Ocean Dynamics. The Equations of Motion 8/27/10. Physical Oceanography, MSCI 3001 Oceanographic Processes, MSCI dt = fv. dt = fu.

Transcription

1 Phsical Oceanograph, MSCI 3001 Oceanographic Processes, MSCI 5004 Dr. Katrin Meissner Ocean Dnamics The Equations of Motion d u dt = 1 ρ Σ F Horizontal Equations: Acceleration = Pressure Gradient Force + Coriolis ρ d + fv ρ d fu dt = 1 ρ ΣF dt = 1 ρ ΣF Or, for a Barotropic Ocean: dw = 1 dt ρ ΣF z dη dt d + fv dη dt d fu Vertical Equation: Pressure Gradient force = Gravitational Force If the onl force acting on a water parcel is the Coriolis force, the Navier Stokes equations can be simplified to: dt = fv dt = fu Northern Hemisphere: If the onl force acting on a water parcel is the Coriolis force, the Navier Stokes equations can be simplified to: dt = fv dt = fu Northern Hemisphere: What does this mean? u > 0 u < 0 What does this mean? u > 0 u < 0 v > 0 v < 0 v > 0 v < 0 1

2 If the onl force acting on a water parcel is the Coriolis force, the Navier Stokes equations can be simplified to: dt = fv dt = fu Southern Hemisphere: f < 0 Horizontal Equations: Acceleration = Pressure Gradient Force + Coriolis ρ d + fv ρ d fu What does this mean? u < 0 u > 0 v < 0 v > 0 If pressure gradients are small: dt = fv dt = fu Inertia currents the water flows around in a circle with frequenc f. T=2π/f T(Sdne) = 21 hours 27 minutes Scaling arguments: 1. If there are no surface slopes or horizontal densit differences then there will be no pressure gradient force (i.e. left with Coriolis, inertia currents) What forces are important in a bathtub? What kind of speeds will the water get up to? What kind of accelerations? What surface slopes? dη dt d + fv dη dt d fu Barotropic! Scaling arguments: What forces are important in a bathtub? What kind of speeds will the water get up to? What kind of accelerations? What surface slopes? dη dt d + fv dη dt d fu Magnitude of the pressure gradient force: dη = 0.1m /1m = 0.1 d g dη d =10ms 2 (0.1) =1ms 2 Magnitude of the Coriolis force: fu = ms 2 = ms 2 Coriolis << Pressure Gradient, so we can neglect rotation effects Acceleration in the bathtub is driven b pressure differences (e to changes in surface slopes) 2

3 Scaling arguments: 1. If there are no surface slopes or horizontal densit differences then there will be no pressure gradient force (i.e. left with /dt=fv, /dt=-fu, inertia currents): dt = fv dt = fu 2. If ou are sitting in our bathtub, ou are in a barotropic environment AND the Coriolis force can be neglected: dη dt d dη dt d But the ocean is not a bathtub. We will conct a scaling analsis on our equations of motion... to find further simplifications for motions with a period greater than ~10 das Scaling Analsis: ρ d + fv T~10 das = s ~ 10 6 s ρ d fu u,v ~ U ~ 1cms -1-1ms -1 f~ 10-4 s -1 Acceleration << Coriolis Geostrophic Balance Acceleration is much smaller than Coriolis and Pressure Gradient Force (this is true almost everwhere in the ocean) The ocean is in Geostrophic Balance (= balance between Pressure Gradient and Coriolis Forces) Ocean is in Stead State (no acceleration) /dt is negligible ρ d + fv ρ d fu 1 ρ d = fv 1 ρ d = fu What does the geostrophic balance mean phsicall? Suppose we have a difference in sea-level height. Water will want to move from the region of high pressure towards the region of low pressure. Geostrophic Balance 3

4 As the water starts to move, the Coriolis effect (rotation) deflects the water to the right () or left (SH). The water keeps getting deflected until the force e to the pressure difference balances the Coriolis force. This balance is called a geostrophic balance and the resulting current is referred to as a geostrophic current. Geostrophic Balance Which direction is the Geostrophic wind? (f <0 SH) PG CF V 13 Geostroph Problem 1 Geostroph Problem 2 In the : Which wa does the current flow if sea level height is increasing towards the South? Which direction does the water flow around this pressure feature if it is in the Northern Hemisphere? You can use the equations, or just think about the forces : f>0 dη/d<0 fv = g dη d fu dη d C P u = g f u=(-)(+)(+)(-)>0 East! dη d 4

5 Geostroph Problem 3 Which direction does the water flow around this pressure feature if it is in the Southern Hemisphere? Anticclone Cclone Northern Hemisphere clockwise counterclockwise Southern Hemisphere Anticclonic circulation: ALWAYS around a high pressure sstem. Geostroph Problem 4 A certain ocean current has a height change of 1.1 m (increasing to the east) over its width of 100 km at 45 N. How fast is the current flowing? clockwise in the Northern Hemisphere counter-clockwise in the Southern Hemisphere fv = g dη d g Δη Δ f = 2Ωsin(φ) = s -1 g = 10 ms -2 Δη = 1.1m Δ = m Cclonic circulation: ALWAYS around a low pressure sstem. counter-clockwise in the Northern Hemisphere clockwise in the Southern Hemisphere V=1 m/s Summar: Geostroph is the balance between pressure gradient forces and the Coriolis force. All major current sstems in the ocean can essentiall be considered geostrophic. Geostroph does not work over short periods of time or small distances (other forces become dominant). Geostroph also fails in regions where friction becomes important. 5

6 Forces on a Parcel of Water d u dt = 1 ρ ( F g + F C + F P + F f +...) d u dt = 1 ρ ( F g + F C + F P + F f +...) Gravit Coriolis Pressure Friction dt = 1 ρ F dt = 1 ρ F dw = 1 dt ρ F z The last force to consider is friction. This is onl important at continental boundaries, at the bottom of the ocean, and at the surface (e to wind). What will the friction term look like? We know that friction alwas tries to retard motion. Effects of Friction A simple model for the frictional force at the sea floor in the and direction is: F = -ru F = -rv Raleigh frictional dissipation, r is a coefficient (r ~ 10-7 s -1 ) Hence the equations of motion become + fv ru ρ d fu rv ρ d 6

7 8/27/10 F = -ru F = -rv Henr Stommel (1948) To eamine the effects of friction, consider the simple balance: i.e no pressure gradient forces and Coriolis + pressure gradient forces dt Harald Sverdrup (1947) no coriolis force. Motion is just decelerated because of friction. u(t) = uoe rt A solution is: wind forcing = ru 1/r represents the time it takes for the speed to drop to 1/e (~1/3) of its initial value. So that at t=0, u(0)=uo U Uo Velocit decreases with time because of friction. Coriolis + pressure gradient forces 1/3Uo Uoe-1 wind forcing (Monda) Mid Semester Break (no class) (Monda) Dr. Caroline Ummenhofer (and Dr. Matthew England) (Monda) Dr. Caroline Ummenhofer (Monda) Dr. Laura Ciasto (Monda) Labour Da (no class) 1/r 26 Time Thermal Wind Balance So far we have assumed that densit ρ is constant (barotropic) Small horizontal changes in ρ can result in large vertical changes in current/ wind e.g. near fronts and eddies 1 = fv ρ d Starting with the geostrophic balance (Wednesda!) Me again on a WEDNESDAY! 1 = fu ρ d We can differentiate the equations with regard to depth (z) 1 ρf d 1 u= ρf d = ρg dz v= d 1 = dz dz ρf d d 1 = dz dz ρf d = ρg dz 7

8 Thermal Wind Balance So far we have assumed that densit ρ is constant (barotropic) Small horizontal changes in ρ can result in large vertical changes in current/ dz = 1 d ( ρf d ρg) dz = g ρf d wind e.g. near fronts and eddies Starting with the geostrophic balance dz = 1 ρf dz dz We can differentiate the equations with regard to depth (z), then substitute for /dz v = 1 ρf d u = 1 ρf d dz = ρg u << ρ dz = d 1 dz ρf d dz = d dz 1 ρf d dz = ρg dz = g ρf d dz = ρg dz = 1 ρf dz = 1 ρf dz = ρg d dz d d dz d dz = ρg dz = 1 ρf dz = 1 ρf dz = ρg d d ρg ( ) d d dz d d dz For geostrophic conditions: Thermal Wind Balance The vertical structure of u and v is related to the horizontal densit gradients dz = g ρf dz = g ρf d d i.e. horizontal densit gradients in temperature (T) and salinit (S) can eplain the change in horizontal velocit with depth (vertical profile of horizontal velocit). Thermal Wind Balance dz = g ρf d d > 0 :, so, dz > 0 Velocit is into the page and increases with depth i.e. v gets more positive with increasing depth 8

9 cold/saline core Potential Densit through an Edd near the Gulf Stream. How does the geostrophic current velocit in the edd change with depth? dz = g ρf 27.9 d g ρf Δρ Δ Estimate the densit gradient at =40km: ρ changes from to kgm -3 over 25 km. Δρ = = Δ 25km dz = g ρf = s 1 dz = g ρf d dz = g ρf d dz = g ρf d g ρf Δρ Δ This means that at z = 500m, dz = s 1 let s see how much v varies over 100m of depth at z=500m (Δz=100m, Δv=?): Δv = s 1 100m = 0.15ms 1 = 15cms 1 i.e the velocit shear ~ -15 cm/s per 100m depth increase. To work out the actual velocities, we need one more piece of information: at depth (sa 2000m) the velocit is 0. This is called the depth of no motion This means that v (velocit into page) is getting more negative as we get deeper

10 Going down in depth, the densit surfaces flatten out. We can assume a level of no motion where there is no longer a change in densit. Hence we can calculate the change in velocit up through the water column. Note that on the other side of the edd, the densit surfaces slope the other wa, so the circulation must also be in the opposite sense. A phsical eplanation Step 1: denser in the middle so the surface will be depressed V= V= V=0+.15 V=0 V is positive into the page 100m 100m 100m v = 1 ρf d P Step 2: Start b figuring out the pressure force e to the surface slope: Pressure increases moving awa from centre Step 3: Forget the barotropic component. What happens at greater depth? Densit is higher at the centre than further awa. barotropic component Remember the fishtank eperiment Remember the fishtank eperiment ρ 1 < ρ 2 10

11 ρ 1 < ρ 2 Remember the fishtank eperiment Step 3: Forget the barotropic component. What happens at greater depth? Densit is higher at the centre than further awa. So the pressure force, just e to densit would be in the positive -direction (on the right hand side of the gre). The densit gradient causes a pressure force that increases with depth. Opposite happens on the left side of the gre. baroclinic component Step 4: Add up barotropic component (black) and baroclinic component (blue). Keep repeating this down the water column, until there is no densit gradient (i.e. at the bottom of the cold core edd) A smaller pressure gradient force means a smaller geostropic velocit with depth v = 1 ρf d Step 5: If we add up the forces e to the surface slope and e to the densit gradient we get a pressure gradient force that decreases with depth. C P P C Finall we need to use the geostrophic relationship: If we have a pressure gradient in the direction, it will create a geostrophic velocit in the direction, proportional in strength to the pressure gradient. v = 1 ρf d 11

12 Eample: Derive the rotation of these cold and warm water eddies, in the SH, using the thermal wind balance. Given that u & v = 0 at 2000m (depth of no motion) estimate the surface current. Thermal Wind Balance For geostrophic flow (i.e. pressure is balanced b Coriolis): Geostrophic flow in the presence of horizontal densit gradients Horizontal densit gradients (T,S) can eplain vertical velocit changes To know absolute velocit we need etra information (i.e. we need to know absolute velocit at some depth) dz = g ρf dz = g ρf d - Can figure out surface velocit from surface heights. - Often we assume that at a certain depth (e.g. 2000m) velocities are zero this is called the depth of no motion. - Once we know the velocit at the surface or the depth of no motion we can calculate velocit at all other depths using the thermal wind equation. d Summar: Ocean Dnamics Most of the motion in the ocean can be understood in terms of Newton s Law that the acceleration of a parcel of water (how fast its velocit changes with time /dt) is related to the sum of forces acting on that parcel of water. We can split the forces, velocities and accelerations into south-north (,v), west-east (,u) and up-down (z,w) components. In the vertical direction the acceleration is related to the difference between the water weight and the bouanc (or pressure) force. When there is a vertical densit gradient this leads to oscillations (Brunt Väisälä frequenc N). The densit gradient tries to inhibit vertical motion (and miing). These vertical accelerations are generall ver weak, so we get the hdrostatic equation. If the hdrostatic equation is integrated over depth, it just sas that the pressure at a point just equals the weight of water above that point. Acceleration in the horizontal can be driven b a number of different forces: (1) The pressure gradient force. This eists whenever there is a surface slope and/ or a horizontal densit gradient. (2) Coriolis (because we live on a rotating planet). It is ver weak, so we onl feel its effect over long times (> few das) and large distances (> 10s of km). Coriolis onl affects moving fluids, deflecting to the right in the and to the left in SH. (3) Friction. Also onl acts on moving water. Alwas acts to slow down motion. Important at the boundaries of the ocean. 12

13 Summar: Ocean Dnamics + fv ru ρ d fu rv ρ d Or for a constant densit (barotropic ocean): dt dt dη + fv ru d dη fu rv d Over much of the ocean, the flow is stead (i.e. /dt=/dt=0) and friction is negligible, so we are left with the geostrophic balance i.e. pressure gradient forces balance coriolis. The current moves at right angles to the pressure gradient. P C 1 ρ d = fv, 1 ρ d = fu When there is a horizontal densit gradient the velocit changes with depth. This can be calculated using the thermal wind equations dz = g ρf d, dz = g ρf d 13