Dynamics Rotating Tank

Transcription

1 Institute for Atmospheric and Climate Science - IACETH Atmospheric Physics Lab Work Dynamics Rotating Tank Large scale flows on different latitudes of the rotating Earth Abstract The large scale atmospheric circulation is driven by the differential warming of the Earth due to its spheric shape. Due to the rotation of the Earth a direct heat exchange between the equator and the polar regions is not possible. The large scale circulation is affected by the Coriolis force which leads to the formation of the typical circulation patterns like the trade winds or the jet stream. In this experiment we therefore look at the large scale circulations in different latitudes which represent different circulation regimes.

2 Atmospheric Physics Lab Work Questions to be answered during the reading of the manual (Will be discussed in a small tutorial ahead of the experiment). What are the main features of the atmospheric circulation and where do they come from? How can you explain the formation of the westerlies in the mid-latitudes and the trade winds? What are the conditions for the increase of the geostrophic wind with height? Page 2

3 Dynamics Rotating Tank Table of Contents 1. Introduction Governing equations Geostrophic wind Thermal wind relation Impact of rotation and gravity on a sloping density surface Experiments Experiment 1: Tropical regions/hadley cell Experiment 2: Mid latitudes/baroclinic instability Experiment 3: Polar regions References Page 3

4 Atmospheric Physics Lab Work 1. Introduction In this series of experiments atmospheric flows on different latitudes on the rotating Earth should be simulated. Three typical regions are concerned, namely the high latitudes (polar regions), mid latitudes and tropics. The flow is simulated with a rotating tank filled with water (simulating the atmosphere), the rotation velocity can be changed; this determines the flow regime Governing equations For all our considerations as a basis we have to concern the governing equations of motion for a rotating fluid: d u dt 2 u = 1 p F (1) Here, = gz 2 r 2 /2 describes the gravitational potential including the centrifugal acceleration. Usually, we neglect the acceleration term and set =gz. The total derivative d /dt can be expressed as d dt = t u = t u t v t w t (2) With the time evolution term / t and the advection term u. 2 u denotes the Coriolis acceleration and p the pressure gradient. For a more rigorous derivation of these equations see standard textbooks for atmospheric dynamics, as given in the appendix. The full Coriolis term Page 4

5 Dynamics Rotating Tank 2 cos w sin v 2 u = sin u (3) cos u is usually approximated in the following way: 2 sin v 2 u sin u = f k u, f = 2 sin (4) 0 where denotes the geographical latitude. We use two approximations: 1. The vertical component competes with the gravity term and can be neglected if. by a scale analysis with u 10 m/s and Ω= s 1 it turns out that uω m/ s 2 g 2. For large scale flows, usually the vertical velocity component w is much smaller than the horizontal wind components, therefore all terms including w can be neglected. Thus the governing equation of motion can be written in this form: d u dt f k u = 1 p F (5) du dt fv = 1 p x F x (6) dv dt fu = 1 p y F y (7) dw dt = 1 p z g F z (8) Additionally, we will often make use of the hydrostatic assumption, i.e. dw dt = 0 1 p z g = 0 p z = g (9) 1.2. Geostrophic wind In the atmosphere large scale motion often can be described quite accurate using geostrophic motion. For the derivation of this type of motion we make a scale analysis of the terms in the equations of motions; here, we use typical values of horizontal length and wind speeds (L~1000 km, U~10 m/s) and a time scale typical for synoptic motion (T~10 5 sec). Page 5

6 Atmospheric Physics Lab Work In our fluid two different kinds of motion compete against each other, i.e. pure horizontal advection vs. Coriolis acceleration. In order to compare these two types of motion we use the ratio of typical values. The advection terms ( du/ dt, u u ) are of the same order of magnitude and can be expressed by d u dt ~ U T The Coriolis term can be estimated by ~ m 10 4, u u ~ U 2 s 2 L ~ 10 4 m s 2 (10) f k u ~ f U (11) thus this leads to the Rossby number, i.e. the ratio of advective terms to rotational terms: R 0 = advection term coriols term = U f L = U 2 L 1 f U (12) For the mid latitudes ( = 45 ) the Rossby number is of the order of 0.1. A small Rossby number implies that the rotational term dominates the advective terms, hence in a first approximation they can be neglected: f k u = 1 p (13) i.e. in this case we have a balance of the pressure gradient by the Coriolis term. The geostrophic wind u g is the exact solution of eq.13 and can be derived as follows: From linear algebra we know that: k k u = u, thus: k f k u = 1 k p (14) u f = 1 k p (15) p= i p y j p x (16) u g = 1 f p y, v = 1 g f p x (17) Page 6

7 Dynamics Rotating Tank Remark: Geostrophic motion in an incompressible fluid and with horizontal distances) is non divergent, i.e.: const (over long u g x v g y = 0 (18) Therefore geostrophic motion is generally horizontal, i.e. there is no change of the vertical wind speeds with height. For a better understanding of this feature, we compare eq.18 with the continuity equation in an incompressible fluid ( u = 0 ), this leads to dw /dz=0. If somewhere in the fluid the vertical component of the geostrophic wind is wg=0, then it is zero everywhere. This two-dimensional property of geostrophic motion is reformulated in the Taylor-Proudman Theorem: If a flow is sufficiently slow and steady ( the equation of motion reduces to: Ro 1 ) and frictional forces are neglected, then 2 u = 1 p (19) Whereas the horizontal component is described by geostrophic wind and the vertical component is given by hydrostatic balance. If additionally the flow is barotropic ( = p ), then: u = 0 u z = 0 (20) Or in other words: For a slow, steady frictionless flow of a barotropic fluid the velocity (horizontal AND vertical) cannot vary into the direction of omega. This could be seen in a first experiment (Slowly stirred dye and/or Taylor columns) using the rotating tank experiment Thermal wind relation Although the geostrophic wind relation can be verified quite often in observations of the large scale flow, we also often observe that isobaric surfaces slope down from equator to the pole, i.e. the slope increases with height. This is obviously a contradiction to the Taylor-Proudman theorem although the horizontal wind itself can follow the geostrophic relation. What is going on? The Taylor-Proudman theorem assumes slow, steady, frictionless barotropic fluids. However, most of these assumptions are fulfilled for large-scale motions but the atmosphere is not really barotropic, i.e. the density (or equivalently temperature) varies along isobaric surfaces. Thus the Taylor-Proudman theorem does not apply strictly. However, from the basic equations of motion the change of the horizontal wind components can be derived assuming a density gradient. For this purpose we start again with the definition of the geostrophic wind and regard the partial derivative in height: Page 7

8 Atmospheric Physics Lab Work u g z = 1 f k p z z 1 1 f k p (21) = 1 f k g u g z 1 (22) For the atmosphere, the assumption of an ideal gas is fulfilled well, hence we can reformulate the equations as follows: u g z = g f T k T u g T T z (23) In the end, we can estimate the different components and end up with the following equation: u g T T z g T k T u g g k T (24) z T While our experiment is dealing with an incompressible fluid, which is certainly not an ideal gas, the relation above can also derived using density variations: = 0 ' whereas ' / 0 1. Then the partial derivation can be derived as follows: u g = 1 z z f k p 1 z 0 f k p = (25) = 1 0 f k p z = 1 0 f k g = g 0 f k = g 0 f k ' (26) For water, we can assume a linear relation between Temperature and density: = 0 1 T T 0 (27) With the thermal expansion coefficient α, thus = 0 T. The thermal wind relation now sounds as follows: u g z = g f k T (28) For a cylindrical geometry, the tangential velocity changes with height depending on the radial temperature gradient can be described as follows: u z = g 2 T r (29) Page 8

9 1.4. Impact of rotation and gravity on a sloping density surface Dynamics Rotating Tank Rotation can also counteract to gravity forces not only to horizontal motions as described above. A prominent example for this balance between rotation and gravity is the formation of sloping density surfaces. A simple theory for such frontal zones was formulated by Margules, see also the sketch: The front is formed between two fluids of different densities ( 1 2 ), y denotes the horizontal axis normal to the discontinuity and gamma is the angle between the surface and the axis y. The pressure is the same at both sides, thus it should be the same calculated along the paths [1] and [2]. This yields the following equation: p z z p y y = p 1 z z p y y (30) 2 And the slope of the front can be described using hydrostatic balance at both sides of the front: tan = dz dy = p 1 y p 2 y g 1 2 (31) Using again the geostrophic approximation for the occurring horizontal winds we end up with the following form of the thermal wind relation: u 2 u 1 = g ' tan 2 (32) Whereas g denotes the reduced gravity. Here, the wind components u are parallel to the front. From this relation one can see how the slope of the front is determined by the balance between rotational forces and gravity. Page 9

10 Atmospheric Physics Lab Work 2. Experiments 2.1. Experiment 1: Tropical regions/hadley cell Description: The tank is filled with water (depth ~10-15cm), in the middle of the tank a pot containing a mixture of water, ice and salt is put. The water is settled for about 10 min, a radial temperature gradient is established. The thermal gradient (surface and bottom of the basin in radial direction) must be determined using the thermometer. In principle a weak circulation should be established due to heat exchange (see figure) Then the tank rotates very slowly (about 1 round per minute, mimicking tropical rotation). Due to the rotation and the circulation for the heat exchange, the thermal wind relation should be established (see figure). Page 10

11 Dynamics Rotating Tank The wind shear should be measured by using small scrap of paper, caliumpermanganate and dye. The values obtained during the experiment should be compared with the theoretical values. How good is the estimation? What causes the differences? Phenomenon in the real atmosphere: In the tropics, a large-scale circulation is observed, like the small circulation transporting heat from the warm regions at the equator to the colder regions in the subtropics. Due to the thermal wind relation, a eastwards drift is established, the so-called trade winds. Explain the analogy in more details Experiment 2: Mid latitudes/baroclinic instability Description: We repeat the experiment of the tropics/hadley cell, but with much higher rotational velocity in the end (ca rounds per minute, mimicking higher rotation at the mid latitudes). The circulation for the heat transport will break down; however, for maintaining the heat transport from the mid latitudes (far away from the cold water in the pot) to the pole (cold water) eddies are formed. The mean diameter of the eddies should be equivalent to the Rossby radius of deformation, i.e. The time scale of the disturbances is given by L eddy ~ L (33) eddy ~ L U (34) Where U = d u dz H (35) Page 11

12 Atmospheric Physics Lab Work is the strength of the upper level flow, d u/dz is the thermal wind and H is the vertical scale of the flow. It can be interpreted as the time it takes a flow moving at speed u to travel a distance L. The upper level flow should be measured by using scrap of paper and the eddy length scale as well as the time scale t should be estimated and compared to the findings in the experiment. How good are these estimations? What causes the differences? Phenomenon in the real atmosphere: The heat transport from the sub tropics to the cold mid latitudes is strongly disturbed by rotation and cannot be maintained only by pure heat transport the circulations breaks down (baroclinic instability). The heat transport is then maintained by the formed eddies, i.e. the high and low pressure systems. Explain this analogy in more details, using values of the middle troposphere in the mid latitudes: N ~ 10 2 s 1, H ~ 7km, U ~ 10m /s Experiment 3: Polar regions Description: The tank is filled with water to a depth of about cm. In the middle a hollow cylinder of radius r 20cm is placed and the cylinder is refilled with salty and dyed water (higher density than pure water). The tank is rotated with an angular velocity of about 10 rounds per minutes and is allowed to settle for about 10 minutes. After the settling the hollow cylinder is removed carefully but rapidly such that the disturbances are minimized (practice!! This is not easy!). Page 12

13 Dynamics Rotating Tank Initially, the cylinder of salty water is not in geostrophic balance but after some time a balance between fluids of different densities is reached (Margules theory). Here the question arises how far has the cylinder to slump side ways before velocity and pressure fields come into equilibrium (Rossby adjustment problem). We assume that during the collapse the angular momentum remains constant, i.e. r 2 ur = const. with the initial radius r 1. If the radius is changed by r, the azimutal speed would be u = 2 r (36) For the upper part, r 0, thus u 0, whereas for the lower part delta r>0, thus u<0. This leads to a vertical wind shear, which should be observed using small scrap of paper. For estimating the slope we can assume tan ~ H r with H = depth of fluid, or equivalently: r ~ L = g ' H 2 = NH 2 (37) Whereas L denotes the Rossby radius of deformation with Brunt-Vaisala frequency: N 2 = g ρ 0 ρ z N 2 g ' H, (38) where ρ 0 is the equilibrium density and ρ/ z the change in density with height. For our experiment we have a density difference leading to about g ~0.2 m/s 2 ( / = 2%). The theoretical values should be compared with the values derived from the experiment. Phenomenon in the real atmosphere: Over the Polar Regions or better over the poles a dome of cold air can be observed. Around this dome, the large-scale flow is circulating due to the slope of the cold air (fronts, see figure). Page 13

14 Atmospheric Physics Lab Work 3. References Helmut Kraus, Die Atmosphäre der Erde, Eine Einführung in die Meteorologie, Springer, 2000 Holton, J.R., An introduction to dynamic meteorology, Academic Press, 1992 Lecture material for the lecture Dynamics of large-scale atmospheric flow by Heini Wernli and Mischa Croci-Maspoli which can be found here: Please also visit the two BAMS-papers on Coriolis effects linked there. Page 14