Atmospheric and Oceanic Modeling - Quasi-Geostrophic Equations

Quasi-geostrophic ocean models 1 Introduction The starting point for theoretical and numerical study of the three dimensional large-scale circulation of the atmosphere and ocean is a vorticity equation that has a form very similar to that employed in the study of 2-d incompressible ‡ow [see eqs.(6) and (7) of section 2.1 of ‘vorticity equation’ notes] except that the vorticity variable - which goes under the name of the ‘quasi-geostrophic potential vorticity’ - is a three-dimensional Laplacian of a streamfunction, rather than a two-dimensional Laplacian. In this chapter we outline the assumptions underlying the quasi–geostrophic equations and describe how they can be integrated forward numerically to study the circulation of the ocean. These numerical studies played a seminal role in the development of baroclinic theories of the wind-driven ocean circulation. 2 The quasi-geostrophic equations Underpinning the quasi-geostrophic equations are two key assumptions: 1. The absolute vorticity of a column of ‡uid is modi…ed chie‡y by vertical stretching or compression. ! () (# + [$%&' v]! ) = # !" (* (1) where # is the Coriolis parameter (for now assumed constant), ) is the vertical velocity and * is a vertical coordinate. By comparison, it is assumed that the change in vorticity due to tilting of the Taylor columns by the thermal wind is small. Note that in eq(1) it has been assumed that the vertical component of [$%&' v]! is small compared to # when multiplying the stretching term, "# : i.e. the Rossby number is "! small +$ = , .. 1 #- 1. where , is the speed of a typical horizontal current which varies over a horizontal scale -. 2. We suppose that the ‡ow is nearly geostrophic. One can use the geostrophic relation to compute [$%&' v]! thus: [$%&' v]! = (/% (%% ¡ (0 (1 where the geostrophic winds are given by # /% ¡ 1 (3 1 (3 = 0; #%% + =0 2 (0 2 (1 (2) and we have supposed that when we divide by 2, it can be replaced by 2, a mean reference density that is constant in space and time (adopting an incompressible baroclinic model). An important consequence of the geostrophic relation is that the horizontal divergence of the geostrophic current vanishes : (%% (/% + =0 (3) (0 (1 Thus if the ‡ow is almost geostrophic it is almost horizontally non-divergent and we cannot reliably use the continuity equation to obtain the vertical velocity - instead we must use the vorticity equation (1). However, because the ‡ow is nearly geostrophic we can use the idea of a streamfunction even though we have 3 dimensional motion. We can think of each level, *, in the ocean as having its own streamfunction from which %% and /% may be computed: r&4/% = /% = (5 (5 ; %% = ¡ (0 (1 (4) This does not completely specify 5 because only horizontal derivatives are involved. So we can add any arbitrary function of * to 5 and we shall choose this arbitrary function in such a way that "' has a useful meaning. "! To this end we write the pressure: 3 = 3$ (*) + 30 (06 16 *6 ") (5) where 3$ (*) is some sort of average value (average over 0, 1 and " at each horizontal level) so that 30 is relatively small at all levels. Thus the 3 that appears in (2) is just 30 (06 16 *6 "), and from (2) and (4) we see that: 30 2# Furthermore we assume that the ‡ow is in hydrostatic balance: 5= (6) (3 73$ (30 = + = ¡82 (* 7* (* = ¡8 (2 + 2$ (*)) ¡ 820 (06 16 *6 ") where we have similarly separated the density …eld out thus: 2 = 2 + 2$ (*) + 20 (06 16 *6 ") (7) Now we choose 2 + 2$ (*) to correspond to 3$ (*) and 20 (06 16 *6 ") to correspond to 30 (06 16 *6 "): i.e. (30 73$ = ¡8 (2 + 2$ ) ; = ¡820 7* (* where 20 is assumed small compared to 2$ . Thus, (6) and (8) imply that: (5 20 9 = ¡8 = # 2 (* 0 where 90 is the buoyancy, with associated reference pro…le 9$ (*). (8) (9) Equation (9) is nothing more than a statement of the thermal wind equation in the incompressible baroclinic model. Indeed taking k £ r (9) (where k is a unit vector in the vertical) and using (4) we immediately obtain the ") ") thermal wind equation: # "( = ¡ "* ; # "+ = ", . "! "! "' "' Thus we see that ", , "* have a meaning as a measure of geostrophic current but also "' has a meaning as a measure of buoyancy. It is this "! double nature of 5 which makes it very convenient in quasi-geostrophic theory. We now go on to determine the equations that govern the evolution of 5. 2.1 The quasi-geostrophic potential vorticity equation We enter the geostrophic approximation in to Eq.(1) and write: where ¢ !% ¡ () # + [$%&' v% ]! = # !" (* [$%&' v% ]! = (10) (/% (%% ¡ = r2& 5 (0 (1 and !% ( = + v% 4r (11) !" (" Note that advection of vorticity in eq.(10) is by the horizontal geostrophic ‡ow (we discuss conditions for the neglect of vertical advection below). Now from the thermodynamic equation we write: !% 90 + : 2) = 0 !" where we have assumed that )0 )! (12) .. 1 and supposed that (9$ 6 (13) (* the static stability, only depends on z. For the moment we have set buoyancy sources to zero on the rhs of eq(12). We can eliminate ) between (12) and (10) to give :2 = Figure 1: A point vortex of strength ‘;’ in a rotating (# = 2­), strati…ed (: ) ‡uid. !% < =0 !" (14) where (using equation 4 and 9): <= r2& 5 ( +# + (* µ # 2 (5 : 2 (* ¶ (15) is the ‘quasi-geostrophic potential vorticity’ (qgpv) and eq(14) is the qgpv equation. Note that the qgpv is conserved in horizontal motion, in the absence of sources and sinks. 2 The factor .- 2 is a pure number which can obviously be interpreted as a scale factor; .- is of order 30 in the ocean, 100 in the atmosphere. So, in analogy with our equation for 2-d barotropic ‡ow, we may write our equation for 3 -dimensional baroclinic ‡ow as: µ ¶ ( (5 ( (5 ( e 2 ¡ + r3 5 = 0 (16) (" (1 (0 (0 (1 2 e 2 = r2& + ( (17) r 3 (e *2 is a modi…ed 3-d Laplacian operator in which (assuming # and : are constant): Figure 2: A point vortex of strength ‘;’ induces a circulation ª given by eq., if #6 : = constant, eq.(21). ( # ( = (e * : (* thus stretching z, the geometric height by, typically, a large factor 3 The invertibility principle . - £ * = *e. We note that < and ª are related to one another through an Elliptic operator eq(17): e 25 <=r 3 (18) e ¡2 < ) 5 r | 3 {z } (19) If the distribution of q is known, then, given suitable boundary conditions, it may be INVERTED for 5 - this is the INVERTIBILITY PRINCIPLE - it e 2 operator: stems from the Elliptic nature of the r 3 /0+123/$0 This elliptic property can be traced back to the assumption of balance that the ‡ow is geostrophically balanced in the horizontal and hydrostatically balanced in the vertical. One can draw a direct and complete analogy between the electric potential induced by a distribution of charge, and the streamfunction induced by a potential vorticity …eld in quasi-geostrophic theory. Moreover, one can employ and exploit all the tricks that are commonly used in electrostatics (image charges, boundary sheets of charge etc), to problems of large-scale quasi-geostrophic dynamics. Once 5 is known then horizontal di¤erentiation yields the geostrophic currents, eq.(6), and vertical di¤erentiation yields the buoyancy perturbation, eq.(9). For example, let us compute the streamfunction induced by a point source of potential vorticity of strength ‘;’, in a strati…ed, rotating ‡uid ( .- assumed constant) of in…nite extent – no boundaries. Of course we are computing the ‘Green’s function’ e 2 5 = ;=(0) r 3 In spherical coordinates 1 ( r = 2 & (& 2 and so µ (20) ( & (& 2 ¶ 1 5 = constant £ 4 & 2 where &2 = 02 + 1 2 + *e2 = 02 + 1 2 + .- 2 * 2 and :># assumed constant. One can deduce the ‘constant’ by integrating eq.(20) over a sphere; the rhs contributes ‘s’, the lhs r5 over the surface of a sphere: we …nd that 5=¡ ; 4? ³ 1 02 + 1 2 + .2 2 * -2 ´ 12 (21) The form of this Green’s function, although simple, is highly instructive. We note that: ² if .- @@ 1 (very strongly strati…ed), then the solution decays away rapidly in the vertical. But if .- .. 1, the in‡uence of the < anomaly can be felt over great depths ² if ; @ 0, then we observe a cyclonic vortex with a ‘pinching together’ of the 9 surfaces: see …g.3. Figure 3: Cyclone ² if ; . 0, then we observe an anticyclonic vortex with a ‘pushing apart’ of the 9 surfaces: see …g.4. 3.1 Boundary conditions A distribution of < can only be inverted for 5 - Eq.19 - given appropriate boundary conditions. At lateral boundaries Dirichlet conditions on 5 are often appropriate i.e. 5, rather than gradients of 5, is typically speci…ed at lateral boundaries. However, at upper and lower boundaries the buoyancy distribution is speci…ed, providing, through (9), inhomogeneous Neumann conditions on 5 i.e. "' speci…ed at upper and lower boundaries. A com"! putational and (depending on one’s point of view) conceptual simpli…cation arises if, as we are liberty to do, the inhomogeneous Neumann condition on 5 are replaced by homogeneous ones thus. Let us de…ne a potential vorticity C is then a typical wavelength; and let D be a scale of vertical variation, so that: We suppose that: ¯ 2¯ ¯r ¯ » C 2 ; & ( 1 » (* D ,6 E are typical horizontal and vertical currents respectively ¢9 ¡ a measure of the horizontal variation in 9 : 2 is the static stability Now we can estimate the magnitude of the above quantities from data, but they are not independent of one-another. We will thus make use of dynamical and kinematic constraints to link them together. 6 » C, , the criterion for geostrophic balance of the horizontal Since 67 # current is that: C, .. 1 (25) # assuming that the vertical advection of momentum is negligible. The criterion for the neglect of vertical advection (which is equivalent relative to "( or "+ in the continuity to the criterion for the neglect of "# "! ", "* equation) is: +$ = " ) "! ()>(* E>D » .. 1 » v4r& (%>(0 C, (26) So what are the appropriate scales to choose for E ? From the buoyancy equation (12): ! C, ¢9 9 » : 2 ) =) E » !"& :2 (27) " ) "! ¢9 E » 2 .. 1 » v4r& C, D : D (28) and eq(26) becomes: Eq.(28) says that horizontal variations in 9 must be much smaller than vertical variations in 9 for vertical advection to be neglected. Thus the quasi-geostrophic equations assume two things: +$ = C, .. 1 # and ¢9 .. 1 : 2D We now want to show that conditions (28) both ‡ow essentially from the largeness of the Richardson number. To restrict the variety of possible systems further we make use of the vorticity equation. If the governing vorticity equation is (1). Thus: ! () #E [$%&' v]! » C 2 , 2 and # » !" (* D For a W scale we use, from eq(27): C, ¢9 #, 2 » :2 : 2D where we have exploited the observed smallness of +$ on the large-scale and hydrostatic balance, and scaled ¢9 as suggested by thermal wind balance: E » #, (29) CD The terms in the vorticity equation are thus of comparable magnitude when: ¢9 » #2 C D » 2 : a balance which can be expressed in terms of +/ and +$ thus: 2 2 +/ +$2 » 1 (30) where : 2D 2 (31) ,2 is the Richardson number. Thus if +/ @@ 1 , then according to eq(30): +/ = 1 +$ » p .. 1 +/ which is (25). Furthermore, using (29), ¢9 1 1 = = p .. 1 2 : D +/ +$ +/ which is criterion (28). Thus we see that: the approximations inherent in the quasi-geostrophic equations ‡ow essentially from the largeness of the Richardson number. The Richardson number takes on a value of » 100 in large-scale atmospheric ‡ows and » 104 in the ocean on the large scale. Finally, note that (30) implies that: +/ +2$ = : 2D 2 C2, 2 »1 , 2 #2 and so C » # 1 = :D -9 the inverse of the Rossby radius of deformation. The vorticity balance (1) thus holds on the deformation radius, -9 , typically 1000 km in the atmosphere and 30 km in the ocean.