The effect of varying forcing on the transport of heat by transient eddies.

Transcription

1 The effect of varying forcing on the transport of heat by transient eddies. LINDA MUDONI Department of Atmospheric Sciences, Iowa State University May Introduction The major transport of heat and momentum in the atmosphere is done by transient eddies in the midlatitudes. In the Northern Hemisphere observed transient eddies transfer much more heat and momentum in winter than in summer (Oort and Rasmusson 1971). In the Southern Hemisphere transient eddies are responsible for at least 90 % of the eddy transport through all seasons (van Loon, 1980). In this paper we focus on the eddies produced in an idealized (simplified) global atmospheric model simulating the Held and Suarez s (1994) test case. The focus is in the middle latitudes, where eddies are strongest. We are going to compare heat flux computed from Stone s (1972) theory to the heat flux from the model runs. Heat flux from Green s (1970) parameterization will also be compared to the flux from the model output. We also investigate the effect of the external heat forcing on the eddy heat flux. In this paper we compare the model generated meridional temperature gradient q/ y and the uilibrium meridional temperature gradient { q / y} from Suarez and Held s test (1994). In addition vertical temperature gradient from theory is compared to the model output. The main objective of our analysis to explore the consistency of the model with theory. A global climate model (GCM) is used to investigate the effects of heat forcing on the eddy transport. The model we used in this paper is very simple with no topography. 1

2 Manabe and Terpstra (1974) have shown that removal of topography from the GCM has the effect of substantially reducing contributions to potential energy conversion at planetary scale. However the zonally averaged climate state and the total energy conversion were relatively unchanged from the realistic topography simulations. We assume a basic model for static stability of a rotating atmosphere where we assume heating by radiative fluxes is balanced by sensible heat and potential energy flux convergences due to large scale eddy motions, as was done in Stone s paper. 2. Method Our model simulates a continuous atmosphere. The continuous theory using pressure or density scaling, weights in favor of the lowest levels. We used the lowest six layers to compute the averages in the vertical and horizontal. A simple GCM was used, with no topography that means the surface is at a constant geopotential. The vertical grid spacing dz is 750m and latitude and longitude both have a grid spacing of The model is forced by an assumed radiative uilibrium temperature distribution by Held and Suarez (1994). The uation for the radiative uilibrium temperature is given by q = max {200K, [315- Dq ( ) y sin 2 f-( q ) z D log (p/p o )cos 2 f]}, (2.0) where p is pressure, p o is surface pressure and f is latitude. We used various values of Dq y and Dq z to compute theoretical uilibrium states that force the model, using q t =K- k T f, p [ ]. (2.1) ( ) q -q The forcing used in this model is thus based on the Newtonian relaxation of the temperature field to a zonal symmetric situation as in Held-Suarez (1994). Forcing GCMs in this way is 2

3 very common particularly for two layer models (Hendon and Hartmann 1985). Held and Suarez (1994) used a standard case in which their horizontal forcing parameter ( Dq y )was 60K, and the vertical forcing parameter ( D q z ) was 10K. In our simulations we use the same standard forcing as a control. However, we used a reference temperature of K instead of the 315K used by Held and Suarez (HS) making our temperature field much cooler than that used by HS. To compare the model behavior with Stone s (1972) theory, we computed the uilibrium static stability ( q z) and the uilibrium meridional temperature gradient ( q y) for the different forcings used by the model. Meridional temperature gradient from the model output was averaged over the lowest six layers for the three latitudes for which the eddy heat flux ([ r v *T *]) was strongest. In our model this meant that the average was taken from the surface to 3750 m above the ground. The computation was done separately for each hemisphere for the different temperature forcings. In this study we will place our focus in the midlatitudes where Stone s (1972) theory applies. The uilibrium horizontal temperature gradient was computed for the surface using q y (1/a) q f = a 2sinfcosf, (2.2) q y where a is the radius of the earth. We computed q z by differentiating (2.0) with respect to z using the hydrostatic uation and ideal gas law, yielding q z = ( Dq ) z 2 cos f g Rq, (2.3) where R is the gas constant for dry air. We apply (2.2) and (2.3) at the latitude for which the heat flux is a maximum in our simulation. 3

4 ( y). Table1 shows the static stability ( q z) and the meridional temperature gradient q for the different values of D q y and z TABLE 1 EQUILIBRIUM VALUES FOR THE GLOBE. Experiment Number D q for the surface at 45 latitude. For the different values of the specified radiative forcing difference in the horizontal (Dq y ) and vertical ( D q z Dq y (K) Dq z (K) ) used in the model, we computed a time average of the zonal averaged temperature and meridional velocity and eddy heat transport ([T], [ r v ] and ([ r v *T * ]). q (K) (45 ) q y x10-6 ( Km ) q z ( Km ) x10-4 2a x10-4 2b x10-3 2c x x10-4 Where [ ] is the zonal average and gives the time average. The eddy heat flux from the model output was compared with two theoretical values; details of their computation follow. We used Peter Stone s (1972) parameterization of the time and zonal averaged eddy heat flux, V J = 0.86g H y Ê y ˆÊ 1+ Ri ˆ q -. (2.4) - ( ) { z }{ q y } Á1 2 Á T f L Ë L Ë Ri o 2 This parameterization assumes that baroclinic instability governs the structure of the flux. Green s (1970) dynamic expression of heat flux was also used for comparison. 1 2 VJ = ( ) { } 2 { } 2 0 gh È g q y Í q z. (2.5) To f ÎTo 4

5 To apply Stone s parameterization to midlatitudes we assume y = 1 2 L. Stone and Green s parameterization become identical when the Richardson number Ri Æ. The assumption made by Stone was used because our focus was in the midlatitudes, which is half way between the pole and the uator. L is the distance from uator to pole and y is the distance from the uator. Where{} denotes the zonal average and the over bar represents the time average. Ri is the Richardson s number, Ri =(To ƒ 2 ) q z { }{ z} g{ q y} 2 q, (2.6) ƒ is the coriolis parameter and H is the density scale height and we used H=7000m. Ri computed from this relationship applies only to the midlatitudes. The coriolis parameter ƒ was computed for midlatitudes from ƒ=2ωsin_ with the rotation rate as (2.7) Ω=7.29x10-5 s. 3.0 Results and Discussions The control in this analysis was integrated over 150 days with the climate statistics being computed by averaging the last 30 days. Our model was always integrated from a previous run which might have had a different forcing. For the additional cases in this study the computations were done for 270 days, with the climate statistics being computed by taking an average of the last 30 days of the run. We hope to dispose of any possible differences that might have cropped up from the different initial conditions by averaging over the last 30 days. 5

6 Horizontal temperature gradient q y from the model output varies almost linearly with the forced temperature gradient q y for the two hemispheres (Fig.1.). The horizontal temperature gradient ( q y ) calculated from the model simulations appears to be consistent with theoretical values. The meridional temperature gradient generated from our model is much less than the radiative uilibrium value. From Fig.1, it appears the eddies have the effect of lowering the meridional temperature gradient of the model. The vertical temperature gradient presents a different setting (Fig.2.) because the model temperature gradient is much higher than the uilibrium case. The highest value of q z occurs when Dq z is 20K. The vertical temperature gradient like the meridional temperature gradient is consistent with uilibrium conditions. However the eddies in this case are acting to increase the vertical temperature gradient. Statistics focused on the latitude of maximum horizontal eddy heat flux. The maximum heat flux occurred at latitude 47 in both hemispheres except for the model in which we had the largest forcing temperature gradient (Dq y =120K ). For this case the maximum eddies were located at 37 in the two hemispheres. For most values of value of the coriolis parameter was the same. But the coriolis parameter computed from the model when Dq y was 120K was different (8.69X10-5 s ). This in it self is consistent with theory in that the theoretical uilibrium statistics were computed for latitude 45 latitude and the statistics for our model falls close to the 45 latitude. Dq y the 6

7 For the different case the maximum heat flux occurred at lower latitudes of 37 for both hemispheres. We have no explanation for the shift in the eddy heat flux maximum, when the horizontal temperature gradient was doubled from a standard one of 60K. Reducing the gradient to 30K did not make the eddies to shift. Nevertheless doubling the horizontal temperature gradient reduced the uilibrium temperature q 0 by 16 K. Halving the horizontal temperature gradient increased q 0 by about 25K. We took the absolute value of the meridional heat flux over the southern hemisphere to make our comparison with that over northern hemisphere easier. The meridional heat flux for both hemispheres increases with increasing temperature gradient (Fig.3. and Fig.4.). The eddy heat flux from the model integration shows that reducing the horizontal temperature gradient by half and maintaining Dq z at 10K results in the eddy heat flux being halved. For the same value of D q y, but varying Dq z, the eddy heat flux [VJ ] changes only slightly. The heat flux for a horizontal forcing of 120K, decreases with increase in the horizontal temperature gradient in both hemispheres. Heat flux over the southern hemisphere is lower than that over the northern hemisphere. Our model has no topography, and moisture, so we would expect the model to be symmetric. Table 4 and 5 is the comparison between heat flux from the model output and that from Stone and Green s theories. The theoretical value varies almost linearly with the value from the model output except for experiment 1 where Dq y =120K. The theoretical value is more than ten times the model output. Heat flux computed from Stone s theory is comparable with one computed from Green s theory tables 4 and 5. The model value of [ *q *] V (6.46Wm -2 ) is inconsistent with both theories for the northern hemisphere case. By 7

8 looking at tables 4 and 5 one would expect the heat flux to at least double when Dq y is doubled. In table 4 where Dq y was increased from 60K (experiment 2a,2b and 2c) to 120K the heat flux was decreased by a factor of two instead. Table 4 and 5 also show the Richardson numbers for different temperature forcings computed for the 45 latitude. The Richardson, number, which is a measure of static stability, varies linearly with Dq y. TABLE2 COMPARISON BETWEEN THE MERIDIONAL TEMPERATURE FROM THE MODEL AND THAT FROM THEORY. Experiment Number q y Km Northern Hemisphere q y Km Southern Hemisphere q y Km E E E-06 2a 9.42E E E-06 2b 9.42E E E-06 2c 9.42E E E E E E-06 TABLE 3 THEORETICAL AND MODEL OUTPUT VALUES FOR THE STATIC STABILITY OVER THE GLOBE..Experiment Number q z Km Northern Hemisphere q z Km Southern Hemisphere q z Km E E- 8.83E- 2a 6.498E E- 3.33E- 2b 1.300E- 3.61E- 3.61E- 2c 3.249E E- 3.21E E E- 2.22E- TABLE 4 RICHARDSON NUMBER,HEAT FLUX COMPUTED FROM STONE S AND GREEN S HEAT FLUX PARAMETERISATION AND THE MODEL VALUE FOR THE NORTHERN HEMISPHERE. Experiment Number Ri [ V *q *] Stone s Theory Green s Theory Model a b c Wm -2 [ V *q *] Wm -2 [ *q *] V Wm -2 8

9 TABLE 5 RICHARDSON NUMBER,HEAT FLUX COMPUTED FROM STONE S AND GREEN S HEAT FLUX PARAMETERISATION AND THE MODEL VALUE FOR THE SOUTHERN HEMISPHERE. Experiment Number Ri [ *q *] V Wm -2 [ V *q *] Stone s Theory Green s Theory Model a b c Wm -2 [ *q *] V Wm -2 Model horizontal temperature gradient vs the forced temperature gradient 2.00E E E E-05 d q /dy (K/m) 1.20E E E E E E E E E E E E E E E E E E- 05 dq/dy (K/m) Northern Hemisphere d_/dy Southern Hemisphere d_/dy d_/dy Fig. 1. The meridional temperature gradient for the northern and southern hemisphere plotted against the forced temperature gradient. 9

10 Model vertical temperature gradient vs forced gradient 1.0E E- 8.0E- 7.0E- dq/dz (K/m) 6.0E- 5.0E- 4.0E- 3.0E- 2.0E- 1.0E- 0.0E E E E E E E- 1.20E- 1.40E- dq/dz (K/m) Northern Hemisphere Vertical d_/dz Southern Hemisphere d_/dz d_/dz Fig2 The vertical temperature gradient for both hemispheres plotted with the uilibrium temperature gradient. Meridional heat flux vs dq/dy (Model Output) Energy Heat Flux{pv*T*} E E E E E E E-05 dq/dy (K/m) Northern Hemisphere pv*t* Southern Hemisphere pv*t* Fig.3. The model meridional heat gradient for both hemispheres plotted against the horizontal temperature gradient from the model. 10

11 Meridional heat flux vs d_/dz (model output) Meridional Eddy Heat Flux[(pv)*T*] /W-m^ E E- 2.00E- 3.00E- 4.00E- 5.00E- 6.00E- dq/dz (model output) 7.00E- 8.00E- 9.00E- 1.00E- 02 Northern Hemisphere (pv)*t* Southern Hemisphere (pv)*t* Fig.4. The model meridional heat flux for both hemispheres plotted against the vertical temperature gradient from the model. 11

12 Further simulations were done to test the effect of damping on the eddies. Table 6 shows the damping parameters, the static stability and the horizontal temperature gradient for the globe. NH and SH denote the northern and southern hemisphere respectively. The velocity damping parameter TABLE 6 EXPERIMENT 1 STATISTICS FOR THE GLOBE FOR VARYING DAMPING PARAMETERS k s AND k. q y k s k f f NH q z NH q y NH [ V *q *] SH q z SH q y E- 7.67E E E SH [ V *q *] E- 5.1E E E E- 1.05E E- 1.02E k s = k f Ê s -s b ˆ max Á0,, Ë 1-s b k f has an effect on the heat flux. Where s = p / ps, p is the pressure and ps is pressure at the surface. Reducing k f by one-twentieth results in a twofold increase in the heat flux. On the other hand, decreasing the temperature damping parameter k s by one fifth causes the meridional temperature gradient to decrease by one order of magnitude. No conclusive reason can be given for the effect of k s on the vertical temperature gradient. We can conclude that the heat flux is dependent on wind damping factor. If we make k f much smaller we should be able to increase the heat flux considerable. There should be a threshold value of k f for which we can get reasonable values of heat flux. 12

13 4. Conclusion Both uilibrium temperature gradients from the model ( q y and 13 q z consistent with theory. The stronger the temperature gradients, the stronger the eddies. The uilibrium temperature q 0 at 45 latitude is taken at the surface, reducing the uilibrium ) are temperature gradients would cause q 0 to increase. Consuently increasing q 0 reduces the heat flux and the strength of the jet streams in middle latitudes. We want to increase the heat flux so we can lower q 0. Eddies are much stronger in the Northern Hemisphere as can be seen Fig. 3. and 4.. This is consistent with the physical world, because the eddies are stronger in the middle latitudes particularly in the Northern Hemisphere where the storm tracks are. Could it be the short averaging length of time and fluctuations in the model that contributed to the difference in the strength of the eddies. But conditions in our model are different from those in the physical world. Moisture and topography are not included in the model; as a result we would expect our model to be symmetric. This analysis leaves us with a lot of question that are not answered such as: 1. What could be strengthening the eddies in the northern hemisphere? 2. Why does heat flux decrease when the temperature gradient increases only for a horizontal forcing of 120K? 3. Why are the eddies shifted for the 120K horizontal forcing? Both damping in the velocity and temperature have very little effect on the strength of the eddies. The frictional effects resulting from damping would reduce the heat flux, as heat is lost to the atmosphere. From our analysis we see that the temperature damping parameter

14 has more weight on the eddies than the velocity damping factor though the effect is of little significance. Stone s theory differs only slightly from Green s theory. References Green,J.S.A.,1970: Transfer properties of the large scale eddies and the general circulation of the atmosphere.quart.j.roy.meteor.soc.,96, Hendon, H.H., and D.L. Hartmann, 1985: Variability in a nonlinear model of the atmosphere with zonally symmetric forcing. J. Atmos. Sci.,42, Isaac M. Held and Max J Suarez 1994: A proposal for the Intercomparison of the Dynamical Cores of Atmospheric General Circulation Model. Bull Amer. teor.soc.,75, Peter H. Stone, 1971:Simplified Radiative Dynamical Model for the Static Stability of Rotating Atmospheres Atmos. Sci., 29, Oort, A. H., and E.M. Rusmusson, 1971:Atmospheric circulation Statistics: NOAA Prof. Pap. 5, [Govt. Printing Office, Washington, DC, No Van Loon, H., 1979: The association between latitudinal temperature gradient and eddy heat transport. Part I: Transport of sensible heat in winter: Mon Wea. Rev., 107,