaCl) in water is 1.110-9 m2 s-1. The diffusion is generally sufficiently slow that we can still treat the flow as incompressible, but the change in density that accompanies the change in temperature (or salinity) is important. If the density is linearly related to temperature (usually true for small temperature perturbations) then we can write = 0 + ′ and T = T0 + T′, with ′= -0T′ (e.g. = 2.110-4 K-1 for water at 20C), and the density obeys the same diffusion equation as temperature,

2

DtD

.


Heating from below z = d z = 0 Consider a fluid which is held at temperature T at some horizontal level z = d, and heated at some lower level to a temperature T + T. A purely conducting state, with no flow and uniform temperature gradient, is an equilibrium solution to the heat and flow equations.

0u , gp and )1(0 cz , where dT

zT

zc

0

1.

We examine the stability of this solution by examining small perturbations about this basic state with perturbation velocity u, pressure p′ and density ′. The linearised equations become

gu pt2

0

wct 02

0 u

(where ∂t means ∂/∂t). These can be manipulated to give a single equation for the vertical velocity, w,

wgcw htt2222

,

where 2

2

2

22

yxh

is the horizontal part of 2

.

Rescaling lengths with d (so that the surfaces are now at z = 0, 1) and times with d2/, this equation becomes,

ww htt2222 RaPr .

Here Pr is the Prandtl number Pr = / (sometimes written as p or ). Typical values at 20C: Pr = 0.72 (air), 7.1 (water). Ra is the Rayleigh number (sometime written as A or R), given by,

3TdgRa .

It can be shown that the system is unstable for sufficiently large values of Ra. If the upper and lower levels are rigid boundaries, the critical value of Ra is 1708, if one of the boundaries is a free surface the critical value is 1101, and for two free boundaries 658. Physical interpretation A "blob" of fluid of dimension heated at the lower boundary will begin to rise. The rise speed is given by a balance between the buoyancy force and viscous forces.

0/ 20 wg , so

0

2

~ gw .

T

T+T


But heat will diffuse out of the blob as it rises giving,

2~

t, so 2/~

t

se .

The vertical velocity decreases with time and the blob only moves a finite distance before "running out of steam."

2

0

2

max ~ sgz .

This distance is greatest for the largest possible blobs ~ d, and for convection we need zmax ~ d or larger. This again implies we need Ra bigger than some order one value for convection. Convection patterns The first convection pattern consists of two-dimensional circular rolls, occupying the full depth of the cavity. Weakly non-linear theory is needed to establish the patterns as Ra increases (and in practice the variations of viscosity and surface tension with temperature have an effect): the next flow pattern is hexagonal cells (flow up the middle and down the sides or vice versa). As Ra is increased still further, a thin heated boundary layer is formed against the lower boundary, and buoyant fluid rises in chaotic turbulent plumes, with plumes more closely spaced as Ra increases. Nusselt number The ratio of the total heat flux to the heat flux in the basic conducting (no flow) state is known as the Nusselt number. At low values of Ra, it can be shown that

crit

critT Ra

RaRaN

21 ,

while for very large Ra, the heat flux is independent of d, and NT = 0.085Ra1/3.

E2 Instabilities in stratified shear flows (Richardson number) Consider the stability of a simple two-layer shear flow, with velocity U1 in the lower layer and U2 in the upper layer (both in x-direction). We assume inviscid, irrotational flow within each layer, with a "vortex sheet" separating the two layers (this allows a jump in velocity across the sheet). We examine the stability of this flow by considering small deviations from a basic state, where the basic state has a horizontal vortex sheet on the plane z = 0. The densities of the two layers are denoted by 1 and 2. Since the flow is irrotational (within each layer) we can write the velocity as the gradient of a velocity potential with

iiu (i = 1, 2). The position of the interface is given by (x, y, t). While the basic state is two-dimensional, we allow for three-dimensional disturbances (i.e. dependent on y). The basic state simply has 1 = U1x and 2 = U2x, with interface position = 0. The continuity equation becomes:

012 (z < ) and 02

2 (z > ). The disturbance is assumed to be confined to a finite region near the interface, so

ii U (i = 1, 2) as z . While there is a jump in the tangential velocity at the interface, the normal velocity above and below the interface must be equal and the interface itself is a material surface. This can be expressed as


yyxxtziii

(evaluated at z = , i = 1, 2)

The Bernoulli equation for irrotational flow (with buoyancy) can be expressed as

ii

ii

i Cgzpt

2

21

throughout the appropriate region, (i = 1, 2).

The pressure must be continuous across the interface (p1 = p2 at z = ). The relationship between the constants is found from the basic flow. The stability of the basic flow is examined by adding small disturbances to the basic state and deriving linear expansions of the governing equations. Wave-like solutions of the form

1101 , with tlykxiez )(11

ˆ are sought for the velocity potentials and the interface height (though of course does not have any z-dependence). It is found that the velocity potentials decay exponentially away from z = 0, with lengthscale 1/|k|, where |k| = (k2 + l2)1/2. After some manipulation, we find a quadratic in , with solution

2/1

21

212

21

22121

2

21

2211

gUUkUUik

k

For instability we need a non-zero real part of i.e. we need

22121

222

21 UUkg k .

For a given total wave number |k|, the most unstable mode propagates in the direction of the basic flow, with k = |k| and l = 0. For U1 ≠ U2, the flow is always unstable to large k (short waves), though in practice these will normally be damped by viscous and (possibly) surface tension effects. Special cases a) Surface gravity waves: U1 = U2 = 0, 2 = 0.

c = Im()/|k| = √(g/|k|) = √(g/2), where = 2/|k| is the wavelength. b) Internal gravity waves: U1 = U2 = 0, 2 ≠ 0.

c = √(g′/2|k|), where g′ = (1 - 2)g/M and M is the mean density. (Similar to surface waves, but the inertia of the upper layer is now important. c) Pure shear: U1≠ U2, 1 = 2.

= -(ik/2)(U1 + U2) ± k(U1 - U2). So the phase velocity is the average velocity of the flow in the direction that the waves are propagating. The roll-up of the vortex sheet can be understood in physical terms by considering the effect of the background flow on crests and troughs of an initial sinusoidal disturbance. Neighbouring crests and troughs are brought together and separated by the background shear. The initial instability is 2D, but


the rolls themselves become unstable to a 3D convective instability (as heavy fluid is brought above lighter fluid). Thorpe (1971: JFM, 46, 299) has some excellent experiments on two-layer shear flow. For single layer flows (beneath an overlying, deep, stationary layer), the stability of the flow can be characterised in terms of a layer Richardson number Ri = g′h/U2. See below for a definition of a local Richardson number for continuously stratified flows and conditions for stability. The Richardson number is the ratio of the PE needed to mix up the fluid to the available KE, when the number is small we expect mixing to be possible. Richardson number Above we showed how shear in layered fluids could produce instabilities and introduced a layer (or bulk) Richardson number. For any general sheared, stratified fluid we can define a Richardson number, based on the local shear and stratification,

2

2

2

zU

N

zU

zg

Ri .

The Richardson number (sometimes denoted by J) can be regarded as a ratio of the potential energy that needs to be overcome to mix up the density profile vs the available kinetic energy due to the shear. Low values of the Richardson number indicate the potential for instability and the Miles-Howard theorem shows that a necessary condition for instability is that Ri < 1/4 somewhere in the flow Miles 1961: JFM, 10, 496; Howard, 1961: JFM, 10, 509; Mile 1963: JFM, 16, 209).

E3 Turbulent length-scales, Monin-Obukhov theory Turbulence and mixing Natural flows are generally turbulent and thus tracers (heat, salt, pollutants, etc) mix more rapidly than just by molecular diffusion. The turbulence is often characterised in terms of an eddy viscosity T and the mixing of other properties represented by a turbulent diffusion coefficient T (possibly different for different tracers). Since it is the same eddies that are mixing both momentum and the other properties you would expect T ~ T. The ratio is called the turbulent Prandtl number PrT = T/T. (Suggested value of PrT = 0.9 for a neutrally stable atmosphere). In many natural flows the fluid has a stable density gradient which tends to limit the vertical extent of turbulent motions and thus the vertical mixing. The result is that horizontal mixing is much faster (over longer length scales) than vertical mixing. Turbulence closure schemes which ignore stratification (e.g. ordinary K- models) are not very accurate for stratified flows. Schemes which include a representative length scale generally do a better job, though the relationship between the scales described below to those used in turbulence closure models is not straightforward. Length scales In what follows, w represents the fluctuations in the vertical velocity, q the rms of the overall velocity fluctuations, the dissipation of turbulent energy, ' the fluctuations in the density. The buoyancy scale is the vertical scale over which all the eddy kinetic energy is converted into potential energy, and is thus the maximum vertical turbulence scale.

NqNwlb /12/12 or estimated from the density profile 1

2/12

z

ld .

The Ozmidov scale is based on the local dissipation rate

2/1

3

NlO .

The Thorpe scale is an attempt to determine the average size of overturns (which put heavier fluid over light) in a measured density profile. The measured profile is rearranged into a profile with the


density increasing monotonically with depth. The rms of the vertical displacements needed to achieve this is called the Thorpe scale, lT. These scales are all closely related. For example, Moum (1996: JGR 101, 14905 and 16500) finds

dObT llll 6.11.1 . and also found that the dissipation rate is linked to the buoyancy scale by

blw 37.0

(though with some variation in the constant 0.7).

Fluctuations that are slow compared with the buoyancy frequency can generate internal waves which carry away and dissipate energy without mixing. If the turbulence is characterised by a frequency fT, then the ratio of this to the buoyancy frequency is given by,

TT Fr

Nlw

Nf

.

In practice, for low turbulent Froude numbers (FrT < 1/3), the mixing is suppressed. Monin-Obukhov similarity theory For turbulent flow near a boundary in a fluid of uniform density (neutrally stable), mixing length theory gives the classic logarithmic profile. The length scale for the turbulent motions is assumed to be proportional to the distance from the surface (i.e. z) and the stress (momentum flux) is assumed to be uniform. For flows with a steady buoyancy flux at the base (such as the atmosphere above a heated or cooled ground) we can use a similar approach, but we now have a new length scale available. The Monin-Obukhov scale is given by

bMO Q

uL

3* ,

where Qb is the buoyancy flux (per unit area), u* the friction velocity and the von Karman constant. The sign convention is that Qb is positive for stable conditions (cooling from below) and negative for unstable conditions. Thus LMO could be positive or negative. The von Karman constant is not always included (but it is convenient to keep it in for comparison with the usual “law of the wall”. We then find that the mean velocity profile must obey,

Lz

zU

uz

*

, where is a function of the similarity variable = z/L.

(Similar equations can be obtained for the fluxes of heat and other quantities.) For neutral conditions, = 0, and (0) = 1 (recovering the usual law of the wall). The behaviour of has been established experimentally and a good fit is given by,

unstable ( < 0): () = (1 – 1)-1/4, stable ( > 0): () = (1 + 1) (where the constants 1 ≈ 16 and 1 ≈ 5). In a stable atmosphere this gives the so-called log-linear wind profile,

Lzz

zz

uU 0

10*

ln1

.


The mixing length does not increase so rapidly with height (eventually limited by the buoyancy scale), and the result is that the wind profile increases more rapidly with height in a stable atmosphere than in a neutral atmosphere. The ratio of LMO to the height, h , of the atmospheric boundary layer (usually capped by a strong inversion) is a useful way of characterising the stability of the boundary layer.

E4 Modelling flows in the natural environment Earlier in the course we saw how integral models could be used to describe turbulent buoyant plumes and how box-models could be used to analyse spreading gravity currents. Sometimes these models can be obtained formally by integrating the equations of motion, typically in planes perpendicular to the main flow direction. The purpose is to produce simplified models that capture the important physical processes but are more easily solved than the full equations. It is often valuable to identify the important scales in the problems and non-dimensionalise the equations to recover more general results. The scalings are useful in themselves, while the equations can sometimes be solved analytically, or at least using simpler numerical methods than full scale CFD. Examples of processes Turbulent plume Gravity current on slope Meltwater plume Dispersion models The mean distribution of a contaminant released from a source in a turbulent flow can be described in terms of a Gaussian distribution. The downstream motion depends on the mean speed while the spread depends on turbulent diffusion. The simplest models use a fixed value for this diffusion, but the effects of stratification and flow around obstacles (e.g. hills, buildings) on the turbulence may be modelled using more sophisticated models for the lateral diffusion. Multiple point sources and line sources (e.g. simulating roads) can also be included, with the mean contamination at a point found by summing the contribution from the various sources. Sub-models describing the initial rise from buoyant sources can be included. This approach is widely used in modelling atmospheric dispersion of pollutants (e.g. www.cerc.co.uk). It is reasonably successful in modelling mean concentrations but is less useful for predicting extreme values (needed when considering toxicity, for example). Scales in natural flows The contrast between the horizontal and vertical scales in natural flows obviously influences the way numerical models are constructed. For example, in ocean models the grids may be spaced at only a few

U

U entrainment

Usin entrainment

Drag -kU2

Ice shelf

meltwater

melting/freezing


metres in the vertical (especially near the sea surface) but at several kilometres (or tens of kilometres) in the horizontal. The difference in horizontal and vertical mixing may be represented by very different horizontal and vertical eddy viscosities. Since the tracers have a direct effect on the dynamics it is important that their transport is represented well (often using high order advection schemes), and finite volume techniques (which conserve fluxes accurately) are preferred. Model types Rectangular grid Often with a fixed set of vertical levels, more closely spaced near the sea surface (ocean models) or ground (atmosphere). The boundaries are often aligned north-south and east-west, so in global models the boxes aren’t perfectly rectangular (especially near the poles). The representation of topography as a series of steps generally leads to enhanced mixing. (e.g. www.soc.soton.ac.uk/JRD/OCCAM/) Sigma-coordinate models Horizontal grid in conventional boxes but the vertical grid adapts to the topography so that the vertical coordinate is a fraction of the total depth (or the depth to some convenient horizontal level). These grids are also known as “terrain-following”. Because the pressures are largely hydrostatic, the tilted grid boundaries can lead to artificial large horizontal pressure gradients. In practice this limits the topographic slopes than can be modelled (often dealt with by some smoothing of the topography). (e.g. http://marine.rutgers.edu/po/models/spem.html) Isopycnic models Used in ocean models. Again a standard horizontal grid but the stratification is represented by a series of layers of fixed density, and the equations are cast in terms of these layer depths. This captures some essential features of the flow (in that the flow and mixing is generally along isopycnic surfaces). Unfortunately the non-linear equation of state makes it hard to construct a global model because density contrasts can change with pressure. (e.g. http://panoramix.rsmas.miami.edu/micom) Finite element models Used more for depth-averaged or single-layer flows (e.g. in tidal analysis). The domain is split into a series of elements (typically triangular) with a greater density of elements near boundaries and other complicated parts of the flow. Recasting the equations for the grid elements (and labelling them) can be cumbersome. Spectral models Instead of using physical grid points these work in a frequency domain, using appropriate sets of basis functions. These have been particularly popular for atmospheric models (since it is relatively straightforward to construct functions for a spherical layer). Less popular for ocean models, though it is possible to use mixed models (e.g. represent the vertical variation in terms of normal modes). Since a finite range of frequencies is used, spectral models are weak when it comes to representing sudden changes such as fronts. Special model elements Boundary layers E.g. ocean surface mixed layer, bottom boundary layers. These are sometimes modelled explicitly, with a depth h(x,y) and properties that also vary with the horizontal coordinate (but assumed to be well-mixed in the layer). There are usually some special model components that deal with the changes in depth of this layer and fluxes to/from the rest of the fluid. They can be put on the top (or bottom) of most of the other model types (rectangular grids, isopycnic, sigma-coordinate). Convection schemes How to deal with cases where dense fluid lies above lighter fluid. Some schemes mix thoroughly (this is probably overdoing it) while at the other extreme the dense fluid is simply placed below the light fluid and the light fluid moved up. Turbulence models In ocean models is it is still common to use simple fixed eddy viscosities, usually fixed sufficiently large to ensure numerical stability. These are generally several orders of magnitude lager in the horizontal than in the vertical. Attempts to improve the representation at least of the vertical mixing range from Richardson number dependent eddy viscosities up to sophisticated closure schemes.


Other processes Various biogeochemical processes can be included in the models for comparison with observed data. Also important are processes such as ice formation and advection. Sometimes (for coarse models) it is necessary to represent small scale processes (e.g. flow through a small strait that can’t be represented properly) by some kind of extra “process model.”

E5 Tutorial: Instabilities and turbulence

environmental and geophysical flows · 2004-01-29 · environmental and geophysical flows...

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