5 Shallow water Q-G theory.

Transcription

1 5 Shallow water Q-G theory. So far we have discussed the fact that lare scale motions in the extra-tropical atmosphere are close to eostrophic balance i.e. the Rossby number is small. We have examined the solutions to the equations of motion in ths shallow water system and found that there were several different types of wave motion that exist under various circumstances. In the presence of rotation the solutions to the linearised shallow water equations ave Poincaré waves (or inertia-ravity waves) and the zero frequency solution which is the eostrophically balanced flow. The Poincaré waves are fast ravity wave motions modified by the presence of rotation that rely on departures from eostrophy for their existence (we need the /t s in the equations of motion). We then went on to discuss Rossby wave motion which is a wave motion that occurs in the presence of a backround radient of potential vorticity. We looked at the particular case of a riid lid situation (i.e. a situation that does not allow for ravity wave motions) and demonstrated usin potential vorticity conservation that this meridional displacement of fluid elements ave a wave motion that propaated to the west relative to the mean flow. This type of wave motion is of a larer scale and has a slower phase speed than that Poincaré wave motions, it is closer to eostrophic balance. It is this Rossby wave motion that dominate in the lare scale extra-tropical atmosphere. We then went on to discuss the Geostrophic adjustment problem i.e. how does an initially unbalanced state equilibrate to a eostrophically balanced state. When examinin this in the presence of rotation we ran into a problem - Geostrophic deeneracy. This problem is that in the steady state, any velocity field that is in eostrophic balance with the heiht field not only satisfies momentum balance but it also automatically satisfies mass conservation i.e. we have too many unknowns and not enouh equations. We can t determine which eostrophically balanced solution is the correct one. In the particular case we examined of an initial state that is at rest and has a small heiht perturbation we were able to make use of potential vorticity conservation which provided us with an additional constraint on the field that allowed us to calculate the equilibrium situation without havin to determine explicitly its evolution toward that state. We did however, discuss that the process of this evolution involved the presence of the Poincaré waves that are produced by the initial departures from eostrophy. These fast wave motions propaate away quickly leavin, on a loner timescale, the eostrophically balanced state. This is why, when we examine the atmosphere on timescales of days to weeks it is predominantly close to eostrophic balance. But what about more enerally? The atmosphere is close to eostrophic balance but not completely. The Rossby number is small but not neliible. It is the small departures from eostrophy that allow the near eostrophic flow of the atmosphere to evolve. The question to be addressed in this section is how can we predict the evolution of a flow that is close to eostrophic balance. The evolution relies on the departures from eostrophy and so we must consider the small departures from eostrophic balance - the quasieostrophic (Q-G) approximation. The usefulness of the Q-G approximation can be demonstrated by discussin the earliest attempts at weather forecastin. It was Bjerknes in 1904 who realised that in order to predict the weather, the differential equations that overn the atmospheric flow had to be solved. Richardson then developed the techniques to solve the equations and performed hist first attempt at weather forecast in durin the first world war. He used measurements 1

2 1080 Pressure (hpa) Time (hours) Fiure 1: An idealised example of surface pressure fluctuations that may be observed over a day (dotted) and the pressure fluctuations that would be associated with the slower near eostrophic flow (i.e. the evolution minus the hiher frequency noise associated with inertia-ravity type motions) (solid). from 7am on 0th May 1910 and attempted to predict atmospheric conditions at a sinle point 6 hours later that could then be compared with the observations made at that time. This calculation took 6 weeks to complete and ave a surface pressure variation over the 6 hours of 145hPa which was over 100 times reater than the surface pressure fluctuation that was actually observed. The problem was that the calculation didn t take into account the small scale fluctuations(noise) in the system associated with the short timescale aeostrophic inertia-ravity motions. Consider Fi. 1 which shows an idealised example of typical variations in surface pressure over a day. What we see here are short timescale fluctuations associated with inertia-ravity wave type motions (dotted). But, this is on top of a loner timescale backround evolution (solid). This loner timescale backround evolution is the evolution of the lare scale near eostrophic flow i.e. the motion that is really of interest for determinin weather conditions on the timescale of hours to days. This fiure illustrates the problem associated with the presence of the hih frequency noise in the system. You could measure a rate of chane of pressure or a velocity field at particular point in time but a lare component of that measurement could be associated with the noise rather than the slower evolvin flow that we are interested in. The quasi-eostrophic approximation developed by Charney in 1947 is an approximation to the full equations of motion that effectively filters out the noise and produces a sinle predictive equation for the time evolution of the eostrophic flow takin into consideration the small departures from eostrophy. This development toether with the development of computers allowed for the first successful weather forecast in Now, computin power has increased so much that numerical weather forecastin models can actually solve the full equations of motion. However, the Q-G approximation still remains

3 important in the initialisation of weather forecasts - it is important not to initialise them with the hih frequency noise. 5.1 Assumptions of the Q-G Approximation There are four main assumptions that are made when formulatin the Q-G approximation. These are (1)The Rossby number issmall (R o = U/fL) i.e. it isnearly ineostrophic balance. The aeostrophic velocities ( v a ) are small compared to the eostrophic velocities i.e. v a v O(R o ) () The scale of the motion is not sinificantly larer than the deformation radius ( ) L O(1) L where L = H/f. This ratio, L /L, is 1/Bu where Bu is the Burer number. It will be shown that this is equivalent to statin that the variations in the fluid depth are small compared to its total depth. (3) Variations in the coriolis parameter are small f = f o +βy βl f o O(R o ) So, the variations in the coriolis parameter over the horizontal lenth scale L are small compared to the coriolis parameter itself. Note that this is not a ood approximation in the tropics. (4) The timescale that we are considerin is the advective timescale. Since the eostrophic velocities are much larer than the aeostrophic velocities, the advection over this timescale is by the eostrophic velocities such that 5. Q-G momentum balance t = t +u x +v y The full shallow water momentum balance is iven by v H t +f v H = h (1) Consider the velocity field to be made up of a eostrophic and an aeostrophic component such that v a v = v + v a where O(R o ) v 3

4 Consider characteristic lenth (L), velocity (U) and timescales (T) of the motion and a typical scale of heiht perturbation ( H). If we were to consider only the order 1 terms of Eq 1 we could just have eostrophic balance f v = h. () This allows us to determine the characteristic scale of heiht perturbations H. Scale analysis of this equation ives f v = h fu H L So, the riht hand side term scales as fu and this allows us to also demonstrate that the assumption of L /L 1 is equivalent to statin that heiht perturbations are small compared to the mean layer depth. H ful This can be rearraned makin use of the definition of the Rossby number and the Rossby radius of deformation to ive H R o H L L Given that we are considerin L /L 1 means that H/H R o. Since the Rossby number is small this implies that variations in the free surface heiht are small compared to the mean layer depth. Now re-writin (1) usin v = v + v a and f = f o +βy and performin a scale analysis ives v t + v a +(f v )+(f v a ) +(βy v )+(βy v a ) = h t U U a fu fu a βlu βlu a fu T T fu fu a βlu βlu a fu U a βl βlu a 1 1 fl fl U f f U R o Ro 1 R o R o Ro 1 U UU a L L U U a It can be seen that the terms of order 1 represent eostrophic balance as expected. By definition of the eostrophic velocity the terms of order 1 exactly balance. Considerin thenthetermsoforderr o andnelectinthehiherordertermsivestheq-gmomentum balance as v t +(f o v a )+(βy v ) = 0 (3) So, to zeroth order there is eostrophic balance. When considerin departures from eostrophy but only retainin terms that are of the order of the Rossby number and nelectin hiher order terms there is acceleration of the eostrophic flow by the coriolis force actin on the aeostrophic velocity and the variation of the coriolis force with latitude on the eostrophic velocity. 4

5 5.3 Q-G Mass Continuity The full shallow water mass continuity equation is iven by h t +h( v H) = 0 (4) Remember the scalin for H ( H R o HL /L ) and consider the heiht field to consist of a mean layer depth and a perturbation h H + H where the time and spatial derivatives of the mean layer depth are zero. The mass continuity equation can be re-written and scalin analysis performed to ive H +H(. v ) +H(. v a )+ H(. v ) + H(. v a ) = 0 t H HU HU a HU HU a 0 T L L L L HU HU HU a HU HU a 0 ( ) L L L L ( ) ( ) L L U HU HU a L U L R o H R L o H L L L L L R Ua oh 0 L U ( ) ( ) ( ) L U a L L Ua R o 1 R L o R U L o 0 L U R o 1 R o R o Ro 0 So, to order (1) we have. v = 0 i.e. the diverence of the eostrophic velocity is zero. This comes from the definition of the eostrophic velocity on an f plane for an incompressible fluid where u = (/f) h/ y and v = (/f) h/ x. Takin the diverence of this clearly ives you zero. To hiher order (nelectin the fourth term which is also by definition 0) we are left with the terms of order R o as follows h t +H(. v a) = 0 (5) This is the Q-G shallow water mass conservation equation. It states that chanes in the free surface heiht are determined by the diverence of the aeostrophic velocity. This makes sense iven that the eostrophic velocity is non-diverent so it doesn t contribute to a chane in the free surface heiht. 5.4 The Q-G PV Equation The shallow water momentum and mass conservation equations are v t +(f k v a )+(βy k v ) = 0 (6) h t +H(. v a) = 0 (7) where /t = / t + u / x + v / y. But, this isn t a closed system of equations. We have equations for the rate of chane of eostrophic quantities but they re interms of 5

6 aeostrophic quantities, which we don t have an equation for. We can obtain an equation for the time rate of chane of eostrophic quantities in terms of eostrophic quantities themselves by formin the Q-G PV equation. Equation 6 can be re-written usin the followin vector identity ( v. ) v = 1 ( v. v ) v ( v ) to ive v t + 1 ( v. v )+( ζ v )+(f o k va )+(βy k v ) = 0 The vorticity equation can then be formed by takin the curl of this ζ t + ( ζ v )+ (f o k va )+ (βy k v ) = 0 Then makin use of the vector identity ( A B) = A(. B) B(. A)+( B. ) A ( A. ) B each of the cross products can be re-written and a lare number of terms are zero to ive ζ t +( v. ) ζ +f o (. v a )+β v = 0 Note that since the diverence of the eostrophic velocity is zero and we are considerin an incompressible fluid then. v a = w a / z. So, the term involvin the diverence of the aeostrophic velocity represents vortex stretchin. So, this equation relates the rate of chane of vorticity to the advection of vorticity, the advection of planetary vorticity and vortex stretchin. We can make use of the continuity equation (7 to replace the diverence of the aeostrophic velocity ζ ( t + v. ) ζ +f o 1 H ) h +β v = 0 t Makinuseofthefactthat(βy)/t = βv andthenroupinupthematerialderivatives ives ( ζ +βy f ) o t H h = 0 Thus we have shallow water potential vorticity conservation of the form q t = 0 q = ζ +βy f o H h (8) This canbewrittenintermsof theperturbationheiht fieldrecallin that u = h f o y and v = h f o to ive x q t = 0 q = f o h +βy f o H h (9) 6

7 Or, it can be written in terms of the eostrophic stream function ψ = h /f o ivin q t = 0 q = ψ +βy 1 ψ L (10) We therefore have a sinle predictive equation for the heiht field or stream function from which we can obtain the eostrophic velocity field. Equation 8 or 9 can be used to find the time evolution of the eostrophic flow. If you know the potential vorticity at an initial time t and you can invert the PV operator to find the eostrophic velocities then you can use those eostrophic velocities to advect the PV to find out the PV at a later time t+ t as a function of x and y (provided you can nelect boundary effects and diabatic heatin) e.. q t = 0 q t = u q x v q y The PV at time t+ t could be calculated by q(t+ t) = q(t)+ q t t t From this new distribution of PV the eostrophic velocities can be found and so on. The Q-G PV iven by Eq. 5.4 can be compared with the full shallow water PV q t = 0 q = f +ζ h It can be seen that if, in this full PV, you consider the heiht field to consist of a mean depth plus a perturbation and you linearise and inore second order terms in Rossby number then you would obtain the Q-G PV. In sections we solved the full linearised Q-G equations where we assumed that the heiht and velocity perturbations were small but we didn t make any assumptions about bein close to eostrophic balance. That ave several different types of wave solutions: Inertia-ravity waves, Kelvin waves and Rossby waves. For Inertia-ravity waves we combined the momentum equations on an f-plane to ive t ( t +f o c ) h = 0 (11) This equation has three time derivatives and ave us three solutions, the zero frequency flow that is in eostrophic balance and the dispersive inertia-ravity waves of positive and neative frequencies. For Rossby waves, we allowed there to be a variation in the backround potential vorticity and solved the PV equation which had a riid lid (i.e. we didn t allow there to be ravity wave motion). This ave us t ( ψ )+U ( ψ ) +β x ( ) ψ Now, in the Q-G approximation the equation to solve is 9. The similarity between this and Eq. 11 is apparent. In the Q-G equation there is only one time derivative (not three as in Eq. 10) so there s only one frequency per wavenumber. In the Q-G PV equation, 7 x (1)

8 the only solution we re left with is the Rossby wave solution. Inertial-ravity waves are no loner solutions to the equations. The Rossby wave motion is the motion that satisties our initial assumptions (1) to (4). By makin those assumptions and modifyin the equations accordinly we have effectively filtered out the faster inertia-ravity wave motion. We are left with a pronostic equation for the evolution of the near eostrophic flow in terms of eostrophic quantities. It is this reduction of the equations of motion to a sinle equation which filters out the hih frequency noise that allowed for the first successful weather forecasts. 8