Environmental and Geophysical Flows

Transcription

1 Environmental and Geophysical Flows Organisation Module coordinator: Dr G F Lane-Serff (Room P/B, etn 646, gflane-serff@umistacuk) The material covered in the module is summarised in these notes As a rough guide, each numbered sub-section will be covered in approimately one one-hour lecture slot, though the tutorial sessions will generally be longer A Introduction and equations of motion A Fluid flows in the natural environment A Equations of motion A3 Buoyancy frequency and internal waves A4 Properties of natural fluids A5 Tutorial: Internal waves B Dense and buoyant sources B Gravity currents B Turbidity currents B3 Tutorial: Gravity currents and bo models B4 Buoyant plumes B5 Plumes in a stratified environment B6 Tutorial: Integral plume models C Shallow flows and hydraulic control C Single-layer flows and hydraulic control C Controlled echange flows C3 Tutorial: Weirs and windows D Rotation D Equations of motion and geostrophy D Rossby adjustment and deformation radius D3 Further approimations and viscous effects D4 Tutorial: Rotating flows D5 Appendi: Waves in rotating flows E Miing and turbulence E Convection, Rayleigh and Nusselt numbers E 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 % of the final mark The remaining 8% is from a conventional unseen, closed-book eamination The eamination 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 53 TRI Turner, J S Buoyancy Effects in Fluids CUP Library classmark 53 TUR Pedlosky, J Geophysical Fluid Dynamics Springer Library classmark 55 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 G F Lane-Serff 9-Jan-4

2 A Introduction and equations of motion A 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) Eamples 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 (eg 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 eternal 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 eample, tracking pollution, chemistry, biology, etc, but also for the dynamics of the flow There are some etra 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 approimate equations for fluid flow with (small) density differences We will consider typical properties of natural fluids, the effects of density stratification and internal waves Net 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 miing and turbulence in natural flows, and various approaches to the modelling of these flows based on the underlying physics A 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 u, where the fluid velocity is u(, t) In practice, treating the flow as incompressible is a good approimation ecept for very deep atmospheric convection (vertical scales ~ 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 G F Lane-Serff 9-Jan-4

3 Conservation of momentum for a fluid with gravitational acceleration g = (,, -g) is given by Du Dt p u g, 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, (, t) If diffusive processes are much slower than advective processes (usually the case) then the density of fluid elements is conserved, D Dt Hydrostatic pressure For a fluid in equilibrium at rest (u = Du/Dt = ), the equations of motion reduce to p p p and g y z This implies that the pressure is a function only of z (constant on horizontal planes), with p( z) p ref g dz, where p ref is the pressure at z = 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 (ie 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 (eg 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 (ie 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 approimations It is often useful to write the density as the sum of a constant mean or reference density and a varying part: (, t) We can then epand the pressure in a similar way, about a reference state of hydrostatic equilibrium such that, p g, ie p gz pref z, where p ref is a reference pressure at z = Thus the pressure can be written as a steady part (function of z alone) and variations around this: Substituting this into the momentum equation gives Du Dt p p( z) p(, t) p u g g p, G F Lane-Serff 3 9-Jan-4

4 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 D Dt It is also possible to use fluctuations about a reference state in which the reference density is a function of height, ie (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: z where w is the vertical component of the velocity D Dt w o, Boussinesq approimation If density fluctuations are small then << 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 approimation Dividing the momentum equation by then gives, Du Dt p u g This approimation is valid for most natural flows under conditions where the incompressibility assumption is valid Reduced gravity Under the Boussinesq approimation, we are epecting accelerations of the order of ( / )g This is the net acceleration eperienced by a fluid element of density ( + ) surrounded by fluid of density (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 approimation 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 approimated by the form valid for stationary fluid, p g z 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 approimation is used in many numerical ocean models The approimation is not valid where there are strong vertical motions, eg deep convection, buoyant plumes, solitary internal waves, and so care must be taken in interpreting model results Further approimations 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 approimate the equations of motion These approimate 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 approimations, that may be made G F Lane-Serff 4 9-Jan-4

5 It is always important to understand the physical basis on which the approimations 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 approimations 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 z N z, where N g z The frequency N is known as the Brunt-Väisälä or buoyancy frequency Periods of oscillation (T = /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 n N cos 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 V N tan H n k k where k V and k H 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: n n c k and g k k sin, cos H kv kh c, phase velocity fluid motion group velocity and energy flu oscillating source G F Lane-Serff 5 9-Jan-4

6 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 fied angles to the vertical Internal waves reflected at boundaries are constrained to have a fied 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 fied 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 miing 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: ik wˆ N wˆ ( z e, k wˆ w ) z 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 nz wˆ sin H U, where n is an integer (nth mode) The wave-number corresponding to the nth mode, k n, satisfies H k n n, where F = U/NH is known as the internal Froude number F 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 < /n (least restrictive for n = ) The phase speed of waves is given by (relative to the fluid), c k n N n H 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 k n - is close to the horizontal lengthscale of the hill, ie the length of the hill, L, matches the wavelength of the free mode Critical flow speed (for eciting the first mode) is thus given by NL L H / U crit / G F Lane-Serff 6 9-Jan-4

7 Lee waves are observed in atmospheric and oceanic flows: see Wurtele et al (996: Ann Rev Fluid Mech, 8, 49) 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 approimately C per 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 - ) and specific humidity (q, mass of water vapour per unit mass of moist air) as follows, p RT ( 678q), where R is the gas constant for dry air: R = 874 J kg - K - The specific humidity is limited by the saturation value q S, which is strongly temperature dependent (eg q S = 38 at C, 4 at 37 C) Pure water is an unusual fluid, with a density maimum of kg m -3 at C (at standard pressure: 35 Pa) Water cooler or warmer than this is less dense, dropping to kg m -3 at C and approimately 9584 kg m -3 at C Even freshwater lakes generally have a small salt content which depresses the temperature of maimum density slightly With increasing salinity, the overall density increases and both the temperature of the density maimum and the freezing point decreases Oceanographers usually measure salinity in "practical salinity units" (psu, approimately equal to parts per thousand) For typical seawater with a salinity of 35 to 36 psu, the density (at sea level) is approimately 6 kg m -3, the freezing point is approimately -95 C and the temperature of the density maimum 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 eample, seawater in the deepest parts of the ocean (depths of ~ km) increases in density by approimately 4 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 4 terms for finding densities at atmospheric pressure, with a further 6 terms for finding the density at other pressures For many applications and models simpler approimations are used with a smaller number of terms, though it is often wise to retain some of the non-linearity G F Lane-Serff 7 9-Jan-4

8 Complicated thermodynamics We will not study the detailed effects of the comple nature of equations of state and other thermodynamic properties in this course, but just list a few eamples of the etra 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 mi, 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 maimum 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 B 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 Eamples 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 (eg 98: Ann Rev Fluid Mech, 4, 3) 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 eample for lava flows and mudflows (the latter can also ehibit non-newtonian flow behaviour) Typical features for a dense current on a solid lower boundary: billows tail head nose 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 (eg 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 mied 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 G F Lane-Serff 8 9-Jan-4

9 h u H 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 (968: JFM, 3: 9), ignoring miing 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/ For the half-height gravity current, there is no raised head and no miing behind the head The speed of the flow is given by Fr = / 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/ The theoretical value of the Froude number increases, with Fr as h/h In practice, eperiments 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 approimately 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 echange eperiments: finite releases If a dense body of fluid is released from a channel section of finite etent ("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 bo model as follows If the cross-sectional area of the current is A (fied), and the length is, then the average height h = A/ We further assume that the speed of the current is controlled by some Froude number condition at the front, with the Froude number constant (= Fr, 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 and so, u Fr gh Fr g A /, ~ t /3, u ~ t -/3 Similar equations can be derived for an aisymmetric 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 approimately twice the incoming gravity current (tail) depth will completely block the steady flow (see Lane-Serff et al 995 for further details - JFM, 9, 39) G F Lane-Serff 9 9-Jan-4

10 B 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-mied 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 miture compared to the ambient fluid Eamples included avalanches (the air/snow miture is denser than the surrounding air, resulting in a rapidly advancing gravity current), turbidity currents (mitures of sediment and sea water flowing, for eample, down continental slopes into the deep ocean) and pyroclastic flows (mitures 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 L U H Consider a two-dimensional turbidity current flow as sketched above, with the density contrast is given by = C where C is a non-dimensional particle concentration (with C = at t = and C as t ) The particles drop out, having a fall speed W S We will assume that those that drop out are not reentrained but those remaining in the current are well-mied 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: which rearranges to give A t W S L, C t W S CL A The velocity condition can be written as L t U Fr g A C L / / o Non-dimensionalisation C is already a non-dimensional quantity We introduce scales for length and time, writing L = L S l and t = T S Substituting these into the equations, we find L reduces the equations to S 3 A Fr W g / 5 S 3 and / 5 T S A Fr W S g And thus C Cl and l C l / / G F Lane-Serff 9-Jan-4

11 C l C l / 3 / This can be integrated to give a relation between the particle concentration and distance If the initial length is small compared to L S (ie l ), then the integration yields the result, 5 / C / l 5 Thus the maimum length (which occurs when C = ) is / 5 3 l 5 or L 5A Fr W g / 5 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 bo models B4 Buoyant plumes Buoyant plumes and the entrainment assumption Here we consider an isolated source of buoyancy (eg 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 r u U ( z) e b, r T T ( z) e b 6, where b z 5 m (constants = 85 and = 6) 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 miing 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 (956) Proc Roy Soc Lond A 34: U U G F Lane-Serff 9-Jan-4

12 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 = 85 (this value is not precisely certain: eg 8 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 = 6 and = 8 (instead of 8 and 6) Top-hat integral plume equations (uniform ambient) For simplicity we will ignore the difference in the spread between velocity and density (ie take = ) Volume flow rate Q R U "Momentum flu" Buoyancy flu M R U B R Ug Qg dq dz dm dz R( U ), R g, db dz We use the convention that g is positive (acting upwards) with g g, where the ambient density is and the plume density is - (a function of Z) We assume that the density contrasts are relatively small, so that a Boussinesq approimation is valid The volume flow rate increases by entrainment around the edge of the plume The momentum flu is increased by the buoyancy force - M is actually the momentum flu divided by the density (sometimes called the specific momentum flu) The buoyancy flu is conserved as the total mass flu (actually a deficit here) remains constant A heat flu of kw in air at typical room temperatures is approimately equal to a buoyancy flu of 8 m 4 s -3 We can rearrange the equations to give, dq dz / / M, dm BQ dz M At Z =, M = Q =, so we can integrate these equations to give 4 / 5/ 6 5 / 3 / 3 M BQ, R Z, U 5 5 B Z 4 / 3 B5 Plumes in a stratified environment Uniform Stratification If the ambient fluid has a uniform stratification with buoyancy frequency N, then the buoyancy flu is no longer conserved: db QN dz G F Lane-Serff 9-Jan-4

13 With two dimensional parameters, N and B (having dimensions T - and L 4 T -3, respectively), it is now possible to define characteristic scales for the problem In particular, we can define a length-scale and also scales for the momentum and volume flu, / 4 3/ 4 L N B N, M N B N, 3/ 4 5/ 4 Q N B N We introduce non-dimensional variables (lower case) such that The equations then become Z = zl N, M = mm N, Q = qq N and B = bb dq dz / / m, dm bq db and q, dz m dz with initial conditions m = q = and b = at z = 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 = at z = 4 -/ ) The upward momentum carries the plume up above this level (to z = 37 -/ ), before it falls back and spreads out at its final level (eperiments suggest this is at z = 376) B6 Tutorial: Integral plume models C Shallow flows and hydraulic control C Single-layer flows and hydraulic control Many flows are much larger in their horizontal etent 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 h() () d() u() G F Lane-Serff 3 9-Jan-4

14 The height of the channel bottom (relative to some arbitrary level z = ) is given by d(), where is the downstream distance The flowing fluid occupies a depth of h(), so that the height of the free surface is = h + d It is assumed that the horizontal length scales over which the channel geometry (ie 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 eample, a slowly varying width or for non-rectangular cross-sections (varying slowly with ) 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(), 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 flu (per unit width) given by Q = uh is constant along the channel: ( uh) Q The -momentum equation can be integrated to give the Bernoulli equation for the free-surface (an energy equation), (Flow is steady and frictionless) u g gh, where H is a constant ("head") Unsteady flow equations For unsteady flows the continuity equation gives the rate of change of the fluid depth caused by the gradient of the flu, ( uh) t The depth-integrated, unsteady momentum equation can be written as, ( uh) ( u h ) t gh (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 and the moving fluid layer has uniform density +, 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 ("D flow") Substituting for = h + d and h = Q/u, the Bernoulli equation becomes an equation to be solved to find u, G F Lane-Serff 4 9-Jan-4

15 u gq u g H d Alternatively, a similar equation can be derived in terms of the fluid depth h The downstream distance effectively only appears in the equation as a parameter, through d() 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 maimum value, say d ma at =, then the RHS has a minimum value there The LHS depends purely on the flow, with LHS as u and as u and has a minimum when u gq / 3 or gh / u, with LHS = ½ (gq) /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 < are said to be subcritical, while flows with Fr > are said to be supercritical LHS ½ (gq) /3 RHS = g(h - d) (gh) u The Bernoulli equation is satisfied when LHS = RHS, ie 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 (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 < If the height of the channel bottom increases as 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 > 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? G F Lane-Serff 5 9-Jan-4

16 Flow over a weir d ma z = = We now consider flow through a channel with a single maimum value of d at =, 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 = d ma ), 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 ), but the flow rate Q must take a particular value Writing we find h crit At the crest the flow is critical (with Fr = ), with H d ma, 3 Q h crit gh crit h h crit and ghcrit u The flow is said to be "controlled" in that the flow rate is forced to take a particular value The approach can be etended 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 etended to multi-layer flows and to include the effects of rotation (to apply to large ocean straits) C Controlled echange flows Eamples 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 etent For eample, it is common to define "water masses" in the ocean, which have particular temperature and salinity properties and occupy welldefined 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 eample of this is the echange 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 eample in the echange of warm and cool air through an open door G F Lane-Serff 6 9-Jan-4

17 Rigid Lid approimation For the gravity current in a channel of finite depth we assumed the upper surface was a fied 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 fied horizontal level cannot be regarded as constant and we must allow for variations in the pressure at the upper surface This approimation is known as the rigid lid approimation 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 approimation 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 etended to multi-layer flows and to rotating flows The development of hydraulic control theory is covered in papers by Gill (977: JFM, 8, 64): basic concepts in hydraulic control theory, Dalziel (99: JFM, 3, 35 and 99: JPO,, 88): two-layer flows and rotating flows, and Lane-Serff et al (: JFM, 46, 69): multilayer (non-rotating) flows A brief summary is given here For the single layer flow over a weir described above, we had (for each location ) 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 epressed in terms of a functional G(h; ), satisfying G(h; ) = For the single layer flow G = is just a rearranged version of the Bernoulli equation (in which appears only as a parameter via the geometry of the channel) For a hydraulic problem, it is necessary that G = has multiple solutions and that the geometry has some etremum (in the single layer flow d had a maimum 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 echange 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 approimation) Fr, Fr Fr where Fr i = u i / (g h i ) 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 eist for flows with three or more layers, ecept now the functional is of the form G(h; ) =, with as many equations as there are interfaces (one less than the number of layers) Two-layer echange flows For these flows (the simplest two-layer flows with no net flow along the channel), we can find the echange flow by using the condition Fr = 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/) For a pure sill, the interface is lowered to approimately h = 375H Other cases lie between these limits The flu in each layer is proportional to HW (g H), with the constant of proportionality being /4 for h = H/ and 8 for h = 375H Internal bores and solitons Unsteadiness in the conditions, for eample through tidal variations in the net flow, may generate internal bores on the interfaces between fluid layers A particularly striking eample 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 G F Lane-Serff 7 9-Jan-4

18 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 D Equations of motion and geostrophy In studying natural flows in the oceans and atmosphere it is convenient to work in a frame of reference fied with respect to the rotating earth However, a rotating reference frame is an accelerating reference frame and thus there are etra contributions to the acceleration terms in the equations of motion (Coriolis 835, Laplace 778 & 779): u D p Ω u Ω Dt Ω g u where velocity, u, and position,, are measured with respect to aes rotating (about the origin) with rotation vector, r The etra terms can be put on the RHS and regarded as etra 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 -/r) The centrifugal force term can also be written as the gradient of a potential since Ω Ω r, where r is the distance from the rotation ais Thus if we write g r, 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 approimation 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-ais point north and the -ais east, with the z-ais corresponding to the local vertical direction Writing out the Coriolis terms in full gives, Ω w v, u w, v u, where,, u y reference frame z z y Ω in the local y z G F Lane-Serff 8 9-Jan-4

19 For shallow flows where vertical velocities are small and vertical accelerations (including that due to the Coriolis terms) are small compared to gravity (ie where the hydrostatic approimation is valid), then Ω u fv, fu,, where f = z = sin(latitude) is the Coriolis parameter The dynamics are then identical to those on a flat-plane rotating at a rate f/ about a vertical ais This is known as the f-plane approimation 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) u u ~ Ω u U Ro, fl 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 fv g and fu g y 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 etra 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 epressions for the vertical shear in terms of the density gradients, v g f and z f u g z y (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, g g T T (hence the term thermal wind) For a two layer system (densities and + ) separated by an interface z = h(,y), the change in velocity across the interface is related to the interface slope (subscript denotes the upper layer), f v h v g and f u u h g y D 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 G F Lane-Serff 9 9-Jan-4