ESCI 343 – Atmospheric Dynamics II Lesson 13 – Geostrophic/gradient Adjustment Reference: An Introduction to Dynamic Meteorology (3rd edition), J.R. Holton Atmosphere-Ocean Dynamics, A.E. Gill Reading: Holton, Section 7.6 GEOSTROPHIC ADJUSTMENT OF A BAROTROPIC FLUID The atmosphere is nearly always close to geostrophic and hydrostatic balance. If this balance is disturbed through such processes as heating or cooling, the atmosphere adjusts itself to get back into balance. This process is called geostrophic adjustment, although it may more accurately be referred to as gradient adjustment, since in curved flow the atmosphere tends toward gradient balance. One method of studying geostrophic adjustment is to first study adjustment in a barotropic fluid using the shallow-water equations. Once we understand adjustment in a barotropic fluid we can easily extend our results to a baroclinic fluid by use of the concept of equivalent depth, studied in a previous lesson. For this we can use the linearized shallow-water equations with zero mean flow, u hfv gt x′ ′∂ ∂′− = −∂ ∂ (1) v hfu gt y′ ′∂ ∂′+ = −∂ ∂ (2) 0h u vHt x y′ ′ ′ ∂ ∂ ∂+ + = ∂ ∂ ∂  (3) If we take ∂/∂x of (1) and add it to ∂/∂y (2) we get ∂′∂+∂′∂−=∂′∂−∂′∂−∂′∂+∂′∂∂∂2222yhxhgyuxvfyvxut. (4) Rearranging (3) we get 1u v hx y H t′ ′ ′ ∂ ∂ ∂+ = − ∂ ∂ ∂ . (5) From the definition of vorticity we know that ζ′≡∂′∂−∂′∂yuxv. (6)2Putting (5) and (6) into (4) we get 0222222=′+∂′∂+∂′∂−∂′∂ζfHyhxhgHth. (7) We need one more equation that relates h′ and ζ′. This is the shallow-water vorticity equation, found by taking ∂/∂x of (2) and subtracting ∂/∂y (1) to get ∂′∂+∂′∂−=∂′∂yvxuftζ, (8) which, using (5), can be written (after some rearranging) as 0=′−′∂∂Hhftζ. (9) Integrating (9) with respect to time gives 0 0hhf H f Hζζ′ ′′ ′− = −, (10) where ζ′0 and h′0 refers to the initial values of relative vorticity and height perturbation. Using this in (7) results in ( )002222222hfHfhfyhxhgHth′−′−=′+∂′∂+∂′∂−∂′∂ζ. (11) Since the quantity gH is the square of the speed of a gravity wave in this fluid, we can denote it by c2 and write this equation as ( )0022222222hfHfhfyhxhcth′−′−=′+∂′∂+∂′∂−∂′∂ζ. (12) Equation (12) governs the geostrophic adjustment process in a barotropic fluid. THE STEADY-STATE SOLUTION Lets simplify things somewhat by assuming the initial state is at rest, and has an abrupt step in the surface height given by )sgn(ˆ0xhh −=′.1 (13) We also assume that there is no dependence in the y-direction. Equation (12) then becomes )sgn(ˆ2222222xhfhfxhcth−=′+∂′∂−∂′∂, (14) 1 The sgn(x) function is defined to be +1 for x ≥ 0, and −1 for x < 0.3a second order, non-homogeneous partial differential equation. The homogeneous form of this equation supports shallow-water inertial-gravity waves (see exercises). After these waves have subsided, there will remain a steady-state solution which obeys the steady state equation )sgn(ˆ2222xhcfhcfdxhd=′−′. (15) Equation (15) is a second-order, non-homogeneous ordinary differential equation with constant coefficients (assuming f and c are constant). The solution to (15) consists of a complementary solution (the general solution to the homogeneous equation) plus a particular solution, )()()( xhxhxhpc′+′=′. (16) The complementary solution is found from the characteristic equation for the homogeneous form of (15), which is ()022=− cfr. (17) Therefore, the complementary solution is then xxcBeAexhαα−+=′)( (18) where cf≡α. (19) For the particular solution we use the method of undetermined coefficients, guessing that the particular solution will have the form )sgn()( xCxhp=′. (20) Putting (20) into (15) shows that hCˆ−= , so that the particular solution is )sgn(ˆ)( xhxhp−=′. (21) Therefore, the general solution of (15) is )sgn(ˆ)( xhBeAexhxx−+=′−αα, (22) or <++≥−+=′−−0ˆ0ˆ)(xhBeAexhBeAexhxxxxαααα. (23) All that remains is to apply the boundary conditions, which require that: • h′′′′ (x) remain bounded asx→ ±∞: This requires that A = 0 for positive x and B = 0 for negative x, so that 00ˆ( )ˆxxBe h xh xAe h xαα− − ≥ ′= + <   (24) • h′′′′ (x) be continuous at x = 0: This means that ˆ ˆB h A h− = + (25) • The first derivative of h′′′′ (x) be continuous at x = 0: This means that B Aα α− = (26) Solving (25) and (26) for A and B yields4ˆˆA hB h= −= so that the steady-state solution is <−≥−=′−0101ˆ)(xexehxhxxαα. (27) ANALYSIS OF THE SOLUTION The figures below show the initial height field, the transient height field, and the steady-state height and velocity fields taken from a 1-D shallow-water numerical model. The transient solution consists of the shallow-water inertial gravity waves. The final height solution is the steady state solution from (27). The figures are striking in that, though the step that was in the initial conditions is smoothed out, there is still a region near the center of the domain with a horizontal pressure gradient, and therefore, with a geostrophic flow out of the page. The initial height field adjusted under the influence of gravity, and set up a flow that is in geostrophic balance with